MEC ANICS
THIRD

L. D. LANDAU

EDITION

by
AND

E. M. LIFSHITZ

INSTITUTE OF PHYSICAL PROBLEMS, U.S.S.R ACADEMY OF SCIENCES

Volume l

of

Course of Theoretical Physics

Translated from the Russian by
J. B. SYKES AND J. s. BELL

U T T E R W O R T H
E l N E M A N N

Butterworth-Heinenann
Linacre House, Jordan Hill, Oxford OX2 8DP
A division of Reed Educational and Professional Publishing Ltd

QA member of the Reed Elsevier ple group
OXFORD BOSTON JOHANNESBURG
MELBOURNE NEW DELHI SINGAPORE

Translated from the 3rd revised and enlarged edition of Medkanika
by L. D. Landau and E. M. Lifshitz, Nauka, Moscow 1993
First published by Pergamon Press plc 1960
Second edition 1969
Third edition 1976

Reprinted 1978, 1982. 1984. 1986, 1987, 1987, 1988, 1989, 1991
1996, 1997, 1998 1999

. 2000

© Reed Educational and Professional Publishing Ltd 1981

All rights reserved. No part of this publication
may be reproduced in any material form (including
photocopying or storing in any medium by electronic
means and whether or not transiently or incidentally to
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Applications for the copyright holder's written
permission to reproduce any part of this publication
should be addressed to the publishers.
British Library Cataloguing in Publication Data
A catalogue record for this title is available from the British Library

Library of Congress Cataloguing in Publication Data
Landau, Lev Davidovich, I 908-68
Mechanics (Course of theoretical physics, v. 1) Translation
of Mekhanika by E. M. Lifshitz. Lifshitz: p.
Includes bibliographical references and index
l . Mechanics. Analytic I. Lifshitz, Evgenii. Mikhailovich,
jbint author H. Title
76-18997
531.0l'515
QA805:L283
1976
ISBN 0 7506 2896 0

Restoration, searchable PDF/ A and bookmarks: Gian Carlo "MaGiCa", 4 July 2023

MECHANICS

[THIS PAGE INTENTIONALLY BLANK]

CONTENTS
CONTENTS
Preface to the third
third English
English edition
edition
Preface

vu
vii
ix
lX

L.D. Landau-a biography
LD.

I. TH
THE
EQUATIONS OF
QF MOTION
MOTION
E EQUATIONS
1.

§1.
§2.
§3.
§4.
§5.

1

Generalised
Generalised co-ordinates
co-ordinates

2

The principle
principle of least action
The
Galileo's
Galileo's relativity
relativity principle
principle
The Lagrangian for a free particle
The
The Lagrangian
Lagrangian for a system
system of particles
The
particles

§6.
§7.
§8.
§9.
§10.

Energy
Energy
Momentum
Momentum

11.
II. CONSERVATION LAWS

Centre of mass
Centre
Angular
momentum
Angular momentum
Mechanical similarity
similarity

4

6

8

12468

13
15
16
18
22

111.
III. INTEGRATION
INTEGRATION OF THE EQUATIONS OF MOTION
MOTION

§11. Motion
Motion in one dimension
dimension
§12. Determination of
of the potential
energy from
from the
the period
of
potential energy
period of
oscillation
oscillation
§13. The
The reduced
reduced mass
§14. Motion
Motion in a central
central field
§15. Kepler's problem

§16.
§17.
§18
§18..
§19.

IV. COLLISIONS BETWEEN PARTICLES
Iv.

Disintegration of particles
particles
Disintegration
Elastic collisions
Elastic
Scattering
Rutherford's formula
formula
Rutherford's
1520.
~20. Small-angle
Small-angle scattering
scattering

v

25

27
29
30
35

41
41
44
48
53
55

vi
Vl
§21.
§22.
§23.
§23
§24.
§25.
§2s
§26.
§27.
§28.
§29.
§30.

.
.

§31.
§32.
§33.
§34.
§35.
§36.
536.
§37.
§38.
§39.

§40.
§41.
§42.
§43.
544.
§44.
§4s.
§45.
§46.
§47.
§48.
§49.
§50.
§51_
§51.
§52_
§52.

v.
V.

Contents
Contents

SMALL OSC
OSCILLATIONS
ILLATIONS

Free
Free oscillations
oscillations in one dimension
dimension
Forced oscillations
oscillations
Forced
Oscillations of
of systems with more than one degree of
of freedom
Vibrations
Vibrations of
of molecules
Damped oscillations
oscillations
Forced oscillations
oscillations under
under friction
friction
Forced
Parametric
Parametric resonance
resonance
Anharmonic
Anharmonic oscillations
oscillations
Resonance
Resonance in non-linear
non-linear oscillations
oscillations
Motion
Motion in a rapidly
rapidly oscillating Held
field

VI. MOTION OF A RIGID BODY

Angular
Angular velocity
The inertia tensor
tensor
The
Angular
Angular momentum
momentum of
~f a rigid body
The equations
equations of motion
motion of
of a rigid body
The
Eulerian
Eulerian angles
equations
Euler's equations
The asymmetrical top
The
Rigid bodies
bodies in contact
Motion in a non-inertial
non-inertial frame of reference
Motion

EQUATIONS
VII. THE CANONICAL EQUATIONS

equations
Hamilton's equations
The Routhian
The
Poisson brackets
brackets
Poisson
The action as a function
function of the co-ordinates
co-ordinates
The
Maupertuis' principle
transformations
Canonical transformations
Liouville's theorem
The Hamilton-]acobi
Hamilton-Jacobi equation
equation
The
Separation·of the variables
variables
Separationof
Adiabatic invariants
invariants
Adiabatic
Canonical variables
variables
Canonical
Accuracy of
of conservation
conservation of
of the
the adiabatic
adiabatic invariant
invariant
Accuracy
Conditionally
periodic
motion
Conditionally periodic motion
Index
Index

Page

58
61
61
65
70
74
77
80
84
87
93

96
98
105

107
110
114
116
122
126

131
133
135
138
140
143
146
14-6
147
149
154

157
159
162
167

P
R E F A C E TO
HE T
HIRD E
NGLISH E
DITION
PREFACE
TO T
THE
THIRD
ENGLISH
EDITION
ons of
revised and
THisS book continues
continues the series
series of
of English
English translati
translations
of the revised
£!nd
THI
augmented volumes in the"Course
the-Course of
of Theoretical
Theoretical Physics,
augmented
Physics, which
which have been
been
appearing in Russian since
since 1973. The
The English
translations of
of volumes 2
appearing
English translations
Theory of
of Fields) and 3 (Quantum 'ill'echanics)
·Mechanics) will shortly both
(Classical Theory
both
have been published.
Unlike those
present volume
not
published. l'nlike
those two,
two, the present
volume 1 has not
required
required any considerable
considerable revision,
revision, as is to be expected
expected in such a wellestablished
theoretical physics
physics as mechanics
established branch
branch of
of theoretical
mechanics is. Only the final
on adiabatic
adiabatic invariants,
invariants, have been revised
Pitaevskii
sections, on
revised by L. P. Pitaevskii
and myself.
The
The Course of
of Theoretical Physics was initiated
initiated by Landau, my teacher
teacher
Our work together
together on these
these books
books began
began in the late 1930s
1930s and
and friend. Our
continued until the
the tragic
tragic accident
accident that
that befell him
him in 1962.
1962. Landau's
Landau's work
continued
science was always such
such as to display
display his striving for clarity,
clarity, his effort to
in science
simple what was
\\"as complex
complex and so to reveal the laws of
of nature
nature in their
their
make simple
true simplicity
simplicity and
and beauty.
beauty. It was this
this aim which
which he sought
sought to instil
instil into
into his
true
pupils, and which has determined
determined the character
character of
of the
the Course. I. have tried
tried
pupils,
maintain this spirit,
spirit, SO
so far as I was able, in the revisions
revisions that have had
to maintain
without Landau's
Landau's participation. It
good fortune
fortune to
to be made without
It has been
been my good
find a colleague for this work in L. P. Pitaevskii, a younger
younger pupil of
of Landau's.
Landau's.
End
The present edition
edition contains
contains the biography
of Landau
Landau which
which I wrote
The
biography of
wrote in
1969 for the posthumous
Russian edition of
of his Collected
Collected Works.
should
1969
posthumous Russian
IVork5. I Should
hope that it will give the reader
reader some
some slight
slight idea
idea of
of the personality
'lf
like to hope
personality of
that remarkable
remarkable man.
that
The English
English translations of
of the Course were begun
Professor
The
begun by Professor
I\1. Hamermesh
Hamermesh in 1951 and continued by Dr. ].
]. B. Sykes and his colleagues.
M,
~o
great for their
their attentive
attentive and careful work, which
which has
No praise
praise can be too great
contributed so much to the success
success of
of our
our books
English-speaking
contributed
books in the English-speaking
wor
world.
ld,
Institute
of Physical
E. M. LIFSH1TZ
LIFSHITZ
Institute of
Physical Problems
[ ·.s.S.R. Academy
of Sciences
l'.S.S.R.
Academy of
Jloscow 1976
1976
31056020

vii
VII

[THIS PAGE INTENTIONALLY BLANK]

LEV
LANDAU
(1908-1968)t
L
E V DAVIDOVICH
DAVIDOVICH L
A N D A U (1908-l968)'t
VERY
YERY little
little time
time has passed since the death
death of
of Lev
Lev Davidovich
Davidovich Landau
Landau on
1 April 1968,
1968, but fate wills that
that even now we view him at a distance,
distance, as it
were. From that
perceive more
that distance
distance we perceive
more clearly not only his greatness
greatness as
the significance of
of whose work
work becomes increasing]
increasinglyy obvious
aa scientist, the
with time, but also that
that he was a great-hearted human being. He
He was
uncommonly just
just and benevolent.
uncommonly
benevolent. There is no doubt
doubt that
that therein
therein lie the
roots of
popularity as a scientist and teacher,
of his popularity
teacher, the roots
roots of
of that
that genuine
genuine
love and esteem
esteem which his direct
direct and indirect
indirect pupils felt for him and which
manifested with
such exceptional
exceptional strength during
during the
the days of
of the
the
were manifested
with such
struggle to save his life following the terrible
terrible accident.
struggle
To
To him fell the tragic
tragic fate of dying twice.
twice. The
The first time it happened
happened was
earlier on 7 January 1962.
1962 when
\Vhen on the icy road, en route
route from
six years earlier
Moscow
~loscow to Dubna,
Dubna, his car
car skidded
skidded and collided
collided with a lorry coming
coming from
the opposite directiondirection. The
The epic
epic story
story of
of the
the subsequent
subsequent struggle
struggle to save
save
the
his life is primarily
primarily a story of
of the selfless labour and skill of numerous
numerous
physicians and nurses.
physicians
nurses. But it is also a story
story of
of a remarkable
remarkable feat of
of solidarity.
solidarity.
The calamitous accident
accident agitated
agitated the entire
entire community
community of physicists,
physicists,
arousing a spontaneous
spontaneous and instant response.
response. The
The hospital
hospital in which
which Landau
Landau
arousing
lay unconscious
unconscious became
became a centre
centre to all those
those his students
students and colleagues
colleagues -stro,·e to make whatever contributions
contributions they
they could to help
help the
the physicians
physicians
who strove
their desperate
desperate struggle
struggle to save
sa,·e I,andau's
Landau's life.
in their
of comradeship
comradeship commenced
commenced on the wry
lllustr:ous
"Their feat of
\ 'try first day. lllustr11ous
scientists
medicine, academicians,
scientists who, however, had no idea of
of medicine,
academicians, correspondcorresponding members
members of
of the scientific
scientific academies,
at:ademies, doctors,
doctors, candidates,
candidates, men of
of the
same generation
generation as the 54-year-old
54-year-old Landau as well
\\·ell as his pupils and their
their
more youthful pupils
act as messengers, chauffeurs,
chauffeurs,
volunteered to act
still more
pupils -- all volunteered
intermediaries,
intermediaries, suppliers,
suppliers, secretaries,
secretaries, members
members of
of the watch
watch and, lastly,
y
porters
a b o r e r s . Their spontaneous
headquarters was
porters and llabourers.
spontaneously established
established headquarters
of the Physician-in~Chief
Physician-in-Chief of
of Hospital ="Jo.
located in the office of
No. 50 and it
became
unconditional and
became a round-the-clock
round-the-clock organizational
organizational centre
centre for an unconditional
immediate
immediate implementation
implementation of
of any instruction
instruction of
of the attending physicians.
physicians.
t By
By E.
E. .\l,
:\1. Lifshitz,
Lifshitz; written for the Russian
Ru"ian edition of
of L.1nd:\u's
L.mdau's Collected
Collected Papers, and
and
'|'
first published in
in Russian in
in Uspeklii
Usp,·khi fi:::ichcskikh
llaul< 97,
97, 169-183, 1969. This translation
translation
j'i:::r'clieskiklz muck
by F.
F. Bergman
Bergm.m (first
(first published
publ"hed in
in Sut'iet
Sv.-iet Physics
Physics Uspelehr'
Cspekhi 12, 135-143,
135-143, 1969), with minor
I.is
S by
and iis
reprinted by
hy kind permission
pennission of
of the
the American Institute of
of Physics.
modifications, and
s reprinted
The reference
reference numbers
numbers correspond
corresponcl to
to the
the numbering
ntunbering in
in the
the Collected
Collected Papers
Papers of
of L.
L. D.
D. Landau
La11dau
The
Press, O
o,.furd
(Pergamon Press,
f o r d 1965).

i1"\.\

Lev
Lev Dawidoifich
Davido·vich Landau
Landau

X

"Eighty-seven theoreticians and
and experimenters
experimenters took part
part in this voluntary
volun tary
rescue
team.
An
alphabetical
list
of
the
rescue team.
alphabetical
of
telephone numbers
numbers and addresses
addresses of
telephone
any one and any institution
institution with which
which contact
contact might
might he
be needed at any
instant
instant was compiled,
compiled, and it contained
contained 2.23
223 telephone
telephone numbers! It
It included
other
hospitals, motor
other hospitals,
motor transport
transport bases, airports,
airports, customs
customs offices, pharmacies,
pharmacies,
ministries,
the places at which
which consulting
consulting physicians
could most
most likely
ministries, and the
physicians could
be reached.
reached.
"During
"During the most
most tragic days when
when it seemed
seemed that 'Dau
'Dau is dying'
dying' - and
there were
were at least
least four such
such days -- 8-10 cars could be found waiting
waiting at
at
any time
time in front of
of the seven-storey
seven-storey hospital
hospital building
building.....
...
"When everything
everything depended
depended on the artificial respiration
respiration machine, on
12 January, a theoretician
theoretician suggested
suggested that
that it should
should be immediately
immediately conconstructed
structed in the workshops
workshops of
of the Institute
Institute of Physical
Physical Problems.
Problems. This was
unnecessary
and
naive,
but
how
amazingly
unnecessary
naive, but ho'h amazingly spontaneous!
spontaneous! The physicists
obtained
obtained the machine
machine from the
the Institute
Institute for the Study
Study of
of Poliomyelitis
Poliomyelitis
carried it in their
their own hands to the ward where
where Landau was gasping
gasping
and carried
saved their
their colleague,
colleague, teacher, and friend.
friend.
for breath. They saved
story could
could be continued
continued without limit. This was a real fraternity
fraternity
"The story
of physicists
physicists.....
. . .""t
"t
Landau's life was saved.
saved. But when after
after three
three months
months he rereAnd so, Landau's
gained consciousness,
consciousness, it was no longer the
the same man whom we had known.
gained
He was not able to recover
consequences of his accident
accident and
He
recover from all the consequences
never again
again completely
completely regained
regained his abilities. The
The story
story of
of the six
stx years
never
that followed is only a story
story of
of prolonged
prolonged suffering
suffering and pain.
that

*

=x=

**

*

=x=

Lev Davidovich
Davidovich Landau
Landau was born on 22 January 1908
1908 in Baku, in the
Lev
of a petroleum
engineer who worked on the Baku oil-fields.
oil-fields. His
His
family of
petroleum engineer
mother was a physician
physician and at one time had engaged
engaged in scientific
scientific work on
mother
physiology.
physiology.
completed his school course at the age of
of 13. Even then
then he already
He completed
attracted by the exact
exact sciences,
sciences, and his mathematical
mathematical ability manifested
manifested
was attracted
itself very early. He
He studied
studied mathematical
mathematical analysis on his own and later he
itself
used
that he hardly remembere•d
remembere""d a time
time when
when he did
did not know
kno\\
used to say that
differentiation
differentiation and integration.
integration.
His parents considered
considered him
him too young to enter
enter a university
university and
and for a
His
lled at
attended the Baku Economic
Economic Technicum.
Technicum. IIn
1922 he enro
enrolled
year he attended
n 1922
niversity where
where he studied
studied simultaneously
simultaneously in
two departments:
departments:
Baku C
University
in two
Chemical. Subsequently
Subsequently he did not
not continue
continue
Physico-mathematical and Chemical.
chemical education
education but
remained interested in chemistry
chemistry throughout
throughout
his chemical
b u t he remained
his life.
grad
par tm ent of
In 1924
1924- Landau transferred to
to the Physics De
Department
of Lenin
Leningrad
In
.r

t

,
.
.
G zeta (Li
(Literary
Gazette), 21
July 1962
1962 •
21 July
ter ary Ga.2:"€tt€),
F rom D.
D. Dam
Dantn,
"Comradeship", Literaturnaya
Lzteraturnaya
Gaazeta
n, "Comradeship",
From

Lev Dawidowieh
Davidovich Landau
Landau
Lev;

XI
xi

University.
physics at that
University. In Leningrad,
Leningrad, the main centre
centre of
of Soviet
Soviet physics
that time,
time,
he first made
the
acquaintance
of
genuine
theoretical
physics,
which
made
acquaintance of genuine theoretical
which was
then
period. He
then going
going through
through a turbulent
turbulent period.
He devoted
devoted himself
himself to its
its study
study with
with
all his youthful zeal
zeal and enthusiasm
enthusiasm and worked
worked so strenuously
strenuously that
that often
often
exhausted that
that at night
night he could not
not sleep, still turning
turning over
over
he became
became so exhausted
formulae in his mind.
Later he used
used to describe how at that
that time
time he was amazed
amazed by the
the incredible
beauty of
credible beauty
of the general
general theory of
of relativity
relativity (sometimes
(sometimes he even
that such
such a rapture
rapture on first
acquaintance with
with
declare that
would declare
First making one's acquaintance
this theory
physicist).
theory should be a characteristic
characteristic of any born
born theoretical
theoretical physicist).
brought on reading
the state of
of ecstasy
ecstasy to which
which he was brought
He also described the
Heisenberg and SchrOdinger
Schrodinger signalling
signalling the birth
of the new
articles by Heisenberg
the articles
birth of
quantum mechanics.
mechanics. He said
said that
that he derived
derived from them
them not
not only delight
delight in
the true
true glamour
glamour of
of science
science but
but also an acute
acute realization
realization of
of the power
of
the
power of
the
whose greatest triumph
the human genius,
genius, whose
triumph is that man is capable
capable of
of apprehending
hending things
things beyond
beyond the pale of
of his imagination.
imagination. And of
of course, the
curvature of
of space-time
space-time and the uncertainty
principle are precisely
precisely of
of this
this
curvature
uncertainty principle
kind.
In 1927
1927 Landau
Landau graduated
graduated from the university
university and enrolled
enrolled for postgraduate study
studv at the
th6 Leningrad
Leningrad Physicotechnical Institute where
where even
even
graduate
1926, he had been
been a part-time research
research student.
student. These years
year:;
earlier, in 1926,
brought his first scientific publications.
1926 he published
published a theory of
of
brought
publications. In 1926
spectra of
of diatomic
diatomic molecules
molecules [l],T
[1 ],t and as early as 1927,
1927,
intensities in the spectra
study of
of the
the problem
of damping
damping in quantum
quantum mechanics,
which first
a study
problem of
mechanics, which
introduced a description
description of
of the
the state
state of
of a system
system with the aid of
of the
the density
density
introduced
matrix.
matrix.
His fascination
fascination with
with physics
physics and his first achievements
achievements as a scientist
scientist were,
His
beclouded by a painful diffidence
diffidence in his relations
relations with
however, at the time beclouded
however,
trait caused
caused him a great
great deal of
of suffering and at times
times -- as he
others. This trait
himself
himself confessed
confessed in later
later years - led him to despair. The
The changes
changes which
which
occurred in him with
with the years
years and transformed
transformed him into a buoyant
bu-oyant and
occurred
gregarious individual were largely a result
result of
of his characteristic
characteristic self-discipline
self-discipline
gregarious
of duty toward
toward himself.
himself. These qualities,
qualities, together
together with his sober
sober
and feeling of
self-critical mind, enabled
enabled him to train
train himself
himself and to evolve into a
and self-critical
person with
person
with a rare
rare ability
ability - the
the ability
ability to be happy.
happy. The same
same sobriety
sobriety of
of
mind enabled
enabled him always to distinguish
distinguish between
between what is of
of real value in
mind
life and wha
whatt is unimportant
retain his mental
unimportant triviality, and thus also to retain
equilibrium during
during the
the difficult moments
moments which
which occurred
occurred in his life too.
equilibrium
1929, on an assignment
assignment from the People's
People's Commissariat
Commissariat of
of Education,
Education,
In 1929,
Landau
Landau travelled
travelled abroad and
and for one
one and
and a half
half years worked
worked in Denmark,
Denmark,
Great Britain
Britain and
and Switzerland.
Switzerland. To
To him
him the
the most important
important part
part of
of his trip
in
Stay
his
Copenhagen
stay
Copenhagen where,
where, at
at the
the Institute of
of Theoretical Physics,
was
e tha
He did
did no
nott know,
k.~ 0 "'• however
hov.ever, at the
the tim
time
thatt these results had
had been already
already published a
Tt He
onI
nl and
year
earlier
by
Honl
and Lond
London.
.
H6
by
r
rlie
year ea
|

xii
xu

Lew
aw idodeh Land
Lev D
Davidovich
Landau
au

theoretical
theoretical physicists
physicists from all Europe gathered round the great Niels
Niels Bohr
and,
and, during
during the
the famous
famous seminars
seminars headed by Bohr, discussed
discussed all the basic
basic
problems
problems of
of the theoretical
theoretical physics
physics of
of the time.
time. This scientific
scientific atmosphere,
atmosphere,
enhanced by the charm of
of the personality
personality of
of Bohr himself, decisively
decisively
enhanced
influenced Landau in forming his own outlook on physics
physics and subsequently
subsequently
he always considered himself
himself a disciple of Niels Bohr. He visited Copenhagen two more times,
times, in 1933 and 1934. Landau's
Landau's sojourn
sojourn abroad
abroad was
the occasion,
particular, of
occasion, in particular,
of his work
work on the
the theory
theory of
of the diamagnetism
diamagnetism
of
of an electron
electron gas [4] and the study of
of the limitations
limitations imposed
imposed on the
measurability of
of physical
physical quantities
quantities in the relativistic
relativistic quantum
quantum region
region (in
measurability
collaboration with Peierls)
Peierls) [6].
collaboration
On his return
return to Leningrad
Leningrad in 1931 Landau worked
worked in the Leningrad
Leningrad
Physicotechnical
Institute and in 1932
Physicotechnical 1'\lstitute
1932 he moved to Khar'kov, where he
became
of the Theoretical Division
Division of the newly organized
organized Ukrainian
Ukrainian
became head of
offshoot of
of the Leningrad
Leningrad Institute. At the
the
Physicotechnical Institute, an offshoot
Physicotechnical
headed the Department
Department of
of Theoretical Physics
Physics at the
the Physics
Physics
same time he headed
and Mechanics
Mechanics Faculty
Faculty of
of the Khar'kov
Khar'kov Mechanics
Mechanics and Machine
Machine Building
Institute and in 1935 he became
became Professor
Professor of
of General
General Physics
Physics at fKhar'kov
'Khar'kov
Institute
.·
University.
Th
Thee Khar'kov
Khar'kov period
period was for
~or Landau
Landau a time
tirne of
of intense
intense and varied
varied
activity.t It was there that
that he began
teaching career
career and estabresearch activity.'t
began his teaching
of theoretical
theoretical physics.
lished his own school of
lished
Twentieth-century theoretical
theoretical physics
physics is rich in illustrious
illustrious names of
of
Twentieth-century
trail-blazing creators,
creators, and Landau
Landau was one of
of these
these creators. But his
trail-blazing
scientific progress was far from exhausted
exhausted by his personal
influence on scientific
personal
contribution to it. He was not only an outstanding
outstanding physicist
physicist but also a
contribution
outstanding educator,
educator, a born
born educator.
educator. In this
this respect one may
genuinely outstanding
genuinely
- Niels Bohr.
the liberty
liberty of
of comparing
comparing Landau only to his own teacher
teacherBohr.
take the
The problems
of the teaching
teaching of
of theoretical
theoretical physics
of physics
physics
The
problems of
physics as well as of
first attracted
attracted his interest while still
still quite a young man. It
as a whole had First
there, in Khar'kov, that
that he first began
work out programmes
was there,
began to work
programmes for the
"theoretical
minimum"programmes
of
the
basic
knowledge
theoretical
"theoretical minimum" - programmes of
knowledge in theoretical
experimental physicists
those who wish to devote
devote
physics needed
needed by experimental
physicists and by those
research work in theoretical
theoretical physics.
addition
themselves to professional
themselves
professional research
physics. In addition
drafting these
these programmes,
lectures on theoretical physics
to drafting
programmes, he gave lectures
physics to
the scientific
scientific staff
staff at the
the Ukrainian
Physicotechnical Institute
Institute as well
well as to
the
Ukrainian Physicotechnical
students of the Physics
Physics and Mechanics
Mechanics Faculty.
Faculty. Attracted
Attracted by the ideas of
students
reorganizing instruction
instruction in physics'
as a whole,
whole, 1he
the Chair
Chair of
reorganizing
physics=as
the accepted the
General Physics
Physics at Khar'kov
Khar'kov State
State' University
\{and subsequently,
subsequently, after
after
General
University l{and
The extent
extent of
of Landau's
L&ndau's scientific
scientific activities
activities at
at the
the time
time can be
be grasped
graspt'd from
from the
list of
of
the list
1't The
the year 1
1936
of second-order phase transitions
t_ransitions
studies he completed during the
936 alone: theory of
.
on IN
nsport equati
[29], theory of
of the
the intermediate state
state of
of superconductors [30], the
tr~nsport
equatron
i_n the
the
the tra
.
case of
of Coulomb interaction [24],
[24] the
the theory
theorv of
of unimolecular
reactrons [23].
[ 2 J], _propertres
of
Pnopertles of
case
unimo lecula f reactlorls
..
.·
IO
•
and abs
absorption
of so d
at very low temperatures [25], theory
theo_ry of
of the
the dlspeI'slot"
drspersron and
un
metals at
opt
n of sound
[22, 28],
28], theory of
of photoelectric effects in
in semiconductors
semiconductors [21][21J[22,

Lrc Da*2.°ido°z'ieh
Da·cido'l·ich Landau
Landau
Lew

XIII
xiii

Fe lectures on general
the
the war, he continued
continued to go
give
general physics
physics at the
the PhysicoPhysicotechnical
technical Faculty
Faculty of
of Moscow
Moscow State
State University).
University).
It was
was there
there also,
also, in Khar'kov,
Khar'kov, that
that Landau
Landau had
had conceived
conceived the idea and
It
began to implement
implement the
the programme
programme for compiling
compiling a complete
complete Course
Course of
of
began
Physics and Course
Course of
of General
General Physics.
Physics. All his life long, Landau
Theoretical Physics
very level dreamed of
of writing
writing books
books on physics
- from school
school textbooks
textbooks
dreamed
physics at eevery
to a course
course of
of theoretical
theoretical physics for specialists.
specialists. In fact, by the time of his
volumes of
of the
the Course of
of Theoretical
Theoretical Physics
Physics
accident, rnearly
fateful accident,
e a l y all the volumes
the first
first volumes of
of the
the Course of
of General
General Physics
and the
P/zysics and Physics
Physics for
for
been completed.
completed. He also had drafted
drafted plans
plans for the
the compilation
compilation
Everyone
Everyone had been
textbooks on mathematics
mathematics for physicists,
which should
should be
he "a
"a guide
guide to
of textbooks
physicists, which
action", should
should instruct in the practical applications
applications of
of mathematics
mathematics to
physics, and should
should be free of
of the
the rigours
rigours and complexities
complexities unnecessary
unnecessary to
physics,
course. He did not
not have time to begin to translate
translate this programme
this course.
programme into
reality.
ty.
reali
Landau always attached great importance
importance to the
the mastering
mastering of
of mathematimathematiLandau
techniques by the
the theoretical
theoretical physicist.
physicist. The
The degree
degree of
of this mastery
mastery
cal techniques
should be
be such
such that,
that, insofar
insofar as possible, mathematical
mathematical complications
complications would
would
should
not
not distract
distract attention
attention from the
the physical difficulties of
of the problem - at least
least
whenever
whenever standard
standard mathematical
mathematical techniques
techniques are concerned. This can be
achieved only by sufficient training.
training. Yet
Yet experience
experience shows
shows that the
the current
current
achieved
style
instruction in mathematics
mathematics for physist
programmes for university instruction
y je and programmes
cists often
often do not
not ensure
ensure such
such training. Experience
EJfperience also shows
shows that
that after
after a
physicist commences
commences his independent
independent research
research activity he finds the study
physicist
of mathematics
mathematics too
too "boring".
of
the first
first test which Landau
gave to anyone
anyone who
who desired
desired to
Therefore, the
Landau gave
become one of
of his students
students was a quiz in mathematics
mathematics in its "practical"
become
calculational
calculational aspects.1'
aspects. t The successful
successful applicant
applicant could
could then
then pass
pass on to the
the
study of
of the
the seven
seven successive sections of
of the
the programme
the "theoretical
study
programme for the
minimum", which
which includes basic knowledge
knowledge of
of all the
the domains of
of theoretical
theoretical
Minimum",
subsequently take an appropriate
appropriate examination.
examination. IIn
Landau's
physics,
physics, and subsequently
n Landau's
opinion, this
this basic
knowledge should
should be mastered
mastered by any theoretician
theoretician
opinion,
basic knowledge
regardless of his future specialization.
specialization. Of
Of course,
course, he did not expect
expect anyone
regardless
to be as universally
universally well-versed
in science
science as he himself.
himself. But
But he thus
thus
to
well-versed in
manifested his belief
the integrity of
of theoretical
theoretical physics
single
manifested
belief in the
physics as a single
science with unified methods.
methods.
science
first Landau
Landau himself
gave the examination
examination for the "theoretical
At First
himself gave
minimum". Subsequently,
Subsequently, after
after the number
number of
of applicants
applicants became
became too
too large,
large,
minimum".
shared with
with his closest associates. But Landau
Landau always
always rethis duty was shared

t The requirements were:
were: ability
ability to
to evaluate any
any indeFinite
indefinite integral that can be
be expressed
'|'
of elementary functions and to
to solve any
any ordinary differential equation of
of the standard
iin
n terms of
of vector analysis and
and tensor
tensor algebra as well
well as of
of the
the principles of
of the
the theory
type, knowledge of
of functions of
of a complex variable (theory of
of residues, Laplace method). It was assumed that
of
fields as
as tensor analysis and group theory would be
be studied together with the
the Fields
fields of
of
such Fields
to which they apply.
theoretical physics to

XIV
xiv

Lei'
z'z'do'z'irh Lan
Le'c Da'
Dm·idm·ich
Landau
dau

served
himself the
un g
served for himself
the first
first test, the
the first meet
meeting
with ea
each
new
young
ch ne
ing with
w yo
applicant. Anyone could
could meet
meet him - it was suffic
sufficient
ring him up and
ient to ring
interview.
ask him for an interview,
Of
Of course,
course, not
not every
every one who began
began to study the "theoretical
"theoretical minimum"
min imu m"
had sufficient
sufficient ability
ability and persistence to complete
complete it. Altogether, between
between
1934
passed this test. The
c1934 and 1961,
1961, 43 persons
persons passed
The effectiveness
effectiveness of this sele
selection can he
be perceived
perceived from the following
follmYing indic
indicative
alone:: of
of these
ative facts alone
persons 7 already have become
persons
become members
members of the Academy
Acadenw of
of Sciences
Sciences and
additional 16, doctors of
of sciences.
sciences.
·
aan
n additional

In the spring
spring of
of 1937 Landau moved
moved to Moscow
l\loscow where
''"here he became
became head
of
of the Theoretical
Theoretical Division of the Institute of Physical Problems which-had
not
not long before
before been
been established
established under the direction
direction of P. L. Kapitza.
There he remained to the end
end of
of his life,
life; in this Institute, which became
became a
home
It was there,
home to him, his varied
varied activity
activity reached
reached its full flowering. It
there, in a
remarkable interaction
interaction with experimental
experimental research,
research, that
that Landau
Landau created
created
remarkable
what
what may be the
the outstanding
outstanding accomplishment
accomplishment of
of his scientific
scientific life -- the
theory
Huids.
theory of
of quantum
quantum fluids.
It
that he received
numerous outward
outward manifestations
manifesta!ions
It was there also that
received the numerous
recognition of
of his contributions.
contributions. In 1946 he was
\Yas elected
elected a full :\I
ember
of the recognition
Member
of the USSR
USSR Academy of Sciences. He was awarded a number of orders
of
(including two Orders
Orders of
of Lenin)
Lenin) and the honorific title of
of Hero
Hero of Socialist
Socialist
Labour -- a reward
reward for both
both his scientific
scientific accomplishments
accomplishments and his contribucontribuLabour
the implementation
implementation of
of important
important practical
He was
tion to the
practical State tasks. He
awarded the
the State Prize three
three times and in 1962.,
1962, the
the Lenin
Lenin Prize.
Prize. '1`here
There
awarded
of honorific awards from other
other countries,
countries. As far back
back as
also was no lack of
member of
of the Danish
Danish Royal Academy
Academy of
of Sciences
Sciences
elected member
1951 he was elected
member of
of the Netherlands
Academy of
of Sciences. IIn
and in 1956, member
Netherlands Royal Academy
n
1959 he became
of the British Institute
Institute of
of Physics
Physics and
1959
became honorary
honorary fellow of
Society and
anCl in 1960, Foreign
Foreign l\'lember
Member of
of the Royal Society of
of
Physical Society
Great Britain.
Britain. in
[n the same
same year he was elected
elected to membership
in
the~
ational
membership
the National
Academy of
of Sciences
Sciences of
of the United
States and the
the Ameri:can
of
Academy
United States
American Academy of
Sciences. In 1960
1960 he became
became recipient
recipient of
London Prize
Prize
Arts and Sciences.
_of the F. London
(United States) and of
of the Max Planck
Planck Medal (\Vest
(West Germany).
Germany). Lastly,
Lastly, in
(United
1962 he was awarded
awarded the Nobel
Prize in Physics "for his pioneering
pioneering theories
theories
1962
N obel Prize
matter, especially
especially liquid helium".
helium".
for condensed matter,
Landau's scientific
scientific influence was, of
of course, far from confined
confined to his own
Landau's
disciples. He was deeply democratic
democratic in his life as a scientist
scientist (and in his life
disciples.
s
that matter; pomposity
pomposity and deference
deference to titles alway
always
as a human being,
being, for that
merits
and
remained
foreign
to
him).
Anyone,
regardless
of
his
scientific
merits
and
remained foreign
Anyone, regardless of
scientific
ask Landau
Landau for
for counsel and
and criticism
criticism (which
were invariably
invariably
title, could ask
(which were
businesslike
be
st be busine~slike
precise and
and clear),
clear), on one
one condition
condition only:
only: the
the question
question mu
must
ptY Phhi1olsoe:
nc
ie
instead
of
pertaining
to
what
he
detested
most
in
scien~e:
_empty
p~Ilo_so­
sc
instead of pertaining
what
most
phizing
vapidity and
and futility
futility cloaked
cloaked in
in pseudo-sCie~ufic
sophistnes.
phizi ng or vapidity
pseudo-sc1e"tI C Sop
Istrlf:s_
approach
his
fro
m
ith
He
had
an
acutely
critical
mind;
this
quality,
along
with
h•s
approach
from
w
He had an acutely critical mind; this quality, along

_

xv

Let: Dazido-vich Landau

the standpoint of profound physics, made discussion with him extremely
attractive and useful.
I n discussion he used to be ardent and incisive but not rude, witty and
ironic but not caustic. The nameplate which he hung on the door of his
office at the Ukrainian Physicotechnical Institute bore the inscription :

L. LANDAU
BEVVARE, HE BITES!

With years his character and manner mellowed somewhat, but his
enthusiasm for science and his uncompromising attitude toward science
remained unchanged. And certainly his sharp exterior concealed a scientifically impartial attitude, a great heart and great kindness. However harsh
and unsparing he may have been in his critical comments, he was just as
intense in his desire to contribute with his advice to another man's success,
and his approval, when he gave it, was just as ardent.
These traits of Landau's person lily as a scientist and of his talent
actually elevated him to the position of a supreme scientific judge, as it
were, over his students and colleaguesft There is no doubt that this side of
Landau's activities, his scientific and moral authority which exerted a
restraining influence on frivolity in research, has also markedly contributed
to the lofty level of our theoretical physics.
His constant scientific contact with a large number of students and
colleagues also represented to Landau a source of knowledge. A unique
aspect of his style of work was that, ever since long ago, since the Khar'kov
years, he himself almost never read any scientific article or book but nevertheless he was always completely au courant with the latest news in physics.

of?
1.

l

IQ?

s

s'

\.

,

v

>&1

clt43&A

t This position is symbolized in A A. Yuzefovich•s well-known friendly car t o n , "Dau

sand", reproduced here

»

XVI
xvi

Lei*
'z'2'.f/1 Land
Le'i: I)ai'ido
Da·cido·cich
Landau
au

He
He derived
derived thisknowledge
this.knowledge from numerous discussions
discussions and from the
the papers
presented
presented at the seminar held under his dire
direction.
ction.
This
This seminar
seminar was held regularly once
once a week for nearly 30 year
years,
s, and in
the
physicists from
the last years
years its sessions became
became gatherings
gatherings of
of theoretical
theoretical physicists
The presentation
of papers at this seminar
seminar became
became a sacred
sacred
all :\loscow.
Moscow. The
presentation of
duty for all students
students and co-workers, and Landau
Landau himself
himself was extremely
extremely
thorough in selecting
selecting the material to be p•resented.
p_resented. He was
serious and thorough
interested and equally competent
competent in every aspect
aspect of
of physics
physics and the partici~
particiinterested
find it easy to follow
folio"· his train of thought
thought in
pants in the seminar did not Find
s"·itching from the
the discussion
discussion of, say, the properties
properties of
of
instantaneously switching
instantaneously
''strange" particles to the discussion
discussion of the energy spectrum
spectrum of electrons
electrons in
"strange"
To Landau
Landau himself
himself listening to the papers was never
ne,·er an empty
empty
silicon. To
not rest
rest until the essence
essence of
of a stL
stL .ly
.iy was completely
completely
formality: he did not
traces of
of "philology"
"philology" - unprm·ed
statements or proposielucidated and all traces
elucidated
unpro\ 'ed statements
propositions
principle of
tions made on the
the principle
of "why
"why might
might it not"
not"therein were eliminated.
- therein
of such
such discussion
discussion and criticism
criticism many studies
studies were
were condemned
condemned
As a result of
''pathology" and Landau
Landau completely
completely lost
lost interest
interest in them.
them. On the other
other
as "pathology"
hand, articles
articles that
that really contained
contained new ideas
ideas or findings were included
included in
the so-called
so-called "gold fund" and remained
remained in Landau's
Landau's memory for ever.
In fact, usually it was sufficient for him to know just
just the guiding idea of
study in order
order to reproduce
reproduce all of
of its findings. As a rule,
rule, he found it easier
easier
a study
obtain them
them on his own than to follow in detail the
the author's
author's reasoning.
reasoning.
to obtain
reproduced for himself
himself and profoundly
profoundly thought
thought out most
most of
of
IIn
n this way he reproduced
results obtained
obtained in all the
the domains of theoretical
theoretical physics.
the basic results
physics.Tt This
probably also was the
the reason
reason for his phenomenal
phenomenal ability to answer
answer practically
probably
practically
concerning physics
that might
might be asked
asked of
of him.
question concerning
any question
physics that
unfortunately fairly wideLandau's scientific
scientific style was free of
of the
the - u:1fortunately
wideLandau's
spread -- tendency
tendency to complicate
complicate simple things
things (often
(often on the grounds
grounds of
of
spread
generality
i g o r which, however,
however, usually turn out
generality and rrigour
out to be illusory). He
He
himself always strove
strove towards
towards the opposite
opposite - to simplify complex
complex things,
things, to
himself
uncover in the
the most
most lucid manner
manner the genuine simplicity
simplicity of
of the
the laws underuncover
underthe natural phenomena.
phenomena. This ability of his, this skill at "trivializing"
"trivializing"
lying the
things as he himself
himself used
used to say, was to him a matter
matter of
of special
special pride.
things
The striving
striving for
for simplicity
simplicity and order
order was an inherent
inherent part
part of the
the structure
structure
The
of Landau's
Landau's mind. It manifested
manifested itself
itself not
not only in serious
serious matters
matters but also
of
in semi-serious
semi-serious things
things as well as in his characteristic
characteristic personal
personal sense
sense of
of
humour.! Thus, he liked to classify
classify everyone,
everyone, from women
women according
according to
humour.1
of their
their beauty,
beauty, to theoretical
theoretical physicists
according to the
the signifisignifithe degree of
physicists according

'I't

Landa u's Papers
the absence of
of certain needed references
papers
references in Landau's
Incidentally, this explains the
reference Off
out
leave
v.hich usually was not
not intentional,
intentional. However, in some cases he could leave out Ei
a reference
on
which
purpose, if
if he
he considered the
the question too trivial,
trivial; and
and he
he did
did have his
his om'nn rather
rather high
high stanstan.· h
dards on
on that matter.
· characteristic,
· · however, that this
· was not
h a b"t
Landau
IS, SO
so
to SPeak,
speak •
to
IS
charactenstrc,
t h"IS trait
tratt
not a habit
I of
o f Landau
d im
"n his,
f disoI
d.
1:t It is
'der"
e of
.
.
.
.
.
II
ate
zone
o
lsorder"
d a "Zon
an
te
ra
cu
ac
outSide life,
hfe, in
m which
whrch he
he was not
not at
at all
all pedantically
pedantlca y accur
everyday outside
would quite rapidly arise around him.

Lp? I)a~:'idui°irl1

\vii

1,0111/uu

_

_

cance of their contribution to science. This last classification was based on a
I ogarithmic scale of Ev e: thus, a second-class physicist supposedly accompushed 10 times as much as a third-class physicist ("pathological types"
were ranked in the fifth class). On this scale Einstein occupied the position
while Bohr, Heisenberg, SchrOdinger, Dirac and certain others were
ranked in the first class. Landau modestly ranked himself for a long time
in class 2% and it u as only comparator.'Q l y late in his life that he promoted
himself to the second class.
He always worked hard (never at .1 desk, usually reclining on .1 divan at
J

xviii
XVlll

Lei'
Da'z~idoi'ifh Lmzdau
Le·c Dm·ido·cich
Landau

home).
home). The
The recognition
recognition of
of the results
results of
of one's
one's work
work is to a greater or lesser
extent
was, of
extent important
important to any scientist
scientist;, it ,vas,
of course,
course, also
also essential
essential to
to Land
Landau.
au .
But it can still
still be said
said that he attached
attached much
much less
less importance
importance to questions
of
questions of
priorit y than is ordinaril
priority
ordinarilyy the case. And at any rate there
there .is no doubt that
that
drive for work was inherently
inherently motivated not by desire
desire for fame but by
his drive
inexhaustible curiosity
curiosity and passion
exploring
of nature in
an inexhaustible
passion for exp
luring the laws of
their large
large and
and small manifestations.
manifestations. He
He never
never omitted
omitted a chance
chance to repeat
repeat
their
"r
elementary truth
truth that one
one should
should never
extraneous purposes,
the elementary
never work
ork for extraneous
purposes,
VV
work
merely for the sake of
of making a great
great discovery,
discovery, for then
then nothing
nothing
or merely
would
would be accomplished
accomplished anyway.
The range of
of Landau's interests
interests outside
outside physics
extremelyy wide.
The
physics also was extremel
Ur
addition to the exact sciences
sciences he loved history
well-versed in it.
IIn
n addition
history and was
as well-versed
He was also
al3o passionately
passionately interested
interested in and deeply
deeply impressed
impressed by every genre
of fine arts, though
though with
with the
the exception
exception of
of music
music (and
(and ballet).
of
ballet).
who had
had the
the good
good fortune
fortune to be his students
students and friends for many
Those who
that our
our Dau, as his friends
friends and comrades
comrades nicknamed
nicknamed hims,
himt, did
did
years knew that
company boredom
boredom vanished.
vanished. The
The brightness
of his
not grow old. In his company
brightness of
dull
and
his
scientific
power
remained
strong.
personality
never
grew
scientific
power
strong.
All
personality never grew
the more
more senseless
senseless and frightful
frightful was the accident
accident which
which put an end
end to his
brilliant activity
activity at its zenith.
brilliant

*

=x=

*

=x=

*

=x=

Landau's articles,
articles, as a rule, display all the features of
of his characteristic
characteristic
Landau's
lucidityy of
of physical statement
statement of
of problems,
style:
clarityy and lucidit
scientific st
problems, the
y je: clarit
shortest and most
most elegant
elegant path
towards their
their solution,
solution, no superfluities.
superfluities.
shortest
path towards
after many years, the greater
greater part
part of
of his articles
articles does
does not require
require
Even now, after
Even
revtswns.
any revisions.
The brief
review below
below is intended
in~nded to provide
tentative idea
idea of
of the
The
brief review
provide only a tentative
abundance and diversity
diversity of
of Landau's
Landau's work
work and to clarify
clarify to some
some extent
extent
abundance
occupied by it in
i~ the history
history of
of physics, a place which
which may not
the place occupied
obvious to the contemporary
contemporary reader.
always be obvious
of Landau's
Landau's scientific
scientific creativity
creativity is its almost
almost
A characteristic
characteristic feature of
unprecedented
which encompasses
encompasses the whole of
of theoretical
theoretical
unprecedented breadth,
breadth, which
hydrodynamics to the quantum
quantum field
theory. IIn
our century,
century,
physics, from hydrodynamics
physics,
Held theory.
n our
paths
which is a century
century of
of increasingl
increasinglyy narrow specialization,
specialization, the scientific
scientific paths
which
of his students
students also have been
graduallyy diverging,
diverging, but Landau
Landau himself
himself
of
been graduall
"
r
unified them
them all, al
always
astounding interest
interest in everything.
everything.
united
ays retaining
retaining a truly astounding
It
It may be that in him physics
physics has lost one of
of the last great
great universalists.
universalists.
examination of
of the bibliography
bibliography of
of Landau's
Landau's works shows
shows
cursory examination
Even a cursory
whi
ch he
cannot be divided
divided into any lengthy
lengthy periods
during which
that his life cannot
periods during
the
surve
y
of
worked
only
in
some
one
domain
of
physics.
Hence
also
tJ:e
su.rvey
of his
worked
some
domain of physics. Hence
them
possible,
in
atic
not in chronological
chronological order
order but, ins
insofar
works is given
given not
ofa r as posstble, m thematic
works
.»--"

.
_
. name originated
. .
h spelling
ell'
Landau himself
himself Inked
liked to
to say
say that
that thus
this
the F
French
off has
his
Landau
name orlgmated from
fnofn the
reno sP
mg o
name:
Landau
=
L'line
Dau
(the
ass
Dau).
name:
L'ane Day
Day).

-l~t

Lev 'Da<tido<t·ich
Landau
Lei:
Da°z1°do°2:z.ch Landau

XIX
XIX

order. We shall begin
begin with the works devoted
devoted to the general
general problems
problems of
of
quafltum
quantum mechanics.
mechanics.
These include, in the
n the course
the first place,
place, several
several of
of his early works.
\VOrks. IIn
course
of his studies
studies of
of the radiation-damping
radiation-damping problem
problem he was the first to introduce
introduce
of
the conc
concept
of incomplete quantum-mechanical
quantum-mechanical description
description accomplished
accomplished
ept of
the
"r
with
which were
were subsequently
subsequently termed
termed the density
density
ith the aid of quantities which
matrix [2]. IIn
n this article the density matrix was introduced in its energy
energy
representation.
Two articles
articles ['7,
[7, 9] are devoted
devoted to the calculation of
of the probabilities
probabilities of
of
quasiclassical
quasiclassical processes.
processes. The
The difficult
difficultyy of
of this problem
problem stems
stems from the fact
of the exponential
exponential nature (with
(with a large imaginary
imaginary exponent)
exponent)
that, hy
by virtue of
of
of the
the quasiclassical
quasiclassical wave functions,
functions, the integrand
integrand in the matrix
matrix elements
elements
n estimate
rapidly Huctuating
fluctuating quantity,
quantity; this greatly
greatly complicates
complicates even
even aan
is a rapidly
of the integral,
integral; in fact, until Landau's work all studies of problems
of this
of
problems of
kind were erroneous.
erroneous. Landau
Landau "rwas
general method
method for
for
as the first to provide a general
the
the calculation
calculation of
of quasiclassical
quasiclassical matrix
matrix elements
elements and he also applied
applied it to
processes.
a num
number
of specific
specific processes.
ber of
Peierls) published
published a detailed
detailed
In 1930 Landau (in collaboration with R. Peierls)
of the limitations imposed by relativistic
relativistic requirements
requirements on the quantumstudy ofthelimitationsimposed
mechanical description
description [6],
[6]; this article
article caused
caused lively discussions
discussions at the time.
mechanical
result lies in determining
determining the limits
limits of
of the possibility
of measuring
measuring
Its basic
basic result
possibility of
the particle
particle momentum
momentum within
within a finite time. This implied
implied that in the relarelameasure any
dynamicall variables
tivistic quantum region it is not feasible to measure
. dynamics
their interaction,
interaction, and that
that the only measurable
measurable
characterizing the particles in their
characterizing
quantities are'
are· the momenta
momenta (and
(and polarizations)
polarizations) of
of free particles.
quantities
particles. Therein
root of
of the difficulties
difficulties that arise
arise when
when methods
methods of
of
also lies the physical
physical root
conventional
quantum
mechanics,
employing
concepts
which
become
conventional
mechanics, employing
which become
meaningless in the relativistic
relativistic domain,
domain, are applied
applied there.
there. Landau returned
retuned
meaningless
published article
article [100],
[100], in Ur
which
expressed his
to this problem
problem in his last published
his he expressed
conviction that
that the
the 1,11-operators,
of unobservable
information,
conviction
up-operators, as carriers of
unobservable information,
with them
them the entire
entire Hamiltonian
Hamiltonian method,
method, should
should disappear
disappear
and along with
from
from a future theory.
theory.

_

One of
of the reasons
reasons for this conviction
conviction was the results
results of
of the research into
One
the foundations
foundations of
of quantum
quantum electrodynamics
electrodynamics which
which Landau
out
the
Landau carried out
1954-1955 (in collaboration
collaboration with A. A. Abrikosov, I. M. Khalatnikov
during 1954-1955
Pomeranchuk) [78-81,
[78-81, 86]. These studies
studies were based
based on the
and I. Ya. Pomeranchuk)
of the point
interaction as the limit of
of "smeared" interaction
interaction when
concept of
point interaction
smearing radius tends
tends to zero. This made it possible to deal directly
directly with
the smearing
finite expressions.
expressions. Further,
Further, it proved
possible
to
carry
out
the
summation
proved possible
out
summation
of the principal
principal terms
terms of
of the entire
entire series of
of perturbation
theory and this
of
perturbation theory
derivation of
of asymptotic
asymptotic expressions
expressions (for the case of
of large momenmomenled to the derivation
fundamental quantities
quantities of quantum electrodynamics
electrodynamicsGreen
ta) for the fundamental
- the Green
functions and
and the vertex part. These relations,
relations, iin
their own turn, were used
functions
n their
derive the
the relationship
relationship between
the true
true charge and mass
mass of
of the
the electron,
electron,
to derive
between the

XX
XX

Leis*
Lev Da-ridozich
Dm·ido'L·ich Landau
Landau

on the one hand, and their "bare" values, on the other. Although
Although these
of smallness of
of the "bare" char
charge,
calculations proceeded on the premise of
ge, it
was argued
argued that the formula for the relation between
between true and bare charges
charges
retains its validity regardless of
of the magnitude
magnitude of
of the bare
bace charge.
charge. Then
retains
of this formula shows that at the limit
limit of
of point
interaction the
analysis of
point interaction
true charge
charge becomes
becomes zero - the theory is "nullif1ed".T
"nullified". t (A review
review of
of the
pertinent
questions is provided
articles [84, 89]).
pertinent questions
provided in the articles
Only the future will show
programme
show the extent of
of the validity of
of the programme
planned by
planned
hy Landau
Landau [100] for constructing
constructing a relativistic
relativistic quantum field
energeticallyy wworking
direction during the
theory. He himself
himself was energeticall
orbing in this direction
last few years prior
prior to his accident.
accident. As part of
of this programme,
programme, in particular,
worked out
out a general
general method
method for.
for_ determining
determining the singularities
singularities of
of
he !'tad
had worked
quantities that
that occur
occur in the diagram technique of
of quantum field
the quantities
theory
theory [98]
[9~]..
response to the discovery
discovery in 1956 of
of parity nonconservation
nonconservation in weak
IIn
n response
interactions, Landau
Landau immediately
immediately proposed the theory
theory of
of a neutrino
neutrino with
interactions,
helicity ("two-component neutrino")
neutrino'') [92]I,
[92]!, and also
also suggested the
fixed helicity
principle of
of the conservation
conservation of
of "combined
"combined parity",
parity", as he termed
termed the
principle
combined application
application of
of spatial
spatial inversion
inversion and charge
charge conjugation.
conjugation. Accordcombined
Landau, the
the symmetry
symm~try of space
space would
ing to Landau,
would in this way be "saved" transferred to the particles
particles themselves.
themselves. This principle
the asymmetry is transferred
principle
"r
indeed proved
proved to be more widely
applicable than the law of
of parity
conservaindeed
idel y applicable
parity conservahowever, in recent
processes not conserving
conserving
tion. As is known, however,
recent years processes
combined pparity
discovered; the meaning of
of this violation
combined
a r t y have also been
been discovered;
violation
unclear.
is at present
present still unclear.
1937 study [31] by Landau pertains
pertains to nuclear physics. This study
A 1937
represents a quantitative
embodiment of
of the ideas
ideas proposed not long
represents
quantitative embodiment
before by Bohr: the nucleus is examined
examined by methods
of statistical
statistical physics
before
methods of
physics
drop of
of "quantum
"quantum Huid".
fluid". It
this study did
did not
not make
as a drop
It is noteworthy
noteworthy that this
of any far-reaching
far-reaching model
model conceptions,
conceptions, contrary
contrary to the previous
use of
previous practice
practice
of other
other investigators.
investigators. IIn
particular, the relationship
relationship between
between the mean
mean
of
n particular,
distance between
of the compound
compound nucleus and the width
width of
of the
distance
between the levels of
established for the first time.
time.
levels was established
The absence
absence of
of model
model conceptions
conceptions is characteristic
characteristic also of
of the theory of
of
The
scattering developed
developed by Landau
Landau (in collaboration
collaboration with
proton-proton scattering
Smorodinskii) [55]. The
The scattering
scattering cross-section in their
their study was
Ya. A. Smorodinskii)
expressed in terms
terms of
of parameters
parameters whose
whose meaning
meaning is not restricted
restricted by any
expressed
specific assumptions
assumptions concerning
concerning the
the particle
interaction potential.
specific
particle interaction
potential.
The study (in collaboration
collaboration with Yu. B. Rumer) [36] of
of the cascade
cascade
The
the aI'ticle
thi. s statement,
the search for
for a more rigorous pno
proot
of th_is
statement,_
ot of
with the
tt IInn connection with
.
. the article
[100] contains
contains the
the assertion,
assertion, characteristic
characteristic of
of Landau,
Landau, that
that "the
"the bre\'ltY
brevitY of
of life
hfe does
~?es not
not allow
allow
[100]
us the
the luxury
luxury o
off spending
spending time
time on
on problems
problems which
lead to
no "et"
nbw results".
r~:rlts
· d
us
which will
will lead
to 'HO
Simultaneously and
and independently,
independently, this
this theory
theory was
was proposed
proposed byY Salarn
am and
an by
by Lee
Lee
I1 Simultaneously
and Yang.
Yang.
and

Lev Davidovich
Davidovich Landau
Landau
Lev

XXI
xxi

theory of
of electron
electron showers in cosmic rays is an example of
of technical
technical
virtuosity,
physical foundations
been earlier
foundations of
of this
this theory
theory had been
earlier formulavirtuosity; the physical
ted
quantitative theory
ted by'
by- a number
number of
of Investigators,
investigators, but a quantitative
theory was essentially
essentially
study provided the mathematical
mathematical apparatus
apparatus which
which became
lacking. That study
lacking.
became the
basis
subsequent work
work in this domain.
domain. Landau
himself took
took part
basis for all subsequent
Landau himself
part in
refinement of
of the shower
shower theory by contributing
contributing two more
the further refinement
articles, one on the particle angular distribution
distribution [43]
[43] and the other
other on
articles,
secondary showers [44]
[44]..
's work dealing with the elaboration
Of ho
no smaller
smaller virtuosity
was Landau
Landau's
elaboration
Of
virtuosity was
of Fermi's
of the statistical
statistical nature of
of multiple particle production
production in
of
Fermi's idea of
represents a brilliant
brilliant example of
of the methomethocollisions ['74].
[74]. This study also represents
collisions
Ur
dological
unit y of
hich the solution
problem is
dological unity
of theoretical
theoretical physics
physics in which
solution of
of a problem
methods from
from a seemingly
seemingly completely
completely different
accomplished by using
accomplished
using the methods
Landau showed
showed that the process of
of multiple
multiple production
domain. Landau
domain.
production includes
stage of
of the expansion
expansion of
of a "cloud" whose
dimensions are large
large comcomthe stage
whose dimensions
pared with
\Yith the mean free path of particles
particles in it,
it; correspondingly, this stage
should be described
described by equations
equations of
of relativistic
relativistic hydrodynamics.
hydrodynamics. The solushould
tion of
of these
these equations
equatir>ns required
required a number
number of
of ingenious
ingenious techniques
techniques as well
tion
thorough analysis. Landau used
used to say that this study
study cost
cost him more
more
as a thorough
effort than any other
other problem
that he had ever
ever solved.
effort
problem that
responded to the requests
requests_-Iand
of the
Landau always ,yillingly
willingly responded
and needs of
experimenters, e.g. by publishing
publishing the article [56] which
which established
established the
experimenters,
energy distribution
distribution of
of the ionization
ionization losses of
of fast particles
particles during
during passage
energy
passage
theory of
of mean energy loss had existed).
through matter (previously only the theory
Turning now to Landau's
Landau's work on macroscopic
macroscopic physics,
physics, we begin
Turning
begin with
several articles
articles representing
representing his contribution
contribution to the physics
of
several
physics of
magnetism.

According to classical mechanics and statistics, a change
change in the pattern
pattern of
of
According
movement of
of free electrons
electrons in a magnetic
magnetic field cannot
cannot result
result in the appearmovement
of new magnetic
of the system.
system. Landau
Landau was
\vas the first to
ance of
ance
magnetic properties of
character of
of this motion
motion in a magnetic
magnetic Held
field for the quantum
elucidate the character
case, and to show that quantization comp
completely
changes the situation,
l et ly changes
resulting in the appearance
appearance of
of diamagnetism
diamagnetism of
of the free electron
electron gas
resulting
("Landau
diamagnetism"
as
this
effect
is
now
termed)
[4].
The
("Landau diamagnetism"
termed)
The same study
qualitatively
q~~litatively predicted the periodic
dependence of
of the magnetic
magnetic susceptisusceptiperiodic dependence
bilit
y on the intensity of
btltty
of the magnetic
magnetic Held
field when this intensit
intensityy is high.
At
0) this phenomenon
At the time (193
( 1930)
yet been
observed by
phenomenon had not vet
been observed
b y anyone,
and it was expe
rimentall y discovered
experimentally
discovered only
onlv later (the De Haas-Van
Haas-Van Alphen
Alp fen
a
quantitath·e theory of
of this effect was presented by Landau in a
effect); a quantitative
later
pape r [38].
later paper
.r

published in 1933 [12] is of
of a significance
significance greatly trantranA short article published
scen
ding the problem
scending
problem stated in its title -- a possible
possible explanation
explanation of
of the field
dependence of
of the magnetic
magnetic susceptibility
susceptibility of
of a particular class of
of substances
substances
dependence

xxn
xxii

Le'l) Da'vido°z:ich
Davido ..cich Landau
Lev

at low temperatures.
temperatures. This article
article was the
the first to introduce the concept of
of
antiferromagnetism
antiferromagnetism (although
(although it did
did not
not use this term)
terrn) as a special
special phase
phase of
of
magnetic
bodies differing in symmetry
phase,
magnetic bodies
symmetry from the paramagnetic
paramagnetic phase;
accordingly, the
the transition from
from one state to the
the othersmust
other .must occur
occur at a
rigorousl
particular model
point. t This article
article examined
examined the particular
model of
of a
rigorouslyy definite point.T
layered
layered antiferromagnet
antiferromagnet with a strong
strong ferromagnetic
ferromagnetic coupling
coupling in each
each
layer and
and a weak antiferromagnetic
antiferromagnetic coupling
coupling between the layers,
layers; a quantitaquantitalayer
tive investigation of
of this case was carried
carried out and
and the characteristic
characteristic features
features
of magnetic
magnetic properties in the neighbourhood
neighbourhood of
of the transition
transition point
point were
of
established. The method
based on
ideas
method employed
employed here by l,andau
Landau was based
on ideas
which
which he subsequent
subsequentlyy elaborated
elaborated in the general
general theory
theory of
of second-order
second-order
phase transitions.
phase
Another
paper concerns the theory
Another paper
theory of
of ferromagnetism.
ferromagnetism. The
The idea
idea of
of the
structure
bodies as consisting
structure of
of ferromagnetic
ferromagnetic bodies
consisting of
of elementary
elementary regions
regions
spontaneouslyy magnetized
magnetized in various directions
directions ("magnetic
("magnetic domains," as
spontaneousl
the
tlie modern
modern term
term goes) was expressed
expressed by P. Weiss
\Veiss as early as in 1907.
However, there
there was no suitable
suitable approach
approach to the question
question of
of the quantitative
quantitative
However,
theory of
of this structure until Landau
Landau (in collaboration
collaboration with E. M. Lifshitz)
Lifshitz)
theory
showed in 1935 that this theory
theory should be constructed
constructed on the basis
basis
[18] showed
of thermodynamic
thermodynamic considerations
considerations and
and determined
determined the form and dimensions
dimensions
of
of the domains
domains for a typical
typical case.
case. The
The same study derived
derived the macroscopic
of
macroscopic
equation of
of the motion
motion of
of the domain
domain magnetization
magnetization vector
equation
vector and, with its
aid,-developed
principles of
magnetic
of the theory
theory of
of the dispersion
dispersion of
of the magnetic
aid,-developed the principles
permeability of
of ferromagnets
ferromagnets in an alternating
alternating magnetic
magnetic field; in particular,
permeability
particular,
ferromagnetic resonance.
it predicted the effect now known as ferromagnetic
resonance.
A sho
short
communication published
published in 1933 [10] expressed
expressed the idea of
of the
rt communication
possibility
of the "autolocalization"
"autolocalization" of
of an electron
electron in a crystal
crystal lattice
lattice within
within
possibility of
the potential
virtue of
of the polarization
polarization effect of
of the electron
potential well produced
produced by virtue
itself. This idea
idea subsequently
subsequently provided
basis for the so-called
so-called polaron
itself.
provided the basis
polaroid
theory
of
the
conductivity
of
ionic
crystals.
Landau
himself
returned
once
theory of
conductivity of ionic crystals, Landau himself returned once
these problems in a later study (in collaboration
collaboration with S. I. Pekar)
more to these
dealing with
with the
the derivation
derivation of
of the equations
equations of
of motion
motion of
of the polaron
[67] dealing
polaroid
external field.
in the external
Another
short communication
communication [14] reported
reported on the results
results obtained by
Another short
Landau (in collaboration
collaboration with G. Placzek)
Placzek) concerning
concerning the structure
structure of
of the
Landau
scattering line in liquids
liquids or gases. As far back as the earl
earlyy 1920s
1920s
Rayleigh scattering
Brillouin
:\landel'shtam showed
showed that,
that, owing
owing to scattering
scattering by sound
sound
Bri
IIouin and I\Iandel'shtam
vibrations, this line must split
split into
into a doublet.
doublet. Landau
Landau and
and Placzek
Placzek drew
drew
vibrations,
attention to the attendantlnecessity
attendant" necessity of
of the existence
existence of
of scattering
scattering by entropy
entropy
attention

t

Roughly aa year
earlier Neel
Nee I (whose
(whose work
was unknown
unknown to
t':' Landau)
Landau). had
had predicted
pre~icted the
the
1' Roughly
year earlier
work was
st of
o
possibility of
of existence
existence o
off substances
substances which,
from the
the magnetic
magnetiC standpoint,
standpomt, consi
con~al•st
of tw
two
possibility
which, from
al state
speci
that a
of
·
·
·
"'
'
1
h
d"d
t
assume
a
spec1
state
of
ume
ass
sublattices
not
sublattlces with
v.· 1th opposite
opposite moments.
moments .. Neel,
P .. ee, however,
o'"·ever, did
1
no
·"th
..
a paramagnet.v~'lth
a posit
ive
·
·
1Y thought
paramagnet · "'
positive
natter is
he
matter
is involved
involved here,
here, and
and instead
mstead
he simple
s1mp
t h oug h. t that
t h at
t"
f
nsistin
g
of
co
.
tructure
cons1s
mg
o
several
e
tur
uc
exchange integral
integral at
at low
low temperatures
temperatures gradually
grad ua II y turns
turns into
mto aa str
s
exchange
magnetic sublattices.
sublattices.
Magnetic
b

Let:
Le'L' Da°z:ido-z:z'ch
Daddu'L·ich Landau
Landau

xxiii
XXlll

fluctuations, not
not accompanied
accompanied by any change
change in frequency,
frequency; as a result,
result, a
fluctuations,
triplet should
should be observed instead
instead of
of a doublet.T
doublet. t
triplet
of Landau's
Landau's works pertain
pertain to plasma physics.
One of
of these
these two
Two of
physics. One
derive the transport
transport equation
equation with
with allowance for Coulomb
Coulomb
[24] was the first to derive
interaction
interaction between
between particles,
particles; the slowness
slowness of
of decrease of
of these
these forces
forces
rendered inapplicable in this case the conventional
conventional methods
methods for constructing
constructing
transport equations.
equations. The other
other work [61], dealing
dealing with plasma oscillations,
oscillations,
transport
showed
under conditions when
particles in the
showed that, even
even under
when collisions
collisions between particles
plasma can
can be disregarded,
disregarded, high-frequency
high-frequency oscillations
oscillations will still attenuate
attenuate
("Landau damping").I
damping").!
compile one of
of the successive
successive volumes of
of the Course of
of
His work to compile
Theoretical
Physics was to Landau
Theoretical Physics
Landau a stimulus
stimulus for a thorough
thorough study
study of
of
hydrodynamics. Characteristically,
Characteristically, he independently
independently pondered
derived
hydrodynamics.
pondered and derived
basic notions
notions and results
results of
of this branch
branch of
of science.
science. His
all the basic
His fresh and
original
parti.cular, to a new
problem of
original perception
perception led, in particular,
new approach
approach to the problem
of
of turbulence
turbulence and he.
elucidated the basic
basic aspects of
of the process
the onset of
he elucidated
process
of the gradual development
d"evelopment of
of unsteady How
flow with increase
increase in the Reynolds
of
number following
following the loss of
of stabilit
stabilityy by laminar motion
motion and predicted
number
predicted
qualitativelyy various
various alternatives
alternatives possible
On investigating
im·estigating
qualitative]
possible in this case [52]. On
qualitative properties of
of supersonic
supersonic flow
around hodies,
arrived at
How around
bodies, he arrived
the qualitative
discovery that in supersonic
supersonic How
flow there
there must
must exist
exist far from
the unexpected discovery
the body
not one
one - as had been
conventional assumption
assumption - but two
the
body not
been the conventional
shock waves, one following
following the other
other [60]. Even
shock
Even in such a "classical" field
theory he succeeded
succeeded in finding
finding a new and
and previously unnoticed
as the jet
jet theory
unnoticed
exact solution
solution for an axially symmetric
symmetric "inundated"
"inundated" jet
of a viscous inexact
jet of
compressible fluid [51].
compressible
In
scientific creative
creative accomplishments
accomplishments an eminent
eminent position is
In Landau's
Landau's scientific
occupied - both from the standpoint
standpoint of direct
direct significance
significance and in terms of
of
the consequent
consequent physical
physical applications
applications - by the theory of
of second-order phase
phase
the
[29]; a first outline of
of the ideas
ideas underlying
already
transitions [29],
underlying this theory
theory is already
contained in an earlier
earlier communication
communication [l'7].ll
[17].11 The concept
concept of
of phase transicontained
of various
various orders
orders had first been
been introduced
introduced by Ehrenfest
tions of
Ehrenfest in a purely
for
ma
l
formal manner,
manner, with respect
respect to the
the order
order of
of the thermodynamic
thermodynamic derivatives
deri,·atiYes
which could
could undergo
discontinuity at the
the transition
question of
of
which
undergo a discontinuity
transition point. The question
exactlyy which
which of
of these
these transitions
transitions can
can exist
exist in reality, and what is their
their
exact
`1`t

No detailed
detailed exposition
exposition of
of the
conclusions and
and results
results oof
this study
study was
was ever
ever published
published in
in
No
the conclusions
f this
article form.
form. ItIt is
is partl
partly
presented
in
the
book
by
Landau
and
Lifshitz,
Electrodynamics
of
article
presented
in
the
book
by
Landau
and
Lifshitz,
Electrodynamics
y
Continuous Ilfedia,
1\fedia, Pergamon,
Pergamon, Oxford
Oxford 1960,
1960, §96.
§96.
Continuous

1

of

-

interesting that
that this
work was
was carried
carried out
out be
by Landau
Landau as
as his
his response
response to
the ""philo1 ItIt isis interesting
this work
to the
philo
logy" present,
present, in
in his
his opinion,
opinion, in
in previous
previous studies
studies dealing
dealing with
this subject
subject (e.g.,
(e.g.; the
the unjustified
unjustified
logy"
with this
replacement of
of divergent
divergent integrals
integrals b) their
their principal
principal values).
values). It
It was
was to
to prove
prove his
his rightness
rightness
replacement
that he
he occupied
occupied hintself
himself with
with this
this question.
question.
that
Landau himself
himself applied
applied this
this theory
theory to
to the
the scattering
scattering of
of X-rays b\
b~ crystal
crystalsS [32]
[32] and
and - in
in
IIII Landau
collaboration with I.
I. IVIl\1. Khalatnikov
Khalatnikov- to
to the
the absorption
absorption of
of sound
sound in
in the
the neighbourhood
neighbourhood ooff the
the
[82].
transition point [82].

b>

-

XXIV
xxiv

Lev Dawido-z~ich
Dm·idu<t·ich Landau
Landau
Lev

physical nature, had remained open, and previous
previous interpretations had been
been
fairly vague
point to the
vague and unsubstantiated.
unsubstantiated. Landau was the First
first to point
the
profound connection
connection between
between the possibility
possibility of
of existence
existence of
of a continuous
continuous
profound
(in the sense
sense of
of variation in the body's state) phase transition
transitton and the jumplike (discontinuous)
(discontinuous) change
change in some
some symmetry
symmetry property
property of
of the body at the
traNsition
transition point.
point. He also showed
showed that far from just any change
change in symmetry
is possible
possible at that
transition point and provided
that transition
provided a method
method which
which makes it
possible
possible to determine
determine the permissible
permissible types of
of change in symmetry. The
The
quantitative
based on
quantitative theory
theory developed
developed by Landau
Landau was based
on the assumption
assumption of
of
of the expansion of thermodynamic quantities in the neighthe regularity of
bourhood
of the transition
transition point.
clear that
that such
such a theory, which
which
bourhood of
point. It is now clear
fails to allow for possible
possible singularities
singularities of
of these quantities
quantities at the transition
transition
point,
does not reflect
reflect all the properties
properties of
of phase
transitions. The
The question
question
point, does
phase transitions.
of the nature
nature of
of these singularities
singularities was of
of great
great interest to Landau and
of
during the last years of
of his activit
activityy he worked
worked a great
great deal on this difficult
during
problem without,
however, succeeding
problem
without, however,
succeeding in arriving
arriving at any definite
definite conclusions.
conclusions.
The
The phenomenological
phenomenological theory of
of superconductivit
superconductivityy developed
developed in 1950
constructed
by Landau (in collaboration with V. L. Ginzburg) [73] also \Vas
was constructed
in the spirit of the theory
became, in
theory of
of phase
phase transitions;
transitions; subsequently
subsequently it became,
particular,
particular, the basis for the theory of superconducting
superconducting alloys. This theory
theory
invokes a number
number of
of variables
variables and parameters
whose meaning
meaning was
\Vas not
not
involves
parameters whose
completely clear
clear at the time
time it was
\Vas originally
originally developed
developed and became
completely
became underunderstandable onl
onlyy after the appearance
appearance in 1957
1957 of
of the microscopic
microscopic theory
theory of
of
standable
superconductivity, which
made possible
rigorous substantiation
substantiation of
of the
superconductivity,
which made
possible a rigorous
Ginzburg-Landau equations
equations and a determination
determination of
of the region
region of
of their
their
Ginzburg-Landau
y. In this connection,
applicability.
connection, the story
story (recounted
(recounted by
hy Y.
Ginzburg)
applicabilit
V. L. Ginzburg)
of
of an erroneous
erroneous statement
statement contained
contained in the original article
article hy
by Landau and
Ginzburg is instructive.
instructi,·e. The hasic
equation of
of the theory, defining the
Ginzburg
basic equation
effecti,·e wave function
function !Jf
of superconducting
su perconducting electrons,
electrons, contains
contains the Held
field
effective
W of
vector
yector potential A in the term

l1 ( •.,
--l lIl Vv
Z1n
Zml
zI
I

6* A
e*A)
I '11
- --

c

1/f

7. J'

which is completely
completely analogous
analogous to the corresponding
corresponding term
term in the SchrOdinger
Schrodinger
which
equation. It might
might be thought
thought that
that in the phenomenological
equation.
phenomenological theory
theory the
parameter eas
e* should
should represent some effective charge which
not have
have
which does not
to he directly
directly related
related to the
the charge
charge of
of the free electron
electron e. Landau,
Landau, however,
refuted this hypothesis
hypothesis by pointing
pointing out
out that the effective charge is not
refuted
uni,Trsal and would depend on various
yarious factors
fa~tors (pre
(pressure,
composition of
of
ssure, composition
universal
.
.
.
.
._
*
ch
ar
e
ge 6e~ would
specimen, etc.);
in an
an lrlhomogeneous
inhomogeneous specimen
specimen th
the cha~ge
would
specimen,
€tc.), then In
be a function
function of
of coordinates
coordinates and this
this would
would disturb
disturb the gauge invariance
mvanance of
of
be
th
l'€ is no reason to co ns id er
the theo
theon.. Hence
Hence the
the article
article stated
stated that
thdt "" .... there
e is no reason to consider
the
*W N O know
ow tha
the
charge
e*
as
different
from
the
electronic
charge"·
thatt
as different from the electronic charge3)- \Vee 00 '' \' kn
the charge
electron
pa
ir,
·in
1·
·
'd
·
h
h
h
f
h
c
per
electron
pair
.1' ·e.,.,
Ill reality
rea tty e* coincides
cotnct es wit
t e charge
c arge of
o the
t e (joopcf
oo
• 1.e
with the
r

Lee Do-vidoivich
Da'[·ido·vich Landau
Le-2:

xxv
XXV

= Ze
2e and not e. This value of
of e* could, of
of course,
course, have been
been predicted
e* =
predicted
of the idea
idea of
of electron
electron pairing
pairing which
which underlies
underlies the micromicroonly on the basis of
theory of
of superconductivity.
superconductivity. But the value Ze
2e is as universal
scopic theory
universal as e and
hence
Landau's argument
argument in itself
itself was valid.
hence Landau's
Another of
Another
of Landau's
Landau's contributions
contributions to the physics
physics of
of superconductivity
superconductivity
elucidate the nature
nature of
of the so-called
so-called intermediate
intermediate state. The concept
was to elucidate
of this state
state was first introduced
introduced by Peierls and F. London
London (1936)
( 1936) to account
account
of
transition to the superconducting
superconducting state
state in a
for the observed fact that the transition
magnetic
phenomenological,
magnetic field is gradual.
gradual. Their theory
theory was pure
purelyy phenomenological,
question of
of the nature of
of the intermediate
intermediate state had
however, and the question
however,
remained
open.
Landau
showed
that
this
state
is
not
a new state and that in
remained open.
showed
state
reality a superconductor
superconductor in that state
state consists of
of successive
successive thin layers
layers of
normal and superconducting
superconducting phases.
1937 Landau [30] considered
considered a
normal
phases. IIn
n 1937
these layers emerge
emerge to the surface of
of the specimen,
specimen; using
model in which these
elegant and ingenious
ingenious method
method he succeeded
succeeded in completely
completely determining
an elegant
the shape
proposed
shape and dimensions
dimensions of
of the layers in such
such a model.'t
model. t In 1938
1938 he proposed
variant of
of the theory,
theory, according
according to which
which the layers repeated]
repeatedlyy branch
branch
a new
new variant
on emerging to the
the surface,
surface; such
such a structure
structure should
should be
be thermodynamithermodynamiout on
favourable, given
gi,·en sufficiently
sufficiently large dimensions
dimensions of
of the
the specimen.I
specimen.!
cally more favourable,
significant contribution
contribution that physics
physics owes to Landau
Landau is his
But the most significant
theory of quantum liquids. The
The significance
significance of
of this new discipline
discipline at present
theory
steadily growing,
growing; there
there is no doubt
doubt that its development
development in recent
recent decades
decades
is steadily
revolutionary effect on
on other
other domains
domains of
of physics
has produced
produced a revolutionary
physics as well -on solid-state
solid-state physics
and even
even on
on nuclear
nuclear physics.
on
physics and
physics.
The
su
perfluidity
theory
was
created
by
Landau during
during 1940-1941
1940-1941 soon
The superfluidity theory
created
Landau
Kapitza's discovery
discO\·ery towards the end
end of
of 1937 of
of this fundamental
fundamental
after Kapitza's
property of
of helium II. Prior to it, the premises
premises for understanding
property
understanding the
physical nature of
of the phase
transition observed in liquid
liquid helium had
had been
physical
phase transition
been
essentially lacking
lacking and
and it is not surprising that
that the previous
previous interpretations
essentially
of this phenomenon
phenomenon now seem
seem even
e\·en naive.
The completeness
completeness with which
which
of
naive.llII The
theory of
of helium IIII had been
constructed by Landau
the theory
been constructed
Landau from the very
beginning is remarkable: already his first classic paper
paper [46] on this subject
subject
contained practically
principal ideas of
of both
both the microscopic
microscopic theory
theory
contained
practically all the principal
of
m IIII and the macroscopic
of heliu
helium
macroscopic theory
theory constructed
constructed on its basis -- the
therm
odynamics and hydrodynamics
thermodynamics
of this fluid.
hydrodynamics of
Underlying Landau's
Landau's theory
theory is the concept
concept of
of quasiparticles (elementary
Underlying
excitations) constituting
constituting the energy
energy spectrum
spectrum of
of helium II. Landau was in
excitations)
pose the question
question of
of the energy
energy spectrum
spectrum of
of a macroscopic
macroscopic
fact the first to pose

1't

Landau hinlself
himself wrote
concerning this
th1s matter
matter that
that "amazingly
"amazingly enough
enough an
an exact
exact detennidetermiLandau
wrote concerning
e l l
nation ooff the
the shape
shape of
of the
the lavers
layers proves
pro,·es to
to be
be possibl
possible"
[30]
nation
[30]

1

A detailed description of
of this work was
was published in
in 1943
19-B [49].
[49].
Thus, Landau
Landau himself
himself in
in his
his work
work on
on the
the theory
theory of
of phase
phase transitions
transitions [29]
[29] considered
considered
I]II Thus,
whether
I is
whether helium
helium III
is aa liquid
liquid crystal,
crystal, et
even
though he
he emphasized
emphasized the
the dubiousness
dubiousness of
of this
this
en though

assu1nption.
assumption.

v

Lev
awidovich Landau
Lev D
Davidovich
Landau

xxvi
XXVI

body in such
such a very
very general
general form, and
and it was he, too, who discovered
discovered the
body
nature of
of the spectrum
spectrum for a quantum fluid of
of the type to which
which liquid
nature
4
helium (He'*
(He isotope)
isotope) belongs
-or,
now termed,
termed, of
of the Bose type.
type.
helium
belongs or, as it is now
In
In his 1941 work Landau
Landau assumed
assumed that the spectrum
spectrum of
of elementary
elementary excitaexcitaof two
two branches:
branches: phonons,
with a linear
linear dependence of
of energy
energy
tions consists of
tions
phonons, with
.e:.
p, and "rotors",
e. on
on momentum
momentum p,
"rotons", with a quadratic
quadratic dependence, separated
separated
ground state
state by an energy gap.
gap. Subsequently
Subsequently he found
found that such
such
from the ground
of spectrum
spectrum is not
not satisfactory
satisfactory from the theoretical
theoretical standpoint
standpoint
a form of
would be unstable)
unstable) and
and careful
careful analysis of
of the more
more complete
complete and
(as it would
exact experimental
experimental data that had by then become
become available led him in 1946
exact
establish the now famous spectrum
spectrum containing
containing only one branch
branch in
to establish
"rotons" correspond
correspond to a minimum
minimum on the curve
curve of
of £(p).
e(p). The
The
which the "rotors"
macroscopic concepts
concept's of
of the theory
theory of
of superfluidity are widely known.
known.
macroscopic
Basically they reduce
reduce to the idea of
of two
two motions
motions simultaneously
simultaneously occurring
occurring
in the Huid
fluid - "normal" motion
motion and
and "superHuid"
"superfluid" motion,
motion, which
which may be
visualized as motions
of two "Huid
"fluid components".T
components".t Normal
motion is
visualized
motions of
Normal motion
accompanied
accompanied by internal
internal friction,
friction, as in conventional
conventiona! fluids. The
The determinadetermination of
of the viscosity
viscosity coefficient
coefficient represents a kinetic
kinetic problem
problem which
which requires
requires
tion
of the processes
of the onset
onset of
of an equilibrium in the
the "gas of
of
an analysis of
processes of
quasiparticles"; the principles
principles of
of the theory of
of the viscosity
of helium IIII
quasiparticles",
viscosity of
developed by Landau
Landau (in collaboration
collaboration with I. M. Khalatnikov) in
were developed
1949 [69, '70].
70]. Lastly, yet another investigation
investigation (carried
(carried out in collaboration
collaboration
1949
Pomeranchuk) [64] dealt with the problem
of the behavior
behaviour of
of
with I. Ya. Pomeranchuk)
problem of
extraneous atoms
atoms in helium,
helium; it was shown,
shown, in particular,
particular, that
that any atom
atom of
of
extraneous
become part
part of
of the "normal
"normal component" of
of the fluid
this kind will become
Huid regardregardof whether
whether the impurit
impurityy substance
substance itself
itself does
does or does
does not
not display
display the
less of
property
of superfluidity
superfluidity -- contrary
contrary to
to- the incorrect view previously
held
property of
previousl y held
literature.
in the literature.
The liquid isotope He3
He 3 is a quantum
of another
another type
type - the Fermi
The
quantum liquid of
type as it is now termed.
termed. Although
Although its properties
properties are not
as.striking as the
type
not as.striking
properties
of liquid He4, they are no less interesting
interesting from the standpoint
standpoint of
properties of
of liquids of
of this kind was developed
developed by Landau and
basic theory. A theory of
presented
papers published
during 1956-1958. The
The First
first two
presented by him in three papers
published during
of these
these [90, 91] established
established the nature
nature of
of the energy
energy spectrum
spectrum of
of Fermi
Fermi
of
liquids,
considered
their
thermodynamic
properties
and
established
liquids, considered their thermodynamic properties
established the
equation for the relaxation
relaxation processes
processes occurring
occurring in these
these liquids. His
kinetic equation
of the kinetic
kinetic equation
equation led Landau
Landau to predict
of vibravibrastudy of
predict a special type of

cription of
t Some
Some of the
the ideas
ideas of the
the "two-component"
''two-component" macroscopic
macroscopic des
description
of fluid
li~l_'id helium
helium
1`
a clear
were introduced
introduced independently
independently o
off Landau
Landau by
by L'.
L. Tisza
Tisza (although
(although without
pro~·,dmg
were
wlthout prow
lfig a clear
e IN
·
) . His
H.1s detailed
d etm·1e d article
- 1e published
bl'•s hed in
m Franc
France
m 19+0 Vv3$I
was •
physical
interpretation
o
f them)phvs1cal
mterpretat10n
of
them
art•c
pu
b
.
f
_
.-'11943
brief
.·-h
and
the
ne
note
of
1938
1
note
of
1938
943 and the I
o'"·•ng to
to wartime
\\·arttme conditions.
conditions. not
not received
received in
In the
t e LL SSR until
unt1
_
owing
.to
t l Y remamed
Thai
d • .- d
S -·
had unfortunate
un
re I ned
in
ends o
Academie
des
Sciences
had infor na. de dY b.
in the
the Courpies
Comptes rrendus
off the Paris
Pans
Aca
em1e
es
c1ences
d un_..
l'
'
fT.
h
r'-T
'\\·as
pro\- I d 1:~d b 5\ I,...and
an au
au IN
•n
noticed. A
A criticism
criticism of
of the
the quantitative
quant1t;:;t1Ye aspects
aspects of
o Tisza
1sza ss the<-:orb
t eo J was prom
•
noticed.
v

f

the article
article [66]
[66].
the
v

u

··ssR

v

Y

u

Lev
.cich Landau
Lew Da·vido
Dawidozfich
Landau

xxxvn
vii

tonal
process in liquid He3
tional process
He 3 in the neighborhood
neighbourhood of
of absolute zero, which
he termed
paper [95] presented
presented a rigorous microtermed zeroth
zeroth sound.
sound. The third
third paper
scopic substantiation
had
substan~iation of
of the transport
transport equation,
equation, whose
whose earlier
earlier derivation
derivation had
number of
contained
of intuitive
intuitive assumptions.
assumptions.
tained a number
con
Concluding this
this brief
complete survey, it only remains
remains to be
Concluding
brief and far from complete
repeated that to physicists
physicists there
there is no need
emphasize the significance
significance of
of
repeated
need to emphasize
Landau's contribution
contribution to theoretical
theoretical physics.
His accomplishments
accomplishments are of
of
Landau's
physics. His
lasting
will for ever
lasting significance
significance and
and will
ever remain
remain part of
of science.

[THIS PAGE INTENTIONALLY BLANK]

CHAPTER
C
H A P T E R II

THE E
EQUATIONS
OF
MOTION
THE
QUATIONS O
F M
OTION

§1. Generalised
Generalised co-ordinates
co-ordinates
§1.

ONE
fundamental concepts
concepts of
of mechanics
mechanics is that
that of
of a Particle
particle.xi't By this
E of the fundamental
ON
bodyy whose dimensions
dimensions may be neglected
neglected in describing
describing its motion.
motion.
we mean a bod
The
possibility of
doing depends,
probThe possibility
of so doing
depends, of
of course,
course, on the
the conditions
conditions of
of the probconcerned. For
For example,
example, the planets
planets may be regarded
regarded as particles
lem concerned.
particles in
their motion about
about the Sun,
Sun, but
but not
not in considering their rotation
rotation
considering their
considering
about their axes.
The
position of a particle
The position
particle in space
space is defined
defined by its radius
radius vector
vector r,
r, whose
The derivative v =
= drldt
drfdt
components are its Cartesian
Cartesian co-ordinates
co-ordinates x, y, z. The
components
of rr with respect
particle, and the
respect to the time t is called
called the 'velocity
velocity of
of the particle,
second derivative
derivative d2r/dt2
d2rfdt2 is its acceleration. In what
what follows we shall, as is
second
customary, denote
denote differentiation
differentiation with respect
respect to time
time by placing
customary,
placing a dot above
r.
a letter: v = in
To define the position
position of
of a system
system of
of N particles
necessary to
To
particles in space, it is necessary
vectors, i.e. 3N
3N co-ordinates.
co-ordinates. The
The number
of independent
independent
specify
number of
specify N radius vectors,
quantities which
which must
must be specified
specified in order to define uniquely
position of
of
quantities
unique y the position
system is called the number of
of degrees of
of freedom;
number is
any system
freedom; here, this number
3N. These quantities
quantities need
need not
not be the Cartesian
Cartesian co-ordinates
co-ordinates of
of the particles,
3N.
particles,
and the conditions
conditions of
of the
the problem
problem may render
render some
some other
other choice
choice of
of coand
convenient. Any s quantities
quantities Q1,
q1, quo,
q2, .....
qs which completely
completely
ordinates more convenient.
ordinates
. . , * Qs
define the position
of a system
system with s degrees
degrees of
of freedom
freedom are called generalised
position of
eo-ordinates of
of the system,
system, and the derivatives
derivatives Q:
qt are called
called its generalised
co-ordinates
generalised
'veloc
velocities.
itzles.
\Vhen the
the values of
of the generalised
generalised co-ordinates
co-ordinates are specified,
specified, however,
however,
\Vhen
the "mechanical
"mechanical state" of
of the system
system at the instant considered
considered is not
not yet
determined in such
such a way that the position
of the system
system at subsequent
subsequent
determined
position of
inst
ants can be predicted.
instants
given values of
of the co-ordinates, the system
system
predicted. For
For given
can
can have any velocities,
and these affect the position
position of
of the system
system after
after an
velocities, and
infinitesimal time interval
interval do.
dt.
infinitesimal
If
all
the
If all the co-ordinates
co-ordinates and velocities
simultaneouslyy specified,
specified, it is
i3
velocities are simultaneousl
known from experience
experience that the
the state of
of the system
system is completely
completely de
determined
known
termined
and that
that its subsequent
subsequent motion can, in principle,
principle, be ca
calculated.
MathematicIculated. Mathematicif all the co-ordinates
co-ordinates q and
and velocities
velocities Q
q are given
gh·en at
ally, this means that, if
some instant, the accelerations
accelerations q'
ij at that instant
instant are uniquely defined.;
defined.~
some
t Sometimes
Sometimes called
called in
in Russian
Russian aa material
material point.
point.
'|'
1t FFor
brevity, we
we shall
shall often
often conventionally
conventionally denote
denote by
by q9 the
the set
set of
of all
all the
the co-ordinates
co-ordinate>
or brevity,
velocities.
91, qa,
92, ...,
... , qa,
9•, and
and similarly by
by q the
the set
set of
of all
all the
the velocities.
qi,
1

A

The Equations of
of .Motion
.11otion

2

§2

The relations
relations between
between the accelerations,
accelerations, velocities
co-ordinates are
The
velocities and co-ordinates
ions
of motion. They are second-order
second-order differential
differential equat
equations
called the equations of
called
for the functions
functions q(t), and
and their
their integration
integration makes possible,
possible, in principle,
principle, the
determination
determination of
of these functions
functions and so of
of the path
path of
of the system.

§2. The
The principle
principle of
of least action

The
The most
most general
general formulation
formulation of
of the law governing
goYerning the
the motion
motion of mechmechof least action or Hamilton's
according
anical systems is the principle of
Hamilton principle,
principle, according
to which every mechanical
mechanical system
system is characterised
characterised b
byy a definite function
function
L(q1,
...,, Qs,
L(qt, go,
q2, ...,
... , is,
q8 , Qi,
qt, QUO.
q2 ••..
q8 , t), or briefly L(q, Q,
q, I),
t), and the motion of the
system is such that
that a certain
certain condition is satisfied.
satisfied.
system
Let the
the system
system occupy,
occupy, at the
the instants £1
It and
and 12,
t 2, positions defined
defined by
Let
by two
2
of values of
of the co-ordinates,
co-ordinates, nu)
tjl> and 9l2\.
q l. Then the condition
condition is that
that the
sets of
sets
system
positions in such a way
way that
that the integral
system moves between
between these positions

I L(2,
L(q, Q,
q, I) dr

(2.1)

t) dt

II

s=

S

t2
£2

tl
al

least possible
The function L is called the Lagrangz"an
of
Lagrangian of
takes the least
possible value.t
va Iue.']' The
system concerned,
concerned, and the Integra
integrall (2.1) is called the action.
the system
The
The fact that
that the Lagrangian
Lagrangian contains
contains only
only q and Q,
q, but
but not
not the higher
higher
derivatives Q,
ij, Q,
lj, etc., expresses
expresses the result
result already mentioned,
mentioned, that the mechmechderivatives
anical state of
of the system is completely
completely defined when the co-ordinates
co-ordinates and
velocities are given.
given.
velocities
Let us now derive the
the differential equations
equations which
which solve the problem
problem of
of
Let
minimising the integral
integral (2.l).
{2.1 ). For
For simplicity,
simplicity, we shall
shall at
at first assume that
that the
the
minimising
system has only
only one degree
of freedom, so that only one function q(t) has to
system
degree of
be determined.
determined.
Let q =
= q(t) be the
the function
function for which
which S
Sis
This means that S
Let
is a minimum. This
increased when q(t) is replaced by any function of the form
is increased
J

q(t) +
+8g(t),
Sq(t),

(2.2)

where 8q(t)
Sq(t) is a function
function which is small
small everywhere
everywhere in the interval
interval of time
where
t1 to £2;
t2; 8q(t)
Sq(t) is called
called a 'vanlztion
variation of
of the
the function q(t). Since,
Since, for
fortt =
= it
t1
from I1
2
and for
fortr =
= to,
t2, all the
the functions
functions (2.2) must take the
the values q<U
q< > respecand
nu) and Q(2)
respectively, it follows that
tively,

8Q(I2l

II

8QU1)

1

(2.3)

0.

ciple of
should be mentioned
mentioned that
that this
this formulation
formulation of
ofthe
principle
of leas
leastt action
action is
is no
nott always
Tt ItIt should
the prin
valid for
the entire
entire path
path of
of the
the system,
system but
but on
only
any sufFlcl¢HtlY
sufficiently short
short segment
segm~nt of
of the
the path
path.
valid
for the
for any
ly for
ecessa rajy a
-· um».
.
. path must
'
butt no
nott hnnecessarily
rn·•mmum.
nm
The integral
mtegral
(2.1) for
the entire
entire
have an
an extreme,
extremum, bu
. a Imn
The
(2.1)
for the
path must have
· fact,
· of
·
d env
· ati
ation
oftthee equations
equations of
of m
on of
ott'io
riv
n,
Th1s
fact, however,
however, is
IS
of no
no importance
Importance
as regards
regards the
t h e de
mo
•on,
This
as
since only
only the
the extreme
extremum condition
condition is
is used.
used.
since

_

.

Q

The-principle of
of least action
Théprinczlble

§2

The
+ Sq is
The change
change in S when q is replaced by qq+

3

-gIJL(q+
L(q+ sq,
Sq, q+
q+ sq,
Sq, al
t) drdt- lJL(q1
L(q, Q,
q, 0
t) dz.
dt.
Hz
t2

£2
t2

rl

r1

When this
this diilerence
difference is expanded
expanded in powers of 8g
Sq and SQ
Sq in the
the integrand, the
When
of the First
first order.
order. The
The necessary
necessary condition
condition for S
S to have a
leading terms are of
minimums'
first variation,
z'ariaminimum( is that
that these terms (called
{called the first
variation, or simply
simply the
the variaof the integral) should
should be zero. Thus the principle of
of least action may
tion, of
be written in the form

fI

8ssS = 8s L(q, Q,
q, 1)
t) dr
dt = 0,
t2
£2

II((

effecting the
the variation,
variation,
or, effecting
2

'
52

I

tl
al

r

tlI

(2.4)

oL
3oL
L
31,
8q+ . 8g
-Sq+-_
sq) do
dt =
= 0.
o.
3oq
3
oq
q
q

Since 8g
Sq =-= d8q;'dt,
dSqjdt, we obtain, on integrating
integrating the second
second term
term by parts,
Since
parts,

L d AL
J(
-----_
] I( - dr- <2. 8q dr = 0.

3oL
L ]t2
£2
8S
. 8q
SS = [-_
Sq +
+
aoqQ *.t1

t2
£2

3oL
3oq
Q

tl
£1

d oL) Sqdt
dt 3oq ,

(2.5)
{2.5)

The conditions
conditions (2.3)
(2.3) show
show that
that the
the integrated
integrated term
term in (2.5)
{2.5) is zero. There
The
remains an integral
integral which
must
vanish
for
all
values
of
Sq.
which
of 8 . This can be so only
If
if the integrand
integrand is zero
zero identically. Thus we have

:t(( ~~)) ~~élan
d aL

3L

dr Ag

=

::

0.
0.

When
the system
system has more than
than one
one degree
degree of
of freedom,
freedom, the
the 1s different
different
\Vhen the
functions QU)
qi(t) must
must be varied
varied independently in the
the principle
principle of
of least
least action.
We then
ntly obtai
then evide
evidently
obtains
equations of
of the
the form
form
n s equations

d AL ) 3L
~(oL)oL
dt ( 3Q5
oqi
oqi
avi

=

o
0

(i = 1,
1,2,
... ,s).
(i
2, ...,s).

(2.6)

These are the
the required
required differential equations,
equations, called
called in mechanics
mechanics Lagrange's
Lagrange?
equa
tiona
If the Lagrangian
Lagrangian of
of a given
given mechanical
mechanical system
system is known,
known, the
the
equations.tt If
equations (2.6) give the
the relations
relations between
between accelerations.
accelerations. velocities
velocities and coordinates, i.e. they
they are the
the equations
equations of
of motion
motion of
of the
the system.
ordinates,
Tt
It

Or, in
in general,
general, an extreme.
extremum.
Or,
In the
the calculus
calculus of
of variations
they are Euler's equations
equations for the
the formal
formal problem of
of deterIn
variations they
min
ing the
mining
the extrema
extrema 'of
"of an
an integral
integral of
of the
the form
(2.1).
form (2.1).

of

4

§3

The
The Equations
Equations of Motion
Motion

Mathematically, the
the equations (2.6)
{2.6) constitute
constitute a set
set of
of s second-order
llvfathematically,
functions ii;(t).
qi(t). The
The general
general solution contains
contains 2s
equations for s unknown functions
equations
arbitrary constants.
constants. To
To determine
determine these
these constants
constants and
and thereby to define
define
arbitrary
the motion
motion of
of the
the system, it is necessary
necessary to know
know the
the initial conditions
conditions
uniquely the
the state
state of
of the
the system
system at some
some given
given instant,
instant, for example
example the
the
specify the
which specify
initial
initial values
values of all the
the co-ordinates
co-ordinates and velocities.
velocities.
Let a mechanical
mechanical system
system consist
consist of two parts A and B which
which would, if
Let
closed, have Lagrangian
Lagrangians LA and LB respectively.
respectively. Then, in the
the limit where
closed,
where
the distance between the
the parts becomes
becomes so large that the
the interaction
interaction between
the
between
may be neglected,
neglected, the
the Lagrangian
Lagrangian of
of the
the whole
system tends to the
the value
them may
whole system

limL = LA-l~LB.

(2.7)

This additivity
additivity of
of the
the Lagrangian expresses
expresses the
the fact that the
the equations of
of moof either of
of the
the two non-interacting parts cannot involve
involve quantities
quantities pertion of
tion
taining to the
the other part.
taining
It is evident
evident that
that the
the multiplication
mdtiplication of
of the
the Lagrangian
Lagrangian of a mechanical
mechanical
It
arbitrary constant
constant has no effect on the
the equations
equations of
of motion.
motion.
system by
system
bY an arbitrary
From this, it might
might seem,
seem, the following important
important property
property of
of arbitrariness
arbitrariness
From
can be deduced: the
the Lagrangian
Lagrangians of
of different isolated
isolated mechanical
mechanical systems
systems
can
multiplied by different arbitrary
arbitrary constants. The additive
additive property,
may be multiplied
however, removes
removes this
this indefiniteness,
indefiniteness, since
since it admits
admits only
only the
the simultaneous
simultaneous
however,
multiplication of
of the
the Lagrangian
Lagrangians of
of all the
the systems
systems by the
the same
same constant.
constant.
multiplication
s corresponds
This
corresponds to the
the natural
natural arb
arbitrariness
the choice
choice of the
the unit of
of mcameaitrariness in the
Thi
of the
the Lagrangian,
Lagrangian, a matter
matter to which
shall return
return in §4.
§4-.
surement of
which we shall
One further
further general
general remark
remark should
should be made.
made. Let
Let us consider
consider two functions
functions
One
tj, Z)
t) and L(q, Q,
tj, Z),
t), differing by the total derivative with respect to time
L'(q, Q,
of some
some function f (q, t) of co-ordinates
co-ordinates and time
time::
of

d

(2.8)

II

L'(q,q,t)
L(q,tj,t)+-f(q,t).
d f ( ,z).
L'(Q»
Q I) = LM,
Q, I) + dr
dt Q

calculated from these two functions are such
such that
that
The integrals (2.1) calculated
II

£1

L'(q,
q•, Z) do
L'(q,tj,t)dt

=I L(q, q•, I) + I ~dt
t2
52

£1

do
L(q,tj,t)dt+

t2
£2

£1

d

-d:
do

=

II

=I

C/g.

S'

t2
£2

S+j(q!2l,f2)-f(qll,tl),
S +f(9(2', to) -f(Q"'» In),

ndithey differ by a quantity
quantity which
which gives zero on variation,
variation, so that
that the co
condithey
ations of
SS' =
= 0 and
and 8S
SS =
= 0 are equivalent,
equivalent, and the form of the equ
equations
tions
ns 8S'
tio
within an
motion is unchanged.
unchanged. Thus the
the Lagrangian
Lagrangian is defined
defined only
only to .within
motion
ti
me,
ordinates and
additive total time derivative
derivative of any
any function
function of coco-ordinates
and ttme.
additive

t.e.
1.e.

§3. Galileo's relativity principle
· IS
· nec
necessary
essary
it
In order
order to
to consider
consider mechanical
mechanical phenomena
phenomena
tt is
.
In
•
· ge
general
different
.on are in
neral different
frame
of reference.
reference. The laws
laws of
of moti
motion
m
frame of
U

e
to oh
.. ch oose a
In form
Ill
form ftoorr

Galileo' s relativity Principle
principle
Calileo's

§3

5

of reference. When
\Vhen an arbitrary frame of reference is chosen,
different frames of
happen that
that the
the laws governing
governing even
even very simple
simple phenomena
phenomena become
it may happen
become
naturally arises
arises of
of finding a frame
of reference
reference
very
complex. The problem
very complex.
problem naturally
frame of
which the
the laws of
of mechanics
mechanics take their
their simplest
simplest form.
in which
If
choose an arbitrary frame of
of reference,
reference, space
space would be ininIf we were to choose
means that,
that, even
even if a body
body interacted
interacted
homogeneous and anisotropic:
anisotropic.- This means
homogeneous
with
other
bodies, its various
various positions
space and
and its different
different orientapositions in space
er bodies,
h no oth
wit
tions
would in general
tions would not
not be mechanically
mechanically equivalent.
equivalent. The same
same would
general be
e, which
truee of tim
time,
which would
would likewise be inhomogeneous,
inhomogeneous; that
that is, different inintru
sta~ts would
would not be equivalent.
equivalent. Such properties
properties of
of space
space and
and time would
stants
of mechanical
mechanical phenomena.
For example,
example,
evidently complicate
complicate the
the description
description of
evidently
phenomena. For
a free body
body (i.e. one subject
remain at rest
subject to no external
external action)
action) could
could not
not remain
rest::
instant, it would begin
begin to move in some
some direcdirecif its velocity were zero at some instant,
tion at the
the next
next instant.
tion
It is found, however,
however, that
that a frame of reference
reference can always be chosen
chosen in
It
which
which space
space is homogeneous
homogeneous and
and isotropic
isotropic and time is homogeneous.
homogeneous. This is
called an inertial
frame. In particular,
body which is at
inertial frame.
particular, in such
such a frame a free body
some instant remains always
always at rest.
rest at some
of the
the
\Ve can now draw some immediate
immediate inferences
inferences concerning
concerning the
the form of
'We
Lagmngian of
of a particle, moving freely, in an
an inertial
inertial frame of
of reference.
Lagrangian
reference.
The
The homogeneity
homogeneity of
of space
space and
and time implies that
that the
the Lagrangian
Lagrangian cannot
cannot conthe radius
radius vector
of the
the particle
particle or the
the time t, i.e. L
explicitly either
either the
tain explicitly
vector r of
mus
mustt be a function
function of
of the
the velocity
velocity v only. Since
Since space
space is
is isotropic,
isotropic, the
the Lagranof the
the direction
direction of
of v,
v, and is therefore
therefore a funcfuncmust also be independent
independent of
gian must
of its magnitude,
mag!litude, i.e. of
of V2
v2 =
= 712:
v2:
tion only of
L(v2)..
L = L(i)2)

(3,
(3.1)
1)
Sinc
e the
Since
the Lagrangian
Lagmngian is
IS independent
independent of
of Ra
r, we
we have 3oLJor
L lEer = 0, and so

ist
Lagrange's equation iT

( )

d 3L
__
~(oL)
dt as
Ov
do

=
= 00

'

wh
ence ala
L v = constant.
whence
oLjov
constant. Since 3Ll3v
oLfov is aa function of
of the velocity only, itit
fol
lows tha
follows
thatt
v = constant.
constant .

(3.2)
(3.2)

.Thus we conclude
conclude that, in an inertia]
inertial frame,
frame, any
any free motion takes place
with a veloc
ity whic
h is constant
With
velocity
which
constant in both
magnitude
and direction.
direction. This is
both magnitude and
the law of
of inert
inertia.
ia.
If
we
consider, beside
s the
. If
~nsider,
besides
the inertial
inertial frame, another frame
frame moving uniformly
uniformly
in
a
In straight
stra1ght line relative
relative to the inertial
inertial frame,
frame, then
then the
the laws of
of free motion
motion in
in

'I't The derivative ooff a scalar
scalar quantity with respect
respect to
to a vector
vector is defined as the
the vector
whose
vector whose
components are
are equal
equal to
to the
the derivatives
derivatives of
of the
the scalar
scalar with
with respect
respect to
to the
the corresponding
corresponding
components
components of
of the
the vector.
vector.
components

of

6

The, Equations
Equations of Motion
Motion
The

§4

the other frame will be the
the same as in the original frame:
frame: free motion
motion takes
takes
place
place with
with a constant
constant velocity.
of free motion
the same
same in
shows that
that not
only are the
the laws of
Experiment shows
not only
motion the
the two frames,
frames, but the
the frames
frames are entirely
entirely equivalent
equivalent in all mechanical
mechanical rerethe
spects.
there is not
not one but an infinity
infinity of
of inertial
inertial frames moving, relative
spects. Thus there
to
properties
to one
one another,
another, uniformly in a straight
stmight line. In
In all these
these frames the properties
of space
space and
and time
time are the
the same,
same, and
and the laws of
of mechanics
mechanics are
are the
the same.
same. This
of
constitutes Galileo's relativity
relativity principle,
of the
the most important
important principles
constitutes
princzple, one of
principles
of mechanics.
above discussion
discussion indicates
indicates quite
clearly that
that inertial
inertial frames
frames of
of referThe above
quite clearly
which they
they should,
should, as a rule, be
ence have special properties,
properties, by virtue of which
used
of mechanical
what follows, unless the
used in the
the study
study of
mechanical phenomena.
phenomena. In
In what
the conspecifically stated,
stated, we shall
shall consider
consider only
only inertial
inertial frames.
frames.
trary is specifically
trary
complete mechanical
mechanical equivalence
equivalence of
of the
the infinity
infinity of
of such frames
frames shows
The complete
of reference
reference which
which should
should be preferred
preferred
that there is no "absolute" frame of
also that
to other frames.
frames.
to
The co-ordinates
co-ordinates rr and
and r'
r' of
of a given point
frames of referreferpoint in two different frames
K', of
of which
latter moves relative to the former
former with
with velocity
ence K and K',
ence
which the latter
V, are related by
rr == r'+
r' + Vt.

Here itit is
is understood
understood that
that time
time is
is the
the same
same in
in the
the two
two frames
frames::
Here

rt

:=

z'.
t'.

(3.3)

(3.4)

assumption that time is absolute
absolute is one of
of the
the foundations
foundations of
of classical
The assumption
mechanics.T
mechanics. t
re called a Galilean transformation. Galileo's
Formulae (3.3) and (3.4)
{3.4) aare
Galileo's
Formulae
formulated as asserting
asserting the
the invariance
invariance of
of the
the mechmechrelativity principle
can be formulated
relativity
principle can
of motion
motion under
under any such transformation.
transformation.
anical equations
equations of
anical
§4.
§4. The
The Lagrangian
Lagrangian for
for aa free
free particle
particle

determine the form of
of the Lagrangian,
Lagrangian, and consider
consider
Let us now go on to determine
simplest case, that of the free motion
motion of
of a particle
first of all the simplest
particle relative to
inertial frame of
of reference. As we have already seen, the Lagrangian
Lagrangian in
an inertial
depend only on the square
square of
of the velocity. To
To discover the form
this case can depend
rtial
of this dependence,
dependence, we make use of
of Galileo's
Galileo's relativity
principle. If
inertial
of
relativity principle.
If an ine
rtial
another ine
inertial
frame K is moving with an infinitesimal velocity eE relative to another
frame K',
then v'
v' = v ++E.
Since the equations
equations of
of motion
motion must have the
the same
K', then
e. Since
verted by
form in every
every frame, the
the Lagrangian
Lagmngian L(1v2)
L(v2) must
must be con
con~erted
by this transform
2
all,
at
if
nly by
-112),
L' which differs from L(
L(v
), 1f at all, o
only
formation into a function L'
bY the
fonnation
total time derivative of
of a function of
of co-ordinates
co-ordinates and
and time
time (see
(see the
the end
end of
of

§2).
I

'I't

•

ics.
· · t• mechanics.

mechan
tivistic
This assmnption
asswnption does
does not
not hold
hold good
good in
m rela
re 1atiVIS
IC
This

§4

7

The
The Lagrangian
Lagrangian for
for aa free
free particle
particle

L' = L(
L(v'2)
L(v2+2v • e-I-62).
E+e2). Expanding
Expanding this
this expression
expression in
in
'v '2) = L(z:2-I-2v
We have L'
powers of
of eE and
and neglecting
neglecting terms
terms above
above the
the First
first order,
order, we
obtain
powers
we obtain
L(v'2)
'2)
L(z)

oL
aL
L(v2)+-2v•
E.
L(@2)
+ Zv • e.
ov2
302

=

II

The
The second
second term on the
the right
right of
of this equation
equation is a total time derivative
derivative only
only
2
of the velocity v. Hence
Hence 3L]3o2
oLfov is independent
independent of
of the
the
if it is a linear function of
velocity, i.e. the
the Lagrangian
Lagrangian is in this case proportional
proportional to the
the square of
of the
velocity, and we write it as
L -- lmv2
(4.1)
2-m@2_•
From the
Lagrangian of
the fact that a Lagrangian
of this
this form
form satisfies Galileo's relativity
relativity
infinitesimal relative
relative velocity,
velocity, it follows at once
once that
that the
the
principle for an infinitesimal
Lagrangian is invariant
invariant for a Finite
finite relative velocity
velocity V of
of the frames
frames K and K'.
Lagrangian
K'.
For
For
L'
+ V)2 = 1.,mw2+mvL' = !mv'2 = §rIm(v
~m(v+
~mv2+mv· V +§m
+!mV2,
V2,
or
= L
L+d(mr·
L' =
+ d(mr- V+!mV2t)fdt.
V-I--§mV2t)/dt.

The second
second term
term is a total
total time derivative
derivative and
and may
omitted.
may be omitted.
The quantity m
m which appears in the Lagrangian (4.1) for a freely moving
The
particle is called the mass of
I agran____
of the barticle.
particle. The
The additive property of
of the Lagranof particles
particles which do not
not interact we haven
havet
gian shows
shows that for a system
system of
gian
' - '

- --

.__J_-.

=

II

L
L

2%%va2~
L)111ava2·

((4.2)
4.2)

It should
should be emphasised
emphasised that
that the
the above
above definition
definition of
of mass becomes
meanIt
becomes meanthe additive property
property is taken
taken into account.
account. As has been
only when
when the
ingful only
been
mentioned
the Lagmngian
can
always
be
multiplied
by
any
constant
mentioned in §2, the
Lagrangian can always
multiplied
of motion.
motion. As regards
regards the function
function (4.2), such
without affecting the equations
equations of
without
multiplication amounts
amounts to a change
change in the
the unit of
of mass
mass;, the
the ratios
ratios of
of the
the masses
masses
multiplication
of different
different particles
remain unchanged
only these
these ratios
of
particles remain
unchanged thereby, and it is only
whic
h are phys
which
physically
meaningful.
ically meaningful.
It is easy to see that
that the
the mass of
of a particle
particle cannot be negative.
negative. For,
For, according
of least
least action, the
the integral
integral
the principle
principle of
to the
=

H

CO

S

dt.
J%nw2
!mv2dt,
2

1

has a minimum for the actual motion of
of the particle in space
space from point
point 11 to
point
If the
the mass were negative,
negative, the
the action integral
integral would take arbitrarily
arbitrarily
point 2. If
the Particle
particle rapidly
motion in which
large negative values for a motion
which the
rapidly left point
point 1I
and rapi
dly approached
there would
would be no minimums
minimum.t
rapidly
approached point
point 2, and there

Tt \Ve
\Ve shall
shall use the suffixes a, b, c,
c, .•• to
to distinguish
distinguish the
the various
particles, and
and i,i, k,
k, I,l, ..• to
to
various particles,
distin
guish the
distinguish
the co-ordinates.
co-ordinates.

m
M

The argument
argument is
is not
not affected
affected by
by the
the point
point mentioned
mentioned in
in the
the Erst
first footnote
footnote to
to §2;
§2; for
for
It The
0, the
the integral
integral could
could not
not have
have aa minimum
minimwn even
even for
for aa short
short segment
segment ooff the
the path.
path.
< 0,
I

8

The Equations of
of .Motion
.llotion

It is
is useful
useful to
to notice
notice that
that
It

v2 == (dl/dz)2
(dlf dt)2
'u2

( d1)2, (dt)2.
(dt)2.
= (dl)2,

§5

(4.3)

Hence,
Hence, to
to obtain
obtain the
the Lagrangian,
Lagrangian, it
it is
is sufficient
sufficient to
to Find
find the
the square
square of
of the
the element
ment of
of arc
arc dl
dl in
in aa given
given system
system of
of co-ordinates.
co-ordinates. In
In Cartesian
Cartesian co-ordinates,
co-ordinates,

for
d[2 == dx2-l-dy?-l-dz2,
dx2+dy2+dz2, and so
for example, dl

L == !m(x2+j2+z2).
m(i2+j2+22).

co-ordinates dl
d[2 == dr'-3+r2
dr2 + r2 d¢>'-2+
drf>2 + day,
dz2, whence
In cylindrical co-ordinates

L

= »1,m(»=2+»2.,*32
!m(f2+r2~2+z2).
=
+,-=;»2).

spherical co-ordinates
co-ordinates dl
d[2 = dr2-l-12
dr2+ r2 d92+r-'?d02+r2 sin"-9
sin20 d¢>2,
drf>2, and
and
In spherical
L
!m( f2 + #92
y2{j2 -!~
+r2¢;'3r2~2 sin20).
L = %m(i2

(4.4)
(4.5)

4-6)
((4.6)

The Lagrangian
Lagrangian for a system of
of particles
§5. The
Let
Let us now consider
consider a system
system of particles
particles which
which interact
interact with one another
another
other bodies. This is called a closed system. It
It is found that
that the
the
but with
with no other
adding to the Lagraninteraction between
between the
the particles
described by adding
interaction
particles can be described
(4.2) for non-interacting particles a certain function
function of
of the
the co-ordinates,
gian (4.2)
gian
which depends
depends on the
the nature
of the
the interaction.T
interaction. t Denoting
Denoting this
this function
function
which
nature of
by - U, we have
=

II

L
L

re, ...),
U(rt,r2,
... ),
2,LJma~·a2---- U(r1,
%?Na7:a

2

(5.1)

where rraa is the
the radius vector
vector of
of the
the ath
ath particle. This is the
the general
general form
form of
of
where
the Lagrangian for a closed system. The sum
= ~
!mava2 is called the
the
sum T =
Z %ma*va2
of the system. The
The significance
kinetic energy, and U the potential
potential energy, of
of these
these names is eexplained
of
6.
XPfained in §§6.
The fact that the
the potential
potential energy
energy depends only
only on the
the positions
positions of
of the
the
The
particles at a given
given instant
instant shows
shows that
that a change
change in the
position of
of any particle
particles
the position
the other particles. We may
may say
say that
that the
the interinterinstantaneously affects all the
instantaneously
necessity for interactions
interactions in
instantaneously propagated.
propagated. The necessity
actions are instantaneously
this type is closely related to the premises
premises upon
upon
classical mechanics to be of this
the subject is based, namely
namely the
the absolute nature
of time and
and Galileo's
which the
Nature of
If
the
propagation
of
interactions
were
not
instantaneous,
relativity
principle.
relativity principle. If the propagation
interactions
not instantaneous,
place with
with a finite velocity,
velocity, then
then that velocity
velocity would
would be different
different in
but took place
of reference
relative motion,
motion, since
since the
the absoluteness
absoluteness of
of
different frames of
different
reference in relative
of composition
composition of
of velocities is
necessarily implies
implies that
that the
the ordinary
ordinary law of
time necessarily
applicable to all phenomena.
phenomena. The laws of
of mot
motion
interacting bodies
bodies would
applicable
ion for interacting
would
would
contradict
inertial frames, a result
then be different in different inertial
result which
which
the relativity principal
principle.
e.
In
§3
only
the
homogeneity
been spoken
spoken of.
of. The
'!'he form
fo~m of
of the
on ly
homogeneity of time has
h as been
In
iso
tro
d
pic
an
s
,
i.e
ou
ne
that time
time is
is both
both ho
homogeneous
and tsotroptc, i.e.. its
its
Lagrangian (5.1)
(5.1) sshows
moge
hows that
Lagrangian
,

- not
ech Amos IS
•
R la
t"vistic
mechanics
is
not cons·d
d
- nc' m
1- ns
-. . Re
• al mech
anics
C°ns1lder
t Th
This
statement is
is valid
valid on
class1cal
mechamcs.
e 1ahv
ereed
.in classic
'r
_ is statement

in this
this
In

book.

§5

The Lagrangian
Lagrangian for
for aa system
system of
of particles
particles
The

9

properties are the
the same
same in both
directions. For,
For, if tis
replaced by -- I,
t, the-=~
th~ Laproperties
both directions.
r is replaced
grangian is unchanged, and
and therefore
therefore so are the
the equations
equations of motion.
motion. In
In other
grangian
words,
given motion is possible
system, then
then so is the reverse motion
motion
words, if a given
possible in a system,
(thatt is, the
the motion
motion in which the
the system
system passes
passes through the
the same
same states
states in
(tha
hich obey
the reverse order).
order). In
In this sense
sense all motions
motions w
which
obey the
the laws of
of classical
classical
the
mechanics are reversible.
reversible.
mechanics
the Lagrangian,
Lagrangian, we can
can derive
derive (the equations. of
of motion
motion::
Knowing the
Knowing

d 3oL
L

Substitution of (5.1) gives

do Eva
Ma dvd/dt

oL
aL

(5.2)

apa

=' -:

2UI"?ra.

{5.3)

the equations
equations of
of motion
motion are called .Vewton's
_Vewton's equations and form
form
In this form the
the basis
basis of
of the
the mechanics
mechanics of
of a system
system of
of interacting
interacting particles.
particles. The vector
vector
the

F

=

- aura
oUJora
to

(5.4)
(5
.4)

which appears
appears on the
which
force on
the right-hand side of
of equation
equation (5.3) is called
called the
the force
the ath
ath particle. Like
Like U, it depends only
only on
on the
the co-ordinates of
of the
the particles,
particles,
the
and not
not on their
their velocities.
velocities. The equation
equation {5.3)
shows that
that the
the acceleraccelerand
(5_3) therefore shows
ation vectors
vectors of
of the
the particles
particles are likewise functions
functions of
of their
their co-ordinates
co-ordinates only.
ation
The potential
energy is defined
defined only
only to
to within
additive constant,
constant, which
The
potential energy
within an additive
which
on the
the equations of
of motion.
motion. This is a particular
case of
of the
the nonhas no effect on
particular case
uniqueness of
of the
the Lagrangian
Lagrangian discussed
discussed at the
the end
end of
of §2. The most
most natural
uniqueness
natural
and most
most usual way of
of choosing
choosing this constant
constant is such that
that the potential
energy
and
potential energy
the distances between the
the particles tend
tends to zero as the
tend to infinity.
If
describe the
the motion,
motion, arbitrary
arbitrary generalised
generalised co-ordinates
co-ordinateti g;
q,
If we use, to describe
instead
instead of
of Cartesian
Cartesian co-ordinates,
co-or~inates, the following transformation
transformation is needed
needed to
obtain
obtain the
the new Lagrangian
Lagrangian::

Ufa

£ ( qt. q2, ... , qs ) , Xa
•
""' ofa . etc.
= fa(Q1»
Ja
Q23
aka etc.
Qs): in = L -qk,
°1

II

Xa
Xa

k ask
oqk

•

k

Substituting
Substituting these
these expressions
expressions in the function
function L = '}?2"2a(X'2
!~ma(xa2 +y'a2
+ya2 -l+ 202)
za2)- U,
we obtai
n the
obtain
the requ
required
Lagrangian
Lagrangian
in
the
form
ired

L

%;¢W-=(Q)Qi<2f:- U(Q)»

(5.5)

where
the flu:
a~,k are
are functions
of the
the co-ordinates
co-ordinates only.
only. The
The kinetic
kinetic energy
energy in
in
where the
functions of
the velocities,
velocities, but
but itit
generalised co-ordinates
co-ordinates is
is still
still aa quadratic
quadratic function
of the
generalised
function of
may depend
depend on
on the
the co-ordinates
co-ordinates also.
also.
may
Hitherto we
have spoken
spoken only
only of
of closed
closed systems.
systems. Let
Let us
us now
now consider
consider aa
Hitherto
we have
system A
A which
which is
is not
not closed
closed and
and interacts
interacts with
another system
system B
B executing
executing
system
with another
a
In such
such aa case
case we
we say
say that
that the
the system
system A
A moves
moves in
in aa given
given
a given
given motion.
motion. In
external
field
(due
to
the
systef!'l
B).
Since
the
equations
of
motion
are
obtained
external field (due to the system B). Since the equations of motion are obtained

The
The Equations
Equations

10

of
of Motion
Motion

§5

the principle
action by
by independently
independently varying
varying each of the cocofrom the
principle of least action
the remainder
ordinates (i.e. by proceeding
ordinates
proceeding as if the
remainder were given quantities), we
Lagmngian LA of
of the
the system
system A
A by using the
the Lagrangian
Lagrangian L of
can Find
find the Lagrangian
can
the whole system
system A
A+
B and
and replacing
the co-ordinates
co-ordinates QB
qB therein
therein by given
given
the
+B
replacing the
functions
of time.
functions of
Assuming that
+B
+
that the system A
A+
B is closed, we have L -=
= T,4(qA,
T A(qA, QA)
qA) +
.|. TB(qB,
- U(qA,
U(QA3 go),
where the first two terms
terms are the kinetic energies
+
T B(q-B, QB)
tiB)qB), where
energies of
the
potential energy.
the systems
systems A and B and the
the third
third term is their combined
combined potential
qB the
the given
given functions of
of time
time and
and omitting
omitting the
Substituting for QB
the term
T[qB(t), QB(i)]
qB(t)] which
and is therefore
therefore the total time
T[q8(t),
which depends on time only, and
derivative
€?A)~ U[qA,
U[QA1 q8(t)].
derivative of
of a function
function of time, we obtain LA =
= TA(Q,-1,
T A(qA, qA)qB(t)].
the motion
motion of
of a system in an external
external field is described
described by a Lagrangian
Lagrangian
Thus the
of the
the usual type, the
the only difference
difference being that
that the potential
energy may
may
of
potential energy
depend explicitly
explicitly on time.
depend
For example, when
single particle moves in an external
external field, the
general
For
when a single
the general
of
the
Lagrangian
is
form
form of the Lagrangian

L

and the
the equation
equation of
of motion
motion is
and

=

}moz:2mv?- U(r, t),
I)1

m
mvin =:=

-aU
uar.
-oUJor.

(5.6)

(5.7)

such that
that the same force F
Facts
A field such
acts on a particle at any point in the field
said to be uniform.
uniform. The
The potential
energy in such
such a Field
field is evidently
evidently
is said
potential energy
II

F-r.
U
U= --F·r.

(5.8)

To conclude
conclude this section, we may make
make the
the following
following remarks
remarks concerning
To
the
equations to various
various problems. It is often
the application
application of Lagrange's equations
mechanical systems in which
the interaction
interaction between
between
necessary to deal with
necessary
with mechanical
which the
the form of constraints, i.e. restrictions on
on
different bodies
bodies (or
{or particles)
different
particles) takes the
their
means of
their relative
relative position. In
In practice, such constraints are effected
effected by means
of
hinges and
and so on. This introduces a new factor
factor into the problem,
rods, strings, hinges
problem,
that the motion
motion of
of the
the bodies
bodies results
results in friction
friction at their
their points of contact,
in that
general ceases to be one of pure mechanics
mechanics (see §25). In
In
and the problem
problem in general
many cases, however,
however, the
the friction
friction in the
the system
system is so slight
slight that
that its effect
effect on
on
many
the motion
motion is entirely
entirely negligible.
negligible. If
the masses
masses of the
the constraining
constraining elements
elements of
of
the
If the
the system
system are also negligible,
negligible, the
the effect
effect of
of the
the constraints
constraints is simply
simply to reduce
the
reduce
number of
of degrees
degrees of
of freedom
freedom s of
of the system
system to a value less than
than 3N.
3N. To
To
the number
the motion
motion of
of the
the system, the
the Lagrangian
Lagrangian {5.5)
used,
determine the
(5.5) can again be used,
with
set of independent
independent generalised
generalised co-ordinates
co-ordinates equal
equal in
in number
number to
to the
the
with a set
actual degrees
degrees of
freedom.
actual
of freedom.
'3°

PROBLEMS

Lagrangian for each of
of the following
following systems
when placed in
in 8a uniform
uniform gravitagravitaFind the Lagrangian
systems when
tional field (acceleration g).

The Lagrangian
Lagrangian for
for aa system
system of
of particles
particles
The

§5
PROBLEM
PROBLEM

11

coplanar double
double pendulum
pendulum (Fig. 1).
1. A coplanar
i'

»/

-

A-A-

I/72

FIG.
FIG. 1

SoLUTION. We take as co-ordinates the angles 961
t/>1 and 962
t/>2 which the strings 11
II and 12 make
SOLUTION.
2.f,t2, U
t}m1l12£12,
$61. In
with the vertical. Then we have, for the particle mi,
mt, T1
Tt == !mtii
U == --m1gl1
-mtgii cos
cos ¢1.
In
order to
to End
find the
the kinetic
kinetic energy
energy of
of the
the second
particle, we
we express
express its
its Cartesian
Cartesian co-ordinates
co-ordinates
order
second particle,
X2, pa
Y2 (with the origin
origin at
at the point
point of
of support
support and the
they-axis
vertically downwards) in
in terms
terms
xg,
y-axis vertically
of the angles $61
t/>1 and 962°
t/>2: xi
X2 ==
= 11
II sin ¢>1
¢1 +12
+l2 sin $52,
¢2, ye
Y2 = 11
II cos $61
t/>1 -1-12
+12 cos 462.
¢2. Then we find
find
2+.Y22)
T2 = !m2(X:!
tl"¢2(=¢22 'l'y•22)

.

-

2.f,t2-H226£22
= !m2[lt
+l22.f,22+2/1/2
+ 21t12 cos(¢1
cos(¢1- ¢>2)¢£1q52]
¢2).f,t.f,2] ·
=
tlm2[/12££12

Finally
Finally

L == §(m1
J.(nu +m2)l12¢£12+t}m2/22{>22+mz/ilzqh£lz
+m2)II 2.f,t2 +!m21~.f,2 2 +m2ltl2.f,t.f,2 COS(151*¢>2)
cos(,Pt-t/>2) -I-(m1
+(mt -I-m2)gl1
+m2)gii cos Q61
.Pt -I-m2g
+m2g12
¢2.
l2 cos ¢>2.

PRoBLEM 2. A simple pendulum
pendulum of mass
mass mz,
m2, with aa mass
mass ml
mt at
at the point
point of
of support which
PROBLEM
on a horizontal line lying in
in the
the plane in
in which m2
m2 moves (Fig.
(Fig. 2).
2).
can move on
I

I
I
I

I
I

I

x
X

*

I7"l

.H72

FIG.
FIG. 2
SOLU
TION.
SoLUTION.

.p

co-ordinate xx of
of m1
m1 and
and the
the angle
angle ¢6 between
between the
the string
string and
and the
the
co-ordinate

Usin
Using
the
g the

vertic
vertical,
have
al, we have

+

II

t-

m2)x.22+§mg(l22
+ mzgl cos ¢>.
L = §(m1
J.(m1 +m2)x
+lma(l2.f,2-I-2lxl¢
cos ¢)
,P)+m~1
.p.
+21x.f, cos

PRO
BLEM 3. A sim
PROBLEM
simple
pendulum of
of mass
mass m
m whose
point of
of support
support (a)
(a) moves
moves uniformly
uniformly
ple pendulum
whose point
On
al circle
on aa vertic
v~rtical
circle with constant
constant frequency
frequency y (Fig.
(Fig. 3),
3), (b)
(b) oscillates
oscillates horizontally
horizontally in
in the
the plane
plane

?f

of mot
ion of
ulum accor
ding to
motion
of the pend
pendulum
according
to the
the law
law xx = aa cos
cos it,
yt, (c)
(c) oscillates
oscillates vertically
accordvertically accordIng
Ing to
to the law
law yy = aa cos
cos it.
yt.
SOLUTION.
SoLUTION. (a)
(a)

The
The co-ordinates
co-ordinates of
of m
mare
x == aa cos
cos it-I-I
yt+l sin
sin¢,
y =
= --a
-a sin
sin ityt+l
cos¢.
are x
¢, Y
+I COS
16-

The
The Lagrangian
Lagrangian is
is

iml".P"+mla-y2
sin(,P-yt)+mgl cos
cos¢;
L = a
m / 2 2 -I-mZay2 sin(<;6--yt)+mg!
45;

here term
ding only on
termss depen
depending
on time
time have
have bee
been
omitted,
together with
the total
total time
time derivative
derivative
n omi
tted, together
with the
of mla-y
mlay cos(<;t>
cos(¢-yt).
Of
-70-

The
ations of
The Equ
Equations
of Mo
Motion
tion

12

§5

are x
. (b)
(b) The
The co-ordinates
co~ord_inates of
of m
mare
x =a
cos ye'
yt+l
sin</>,
y =
= Il cos
cos</>.
The Lagrangian
Lagrangian is
is (omit(omit= a cos
+1 sin
¢, :v
¢>. The

denvattves)
ting total derivatives)
(c)) Similarly
(c

2 22+nday2
L
t}ml2£
L == !ml
tf, +mlar cos ityt sin go-I-m
t/>+mgl
cos</>.
g! cos
go.

L = §ml2.l£2+mlay2 cos it us g6~l~m.gl cos go.

x

,m

FIG.
FIG.

3

PROBLEM 4. The system
in Fig.
Fig. 4. The particle
particle m2
n12 moves on a vertieal
and the
the
PROBLEM
system shown in
vertical axis and
whole
system rotates
rotates about
about this
this axis
axis with
a constant
constant angular
angular velocity
velocity Q,
0.
whole system
with a

.A
A

'o

0'

m4

mI

I

U

I m2
FIG.
FIG.

4

SoLUTION. Let
Let 6(J be
be the
the angle
angle between
between one
one of
of the
the segments
a and
and the
the vertical,
vertical, and
and ¢,
r/> the
SOLUTION.
segments a
the
•ngle of
of rotation
rotation of
of the
the system
system about
about the
the axis;
axis; ¢
t/> = Q.
0. For
For each
each particle
particle mi,
m1, the
the infinitesimal
infinitesrmal
angle
by c1/12
dlt2 =
= az
a 2 d92-1-a2
d82+a2 sing
sin2 6(J d¢2.
drf> 2• The distance of
of m2
m2 from the point
point
displacement is given by
of support
suppart A
A is
is 2a
2a cos
cos 6,
8, and
and so
dl2 so -2a
-2a sin 08 d6.
dB. The Lagrangian is
is
of
so dl

=

=

28)+2n12a21)
L == m1a2(62-|m1a2(1l2+ Q2
sin29)
sin2 8+2(ml +m2)ga cos 0.
8.
0 2 sin
-I-2m2a2622 sin20-I-2(m1-I-m2)ga

C
CHAPTER
H A P T E R IIII

CO:-.JSERVATION
C
O N S E R V A T I O N LAWS

§6.
§6. Energy
DURING
DliRING the
the motion of
of a mechanical system, the
the 2s
2s quantities Qi
qi and
and Qi
q,
(i =
= 1, 2, ...,
... , s) which specify
specify the state of the system
system vary with time. There
exist, however,
however, functions of
of these quantities
quantities whose
whose values remain
remain constant
during
during the
the motion, and
and depend only on the
the initial
initial conditions.
conditions. Such
Such functions
of the motion.
are called integrals of
number of
of independent integrals
integrals of
of the
the motion for a closed
closed mechanical
mechanical
The number
degrees of
of freedom
freedom is 252s-1.
evident from the
the following
following
system with
with s degrees
system
1. This is evident
simple
simple arguments. The general solution
solution of
of the
the equations
equations of
of motion
motion contains
arbitrary constants
constants (see the discussion
discussion following
following equation
equation (2.6)).
(2.6}). Since
Since the
2s arbitrary
equations
equations of
of motion for a closed
closed system do not
not involve
involve the
the time
time explicitly,
of the
the origin
origin of
of time
time is entirely
entirely arbitrary,
arbitrary, and
and one
one of
of the
the arbitrary
choice of
the choice
the solution
solution of
of the equations
equations can
can always
always be taken as an additive
additive
constants in the
to,
constant to in the time. Eliminating
Eliminating t +
+to
the 2s functions Q-i
qi =
= Q1:(£
qi(t +
+to,
constant
to from the
C1,
£0, C1, Co,
€XPrCSS the 2$-Ct, C2, ...,
... , C28-_1),
C2s-I}, Qi
qi =
= Qi(t+
qi(t +to,
C2, ...,
... , C23_1),
Czs-1}, we can e:ll..-press
2s-11
constants C1,
Ct, Co,
c2, ...,
... , C2s_1
C2s-I as functions
functions of q and Q,
q, and these functions
arbitrary constants
of the
the motion.
will be integrals of
Not all integrals
integrals of
of the motion,
motion, however,
however, are of
of equal importance
importance in mechmechNot
some whose constancy
constancy is of
of profound
significance, deriving
deriving
anics. There are some
profound significance,
from
and isotropy
isotropy of
of space
space and
and time.
time. The
The
from the
the fundamental
fundamental homogeneity
homogeneity and
quantities represented
represented by
by such
such integrals
integrals of the motion
motion are said
said to be eonseroed,
conserved,
quantities
and have
have an
an important
important common
common property
property of
of being
additive: their
their values for a
and
being additive:
of several
several parts whose
whose interaction
interaction is negligible
negligi,ble are equal
system composed
composed of
system
sums of
of their values
values for the
the individual
individual parts.
to the sums
this additivity
additivity that
that the quantities
quantities concerned
concerned owe their
their especial
especial
It is to this
mechanics. Let
Let us suppose, for example,
example, that two bodies
importance in mechanics.
Importance
during a certain interval
interval of
of time. Since each
each of
of the additive
additive integrals
interact during
both before
and after the interaction,
interaction, equal
equal to the
the
of the whole
whole system is, both
of
before and
sum of
bodies separately, the
of its values
values for the
the two bodies
conservation laws for these
the conservation
the state
quantities immediately
immediately make
make possible
possible various
various conclusions
conclusions regarding
regarding the
quantities
of
bodies after
of the
the bodies
after the
the interaction,
interaction, if their
their states before
before the interaction are
know
known.
n.
Let
Let us consider first
first the conservation law resulting
resulting from the
the homogeneity
of
of time. By virtue of
of this homogeneity,
homogeneity, the
the Lagrangian of
of a -''used
"JSed system
does not depend
depend explicitly on time. The total
total time derivative
derivative of
of the
the Lagrandoes
gian can therefore
therefore be written
gian
oL
cL
dL
5'L

_

-,

I'

dt =
dt

+ ~ cqi qj.
~
oqi l'Ji+
QQ 9"
i
'i
. C

l

13

•

14

§6

Co
nservation La
Conservation
Laws
ws

If L
If
L depended explicitly on time, a term 3Ll3t
oLJot would have to be adde
added
on
d 011
the right-hand
right-hand side. Replacing
Replacing oLfoq~,,
accordance with
with I_,agrange's
Lagrange's equathe
HLlZig1, in accordance
(dfdt) oLfoq~,,
tions, by (d/dt)
BLlEqi, we obtain

2%-; (3521)

Zd

11

dL
dt

i

11

i

or

Q

d

Hence we see that the
the quantity

.

dt(q

(Ii

aL

355;

avi)
AL

+

-L

III
F11

Z g aL
.

Ag'

2

AL

" a l
391

i

0.

L

(6.1)

I

i

_

remains constant
constant during
during the
the motion
motion of a closed
closed system,
system, i.e. it is an integral
integral
remains
of the
the motion;
motion; it is called
called the
the energy of the
the system.
system. The
The additivity
additivity of the
the
of
energy follows immediately
immediately from that
that of
of the
the Lagrangian,
Lagrangian, since
since (6.1) shows
shows
energy
of the
the latter.
that it is a linear function of
of conservation
conservation of
of energy
energy is valid not only for closed systems,
systems, but
The. law of
The
those in a constant
constant external
external field (i.e. one independent
independent of time):
time): the
the
also for those
only property of
of the
the Lagrangian used
used in the
the above
above derivation,
derivation, namely
namely that
only
does not
not involve
involve the
the time
time explicitly,
explicitly, is still valid,
valid. Mechanical
Mechanical systems
systems whose
whose
it does
conserved are sometimes
sometimes called
called conservative
conservative systems.
systems.
energy is conserved
system (or
(or one in a
seen in §5, the Lagrangian of a closed system
As we have seen
of the form L =
= T(g,
T(q, q•)tj)- U(q), where T is a quadratic
quadratic
constant field) is of
constant
of the
the velocities.
velocities. Using Euler's theorem on homogeneous
homogeneous functions,
function of
we have

.

aL

in (6.1) gives
Substituting this in
Cartesian co-ordinates,
co-ordinates,
in Cartesian
E

Z . IT
¢

ia

q-T`=

E =
= T025
T(q;q)+
U(q);;
E
<2)+ U(Q)

=

QT

•

L)mava
U(r1, 12,
r2, ...).
... ).
2
mama?2 + U(I'1,
a
a

(6.2))
(6-2
(6.
(6.3)
3)

§'7
§7

15

Momentum
Momentum

Thus the
the energy of
of the
the system
system can
can be written as the sum
sum of
of two quite
quite different
different
the kinetic
kinetic energy,
energy, which
which depends
depends on
on the
the velocities,
velocities, and
and the
the potential
terms: the
potential
energy, which
which depends
depends only on the co-ordinates
co-ordinates of
of the particles.
energy,
particles.

§7. Momentum
second conservation
conservation law follows from the homogeneity of
of space.
A second
space. By virtue
of this
homogeneity, the
the mechanical
mechanical properties of
of a closed
closed system
system are unthis homogeneity,
changed by any
any parallel
displacement of
of the entire
entire system
system in space.
space. Let
Let us
changed
parallel displacement
consider an infinitesimal
therefore consider
infinitesimal displacement
displacement e,
E, and
and obtain
obtain the condition
condition
for the
unchanged.
the Lagrangian
Lagrangian to remain
remain unchanged.
A parallel
which every
particle in the
parallel displacement
displacement is a transformation
transformation in which
every particle
the
system
r + e.
system is moved
moved by the same amount,
amount, the radius
radius vector
vector r becoming
becoming r+
E.
change in L resulting
resulting from an infinitesimal
infinitesimal change
change in the
the co-ordinates,
co-ordinates,
The change
the
particles remaining
the velocities
velocities of the
the particles
remaining fixed, is
=

oL
CL
AL
EL
- or
L:-·
Sru = E·L:-,.,
Era
Ora
ora
ilrl
6.

u

11

8L
SL

a

a
a

1

where the
the summation
summation is over
over the
the particles in the system. Since 6E is arbitrary,
arbitrary,
where
condition 8L
SL =
= 0 is equivalent
equivalent to
the condition

2
aL bara =
'))Lfcra
a
a

0.

equations (5.2) we therefore
therefore have
From Lagrange's equations
a
a

oL
aL

d
Ova = dt
dt
8'va

0.
L 3%
Ova = O.

8oL
L

a

II

L

d
dt
do

Thus we
we conclude
conclude that,
that, in aa closed
closed mechanical
mechanical system,
system, the
the vector
vector
Thus

P

E

remains
remains

(7.1)
(7.
1)

Z 2LlEva

(7.2)

mava.

(7.3)

a
a

during the
the motion
motion;, it is called
called the momentum
momentum of
of the
the system.
constant during
( 5.1 ), we find that
that the
momentum is given
given in
Differentiating the Lagrangian (5.1),
Differentiating
the momentum
of the
the velocities
velocities of
of the particles by
terms of

P=

a

The
vity of
The additi
additivity
of the momentum
momentum is evident.
evident. Moreover,
Moreover, unlike the
the energy,
energy,
the momentum
momentum of
of the
the system
system is equal
equal to the
the sum
sum of
of its
its values
= mava
mav a for
the
values Pa =
the individual
individual particles, whether
whether or not
not the interaction between
them can
can be
the
between them
neglected.
The three
three components
components of
of the
the momentum
momentum vector
vector are
are all conserved
conserved only
only in
the absence
absence of
of an external
external field. The individual
individual components may
may be conserved
conserved
the
the presence of
of a field, however,
the potential energy in the
field does
does
even in the
however, if the
the field
not
on all the
the Cartesian co-ordinates. The mechanical properties of
of
not depend on

16

§8

Co
nserrxiztion Law
Conserv.:ation
Laws
s

the
the system
sy~tem are
are_ evidently uncha
unchanged
displacement along
along the
the axis of
of a
nged by a displacement
co-ordinate
which
does
not
appear
in
co-ordmate whtch does not
the potential
energy, and
and so the
the correcorrethe
potential energy,
sponding
component
of
the
mo
a unimen
in
sponding component of
momentum
conserved. For
For exam
example,
uniple,
tum is conserved.
form field in the z-direction,
z-direction, the x
x and 3°
y components
components of momentum
momentum are
conserved.
conserved.
The
The equation (7.1) has a simple physical
physical meaning.
meaning. The derivative
derivative
3LlZlra
U/31-a is the force Fa
oLfora == -- 3oUJora
Fa acting on the
the ath
ath particle.
equation
particle. Thus equation
(7.1
(7.1)) signifies
signifies that the
the sum
sum of
of the forces on all the particles
particles in a closed
closed system
is
ts zero
zero::

:LFa
Fa

0.

(7_4)
(7.4)

Pt = 3 L,»'aQs

(7.5)
(7-5)

aL,/go

(7.6)

a

=

In
+F
F22 == 0: the
In particular, for
f9r a system
system of
of only
only two particles,
particles, F1
F1 +
the force exerted
exerted
by
particle on the
by the
the first particle
the second
second is equal
equal in magnitude,
magnitude, and
and opposite
opposite in direction, to that
the second
second particle
particle on the
the first. This is the
the equality
equality
that exerted
exerted by the
of action
action and reaction
reaction (Newton's
{1\rewton' s third
third law).
of
If
If the motion
motion is described
described by generalised
generalised co-ordinates
co-ordinates ii,
qi, the
the derivatives
derivatives
of the
the Lagrangian
Lagrangian with
with respect
respect to the generalised
generalised velocities
velocities
of
are
are called
called generalised
generalised momenta,
momenta, and
and its
its derivatives
derivatives with
with respect
respect to
to the
the generalgeneralised co-ordinates
co-ordinates
ised

11

are called
called gen
generalised
forces.
are
eralised forces.

F

In this
this notation,
notation, Lagrange's
Lagrange's equations
equations are
are
In

Pi

(7-7)
(7.7)
In Cartesian
Cartesian co-ordinates
co-ordinates the
the generalised
generalised momenta
momenta are
are th
thee components
components of the
In
vectors
Pa· In
In general,
general, however;
however,- the
the Pi are linear
linear homogeneous
homogeneous functions
functions of
of
vectors Pathe
the generalised
generalised velocities
velocities Qu
qi, and
and do
do not
not reduce
reduce to
to products
products of
of mass
mass and
and velocity.
-.

=

i.
Fi.
'.'*|

r

P
ROBLEM
PROBLEM

A particle of
of mass
m moving with velocity
v1leaves
half-space in
in which its
its potential energy
energy
mass m
velocityvl
leaves a half-space
constant U1
U1 and
and enters
enters another
another in
in which its
its potential energy is a different constant
constant U2.
is a constant
Determine the
the change
change in the
the direction
direction of
of motion
motion of
of the
the particle.
particle.
Determine

SoLUTION. The potential energy
energy is independent of
of the
the co-ordinates
co-ordinates whose axes are parallel
SOLUTION.
to the
the plane
plane separating
separating the
the half-spaces.
half-spaces. The
The component
component of
of momentum
momentum in
in that
that plane
plane is
is
to
therefore conserved.
conserved. Denoting
Denoting by
by 91
01 and
and H2
02 the
the angles
angles between
between the
the normal
normal to
to the
the plane
plane and
and
therefore
the ,.·elocities
VI and
and vz
V2 of
of the
the particle
particle before
before and
and after
after passing
passing the
the plane,
plane, we
we have
have u1
V1 sin
sin 61
01
the
velocities V1
= U2
V2 sin
sin 02.
02. The
The relation
relation between
between 1:1
"VI and
and kg
t'2 is
is given
given by
by the
the law
law of
of conservation
conservation of
of energy,
energy,
=

and the
the result
result is
is
and

§8. Centre
Centre of
of mass
mass
§8.

sin 61
sin 62

J

2

1.

tnvlz (U1 - U2)

-

h

d"fferent
values in
1 ferent values
if
d

.h
momentum of
of a closed mechanical
mechamcal system has
as
.
The momentum
.
.
.
'th ve
1o
C1
.
.
.
f
f
f
f
K'
ves
w1t
ve
OCtty
reference.
If
a
frame
K
n*1OVCS
w1
TY V
V
dtfferent (merttal) frames
o re erence. I
rame
rno
es of
deferent
(mertnal) fram

I

Centre of
of mass
mass
Centre

§8

17

relative to another
another frame K, then the
the velocities
velocities Va'
va' and Va
Va of
of the
the particles
relative
relative
P and
relative to the
the two
two frames are
are such that Va
Va =
= Va'
va' +
+ V. The momenta P
and P'
P'
in the
the two frames
frames are therefore related
related by
by

or

P
P

=

21
a
a

7?Iava
'2.,rnaVa
=

L11laVa'
+VV '2.,ma,
Maya '+
My:
a

(L

Emu.

p' + v '2.,ma.
P'+V

11

P

=

a
a

a
a

(8.1)

particular, there is always a frame of reference
reference K'
which the total
total
In particular,
K' in which
P'
momentum is zero. Putting
Putting P' =
= 0 in (8.1), we Find
find the velocity of this frame
frame::
momentum

V = P /2%

2m¢7,va/ imp.

(82)
(8.2)

If
total momentum
momentum of a mechanical
mechanical system
system in a given
given frame
frame of
of reference
If the total
reference
is zero, it is said
natural generalisaid to be at rest relative
relative to that
that frame. This is a natural
generalisation
sation of
of the term
term as applied
applied to a particle. Similarly,
Similarly, the
the velocity
velocity V given
given by
the velocity
velocity of
of the "motion as a whole" of
of a mechanical
mechanical system
system whose
whose
(8.2) is the
momentum is not
the law of
of conservation
conservation of
of momenmomenmomentum
not zero. Thus we see that the
tum makes
makes possible a natural definition
definition of
of rest and
and velocity,
velocity, as applied to a
tum
mechanical system as a whole.
mechanical
Formula (8.2)
(8.2) shows that
that the
the relation
relation between
the momentum P and
and the
the
Formula
between the
velocity V of
of the
the system
system is the
the same as that between
the momentum and
and velovelocity
between the
city of
of a single
single particle
of mass
mass fL
= Ema,
~ma, the
the sum of
of the
the masses
masses of the
the particles
citv
particle of
p. =
the system. This result
result can
can be regarded
regarded as expressing
expressing the additivity
additivity of
of mass.
in the
side of
of formula
formula (8.2) can
can be written
the total
total time
time derivaderivaThe right-hand side
written as the
of the
the expression
expression
tive of

R

E

EmarG/ Et.

(8.3)

\Ve can
can say
say that
that the velocity
velocity of the
the system
system as a whole
the rate
rate of
of motion in
whole is the
of the
the point
radius vector
vector is (8.3). This point
point is called
called the centre
space of
point whose radius
of mass of
of the system.
system.
of
of conservation
conservation of
of momentum
momentum for a closed
closed system
system can
can be formuformuThe law of
lated as stating that
that the
the centre of
of mass
mass of the system
system moves
moves uniformly
uniformly in a
straight line. In this
this form it generalities
generalises the law of
of inertia
inertia derived
derived in §3 for a
straight
single free particle,
whose "centre of
of mass" coincides
coincides with
with the
the particle itself.
single
particle, whose
In considering the
the mechanical
mechanical properties of a closed
closed system
system it
it is natural
natural
of reference
reference in which
which the centre of
of mass
mass is at rest. This elimielimito use a frame of
nates a uniform
of the
the system as a whole,
whole, but
uniform rectilinear motion of
but such motion
is of no interest.
interest.
energy of
of a mechanical
mechanical system
system which
which is at rest
rest as a whole
whole is usually
The energy
usually
called
internal energy Et. This includes
includes the kinetic
kinetic energy
energy of
of the relative
relative
called its
its internal
motion of
motion
of the
the particles in the
the system
system and
and the potential
potential energy
energy of
of their
their interaction. The total energy of
of a system moving as a whole
with
velocity
V can
whole
written
be written
E =

pl/2-I-Ei.

(8.4)
(8-4)

18

Although this
~his formula
The
The energies
energtes E
E_ and E'
E'
K' are related
related by
and K'

§9

Con
servation Law
Conservation
Laws
s

is fairly obv
obvious,
may give
give a direct proof
proof of
of it.
ious, we may
of a mec
hanical syst
reference K
mechanical
system
frames
K
es of reference
em in two fram

Za

2 U
E=
E
=!12 '.Lmava
+U
Ma'7102 -I-

=!12
=

'.Lma(va' + v)2
V) 2 + U
Emma'
a
a
a

ifLV2+
V · '.Lmava'
%UV2
+ VZ Maya' ++!é
a
a

E' ~l- V. p' -1-'pV2.

Zaa

2
'.Lmava'
U
Ma'?~'a + U

(8.5)

This formula gives the
the law of
of transformation
transformation of
of energy
energy from one frame to
another,
If the centre
centre of mass
another, corresponding
corresponding to formula (8.1) for momentum.
momentum. If
p'
at rest
rest iN
inK',
then P' = 0, E'
E' = E
E,, and we have (8.4).
is at
K', then
PROBLEM
PROBLEM

Find the
the law
law of
of transformation
transformation of
of the
the action
actionS
one inertial
inertial frame
frame to
to another.
another.
Find
S from
from one

SoLUTION. The
The Lagrangian
Lagrangian is equal
equal to
to the
the difference
difference of
of the
the kinetic
kinetic and
an'd potential
potential energies,
energies,
SOLUTION.

and is evidently transformed
transformed in
in accordance
accordance with a formula analogous to (8.5)
(8.5)::
and

= L'
L' +V •• P'+%pV2.
P' +!tLV2•

L

=---

Integrating this
this with
with respect
respect to
to time,
time, we
we obtain
obtain the
the required
required law
law of
of transformation
transformation
Integrating
action::
action
where
where

S

=

S"+#V • R'+%»»v2¢,

the
of the

R'
R' is
is the
the radius
radius vector
vector of
of the
the centre
centre of
of mass
mass in
in the
the frame
K'.
frame K'.

§9. Angular
§9.
Angular momentum

Let us now
now derive
derive the
the conservation
conservation law which
which follows
follows from
from the
the isotropy of
of
Let
space.
isotropy means
means that
that the
the mechanical
properties of a closed
closed system
space. This isotropy
mechanical properties
it is rotated
rotated as a whole in any manner
manner in space. Let
Let us theretheredo not vary when it
an infinitesimal rotation of
of the
the system, and
and obtain the
the condition
fore consider an
for the
the Lagrangian to remain
remain unchanged.
shall use the
the vector
vector 84>
Sc~> of
of the infinitesimal
infinitesimal rotation, w
whose
magnitude
\Ve shall
hose magnitude
the angle of
of rotation
rotation 8¢>,
Scf>, and whose direction
direction is that
that of the
the axis of
of rotation
rotation
is the
(the direction
direction of
of rotation
rotation being
that of
of a right-handed
right-handed screw
screw driven
driven along
along 5<l>).
Sc~> ).
(the
being that
Let us Find,
find, first of
of all, the
the resulting
resulting increment
increment in the radius vector from
Let
an origin on
on the
the axis
axis to any
any particle in the
the system undergoing
rotation. The
undergoing rotation.
of the
the end
end of
of the
the radius
radius vector is related
to the
the angle
angle by
linear displacement of
related to
bY
plane
the
to
r
!Sri =
= r sin
sine6* 5.Scf>
The direction of 8r
Sr is perpendicular
l8rl
1> (Fig. 5). The
perpendicula to the plane
of 1°r and 84>.
Sc~>. Hence
Hence it is clear that
of
(9
.1)
(9.1)
81'
r.
Sr = 8¢
Set> x
xr.

§9

19

Angular momentum
Angular
momentum

When the
\Vhen
the system
system is
is rotated,
rotated, not
not only
only the
the radius
radius vectors
vectors but
but also
also the
the velocities
velocities
of the
the particles
particles change
change direction,
direction, and
and all
all vectors
vectors are
are transformed
transformed in
in the
the same
same
of
The velocity
velocity increment
increment relative
relative to
to aa 'fixed
fixed system
system of
of co-ordinates
co-ordinates is
is
manner. The
manner.

iv
8v = 6q>
84> xv.
xv.
"8

¢

(9.2)

805

Of
0

FIG.
FIG.

5

If
expressions are substituted
substituted in the condition that the Lagrangian is
If these expressions
unchanged by the
the rotation
rotation::
unchanged
3oL
3oL
L 8
L 8 )
V
.
t a .|.
8L =
=
8ra+-·
i
a
v
s
( Oraa
a 8vaa l = 0
Ova
a
a

/

Z
L (-·

.»

and
L/ 3 r a by
ative 3
and the deriv
derivative
oLf8va
replaced by
by pa,
Pa• and
and 3oLfora
by 15
Pa•
the result
result is
is
L 8% replaced
(19 the

L(Pa•84>Xra+Pa•84>Xva)
§(15a~
84> X1.a'*'Pa• 8¢» x v )
a
a

;

=

=:.

0

or, permuting
permuting the
the factors
and taking
taking 8q>
84> outside
outside the
the sum,
sum,
or,
factors and

2(ra

884>
X 15a -1- Va Xpa)
4> L(raXPa+vaXPa)
a

=

d
8¢xpa = 0.
84>· dt L:raXPa
a

to

Since
Since 5<l>
84> is arbitrary,
arbitrary, it follows that (djdt)2111
(dfdt) ~ra x
Xpa
0, and
and we
conclude
we conclude
Pa = 0,
that the
the vect
vector
or
Zta Xpar
a

III

M

(9.3)

called the angular momentum or moment of
of momentum of the
the system,
system, is conCalled
the motion
motion of
of a closed
closed system. Like the
the linear
momentum, it is
served in the
served
linear momentum,
additive, whether or
or not
not the
the particles in the
the system interact.
There are no other add
itive integrals
additive
integrals of
of the motion.
motion. Thus every
every closed
system
energy, three components of
of momentum,
momentum,
system has seven
seven such integrals: energy,
and three components
components of angular [noI'I'1CI1tU1'I'1•
momentum.
Since the
the defllrliti0r1
definition of angular momentum
momentum involves
involves the
the radius
of
radius vectors
vectors of
the particles, its vvalue
the choice of
of origin.
origin. The radius
the
a l e depends
depends in general on the

20

§9

Co
nservation Law
Consen:ation
Laws
s

vectors ra
r a and
and ra'
r a' of
of aa given
given point
point relative
rdative to
to origins
origins at
at aa dista
distance
a apar
apartt are
nce a
related
by to
ra = ra'
ra' +
+a.
Hence
related by
a. Hence
=

Z

'.LraXPa
to Xpa

ra' xpa
+ a xX LPa
Pa
'.Lra'
Xpa+a
2
a
a

11

M
M

11

-

a

=
= M
M'' +
+axP.
a X P.

a

(9.4)

It
seen from this formula that the angular momentum
momentum depends
depends on the
It is seen
choice
choice of
of origin except when the system
system is at rest as a whole (i.e. P =
= 0).
indeterminacy, of course,
course, does not
the law of conservation
conservation of
of
This indeterminacy,
not affect the
angular momentum,
momentum, since momentum is also
also conserved
conserved in a closed
closed system,
system.
angular
\Ve may
may also derive
derive a relation
relation between
angular momenta in two inertial
We
between the angular
frames of
of reference
reference K and K',
of which
latter moves with
with velocity
vdocity V
frames
K', of
which the latter
rdative
former. We shall
shall suppose that the
the origins
origins in the
the frames K and
and
relative to the former.
K'
coincide at
at a given
given instant.
instant. Then the radius
radius vectors of
of the particles
the
K' coincide
particles are the
the. two frames,
frames, while their velocities
velocities are related
related by
by Va
va = va'
+ V.
same in the
va'+V.
Hence we have

Zi

M = '.LmaraXVa
Efnara Xva = '.LmaraXva'+
Z7n@;-G xva' + '.LmaraxV.
#Zara X V.
M
aa
aa
aa
II

i'

The 1r'first
sum on the
the right-hand
right-hand side
side is the
the angular momentum
momentum M'
M' in the
The
*l-1rst sum
using in the second
second sum
sum the radius
radius vector
the centre of
of mass
frame K';
K', using
vector of the
(8.3),
obtain
(8.3), we obtain

M =
M ' ++fLR
p R xxV.
V.
= M'

(9.5)

M

(9.6)
(9.6)

formula gives the law of
of transformation
transformation of angular
angular momentum
momentum from one
This formula
corresponding to formula
formula (8.1) for momentum
momentum and
and (8.5)
frame to another, corresponding
frame
energy.
for energy,
If the
the frame K'
that in which
system considered
considered is at rest as a whole,
If
K' is that
which the system
the velocity
velocity of
of its centre of
of mass, PV
fL V its total
total momentum
momentum P relative
relative
then V is the
K, and
and
to K,

M'
+ RxP.
M'+RxP.

II

=

In other
other words, the
the angular
angular momentum
momentum M of a mechanical
mechanical system
system consists
consists
In
''intrinsic angular
angular momentum" in a frame
frame in which
which it is at rest, and
and the
of its "intrinsic
angular momentum
momentum R X
x P due
due to its motion
motion as a whole.
whole.
Although the
the law of conservation
conservation of
of all three
three components
components of
of angular
angular
Although
momentum (relative
(rdative to an
an arbitrary origin) is valid
valid only
only for a closed
closed system,
momentum
the law of
of conservation
conservation may hold
hold in a more
more restricted
restricted form even
even for a system
the
fidd. It
from the
the above derivation
derivation that the
the component
component
in an external Field.
It is evident from
is
sYmm€t"ca1
of angular
angular momentum along an
an axis about which the
the field is symmetrical IS
is
of
are
unaltered
the
mechanical
of
the
system
are
unaltered
always
conserved,
ways
conserved,
for
the
mechanical
properties
of
the
system
al
·
· Here
of
must,• of
by any
any rotation
rotatton
about that axis.
runs.
Here the
th e angular
angu 1ar •M
.momentum
by
about
omentum must
course, be defined
defined relative to
to an origin lying
lying on
on the
the ax1saxts.
course,

•*

§9

Angular
Angular momentum

21
21

The most
most important
important such
such case
case is
is that
that of
of aa centrally
centrally s*ymmetric]'ield
symmetric
field or
or centre
centralI
The
._

field, i.e. one
potential energy depends only
field,
one in which
which the
the potential
only on the
the distance
distance from
some
point (the
It is evident
(the centre).
centre). It
evident that
that the
the component
component of
of angular
angular
some particular point
momentum
along
any
axis
passing
through
the
centre
is
conserved
in
momentum
any axis
through the
conserved motion
motion
in such
NI is conserved
such a field. In other
other words,
words, the
the angular
angular momentum
momentum l\11
conserved provided that
that it is defined with
with respect
respect to the centre
centre of
of the Held.
field.
vided
Another
homogeneous Field
that of
of a homogeneous
field in the of-direction
z-direction;, in such
such
Another example
example is that
a field, the component
component BIz
Nlz of
of theangular
tbe. angUlar momentum
momentum is conserved,
conserved, whichever
whichever
point is taken as the origin.
point
The
The Component
component of
of angular
angular momentum
momentum along
along any axis (say the
the z-mis)
z-a.•ds) can
can
be found by differentiation
differentiation of
of the
the Lagrangian
Lagrangian::
My
Mz=

2L:-·,
2cL
L
N>a
¢¢>a
a

q • 1

(9.7)

where
the co-»ordinate
co-ordinate ¢,
cf> is
is the
the angle
angle of
of rotation
rotation about
about the
the :as-axis.
z-axis. This
This is
is
where the
e\·ident from
from the
the above
above proof
proof of
of the
the law
law of
of conservation
conservation of
of angular
angular momentum
momentum,,
evident
but can
can also be proved directly.
directly. In
In cylindrical co-ordinates
co-ordinatt:s r,r, ¢,,
cf>, 2z we have
but
(substituting xa
pa == rpa
ba)
Xa =
= pa
ra cos ¢,a,
cf>a, Ya
a sin
sin c/>a)
.TVlz ""'

1 2

2
a
EMa7a2 a-

ma(XaYa)'a.Xa)
2111a(xayla
- Jwéa)
[L

[L

The Lagrangian
Lagrangian is,
is, in
in terms
terms of
of these
these co-ordinates,
co-ordinates,
L =

!_1_2

(9.8)

Za

2 +ra2<}a 2 +za2 }- U,
2ma(ia
I'a2*?a2 -1- ;:§'a2)7"a(?a2 +

substitution of
of this
this in
in (9.7)
(9. 7) gives (9.8).
and substitution

PROBLEMS
P
ROBLEMS

PRO
BLEM 1,
PROBLEM
1. Obtain
Obtain expressions
expressions for the
the Cartesian
Cartesian components
components and
and the
the magnitude
magnitude of
of the
the
angular momentum of
of a particle in
in cylindrical co-ordinates r,
r, gb,
r/>, z.
z.
SOLU'rION.
SOLUTION.

m,
A1x =
= m(r2
m(r:i:-zr)
sin 1,6-mrzé
r/>-mrzJ,. cos ¢,
r/>,
-2:r=.) sin
My=
-m(rz-zr)
q,-,;rzJ, sin r/>,
My
= -m(rai
-or') cos 96-mrz"l
glow,

=

!VIz =
JVIz
2
1\1
£
//2

f,

'

mr2
mr2¢',

2(rz-zf)2.
m 2 r 2 ,62(r2 -l-z8)
+z2)+m
= m2r2¢>2(r2
-I-m2(ra'
-zi)3.

PRODLE:.\1 2.
2. The
The same
same as
as Problem
Problem 1,
1, but
but in
in spherical
spherical co-ordinates
co-ordinates r,r, 19,
0, ¢.
rf>.
PROBLEM
SOLUTION.

.Mr
lt4x ==
/VI!, =
!Vly
=
11/2
i'\.Iz =
M
.'~1
Q-2 ==

-mr2(9
+ ¢ sin
-mr2((j sin ¢
rf>+~
sin 90 cos B
0 cos 56),
r/>),
mr2(€l
mr2(1i cos ¢,-9I
.P-~ sin
sin 6*
0 cos 0 sin
sin go),
r/>),
mr 2 ~ sin29,
sin20,
znrgql
m 2 r 4 (fP +¢2 sin29)
sin20).
m'3r'*(@L2

+9<2

.

V.Thich components
components of
of momentum P and
and angular momentum
momentum M
Mare
PROBLEM 3. Which
are conserved
in motion
motion in
in the
the following
fields??
in
following fields
(a) the
of an
the Held
field_of
a'! iniinlte
infinite homogeneous
homogene~ms plane,
plane, (b)
(b) that
that of
of an
an infinite
infinite homogeneous
homogeneous cylinder,
cylinder,
(c)
(c) that of
of an
an inFlnlte
mfimte homogeneous prism,
pnsm, (d)
(d) that
that of
of two
two points,
ppints, (e)
(e) that
that of
of an
an infinite
infinite homohomogeneo
us half-plane,
geneous
half-plane,, (f)
(f) that
that of
of aa hoITlol;eneous
homogeneous cone,
cone, (g)
(g) that
that of
of aa homogeneous
homogeneous circular
circular torus,
torus,
ite homogeneous
(h) that
that of
of an
an infin
infimte
homogeneous cvlln
cvlindrical
helix..
(h)
drical helix

22

1

§10
§10

Co
nsetwatioiz Law
Conservation
Laws
s

SOLUTION.
i f _the
the plane
i f the
~OLUTI<;>N. (a) Pa,
Px,_ PIID
Pu, IL/,
l'.fz ((if
plane is the xv-plane),
xy-plane), (b)
(b) Ma,
Mz, PZ
Pz ((if
the axis
axis of
of the
~he
cylinder
cylmder is
Is the
the 2-axis),
z-ax1s), (c)
(c) P;
P. (if
(1f the edges of
of the
the prism
prism are
are parallel
parallel to
to the
the z-axis),
z-ax1s),
If(d) M;
Mz (if
(if the
the line joining the
the points is the
the z-axis),
z-axis), (e)
(e) P,
p 11 (if
(if the
the edge of
of the
the ha
half(d)
plane is the
the y-axis),
y-axis), (f)
(f) M;
Mz (if
(if the
the axis of
of the cone is
is the 2-axis),
z-axis), (g)
(g) M;
Mz (if
(if the
the axis
axis
of
of the torus is
is the
the z-axis),
z-axis), (h)
(h) the Lagrangian is
is unchanged by
by aa rotation
rotal;ion through
through an
an angle
angle
89b
8rf> about the
the axis
axis of
of the
the helix (let
(let this be
be the
the z-axis) together with 3a translation through
through a
a
distance
distance a8¢12
h8rf>f27T along
along the
the 2lxis
axis (A
(h being
being the
the pitch
pitch of
of the
the helix).
helix). Hence
Hence 8L
8L := '82
8z 3L[3z+
oL/oz+
+8r/> aLlay
C!Lfor/> = s¢»(nP,/2a»+M¢)
8rp(hP./2TT+Mz) = 0, so
so that
that mz+npl2-ff
Mz+hP./2TT == constant.
constant.
+8¢

.

=

§10.
§1
0. Mechanical similarity

Multiplication of
of the Lagrangian
Lagrangian by any constant clearly
cleany does not affect
affect
Multiplication
equations of
of motion.
motion. This fact {already
mentioned in §2) makes possible,
the equations
(already mentioned
possible,
'in
number of
of important
important cases, some
some useful
useful inferences
inferences concerning
concerning the
in a number
the proof the
the motion, without the
the necessity
necessity of
of actually
actually integrating the
the equaequaperties or
tions.
Such cases include
include those where
where the potential
energy is a homogeneous
homogeneous
Such
potential energy
function of
of the co-ordinates,
co-ordinates, i.e. satisfies
satisfies the condition
condition
function
\

U(()(rt, 0£.1'2)
()(r2, ... , urn)
()(rn)
U(ar1,
-°-)

=
=

()(kU(rt, 1.2)
r2, °...
rn),
a*IU(r1,
° ° », iN):

(10.1)
(10.1)

where a()( is any constant
constant and k the
the degree
degree of
of homogeneity
homogeneity of
of the function.
function.
where
Let us carry
carry out
out a transformation
transformation in which
which the co-ordinates
co-ordinates are changed
changed by
Let
factor()(
the time
time by a factor
factor {1:
ra-+
()(ra, t-+
the velocities
velocities
a factor
a and the
B: Ra
-> are,
t -> {Jt.
Br. Then all the
changed by a factor alB,
()(/{1, and the kinetic
kinetic energy
energy by
by a factor
Va = ddra/dt
r / d t are changed
()(~/{12. The
The potential
energy is multiplied
multiplied by a*.
()(k.
If
such that
m2/B2.
potential energy
If a()( and {1
B are such
()(2ff12 =
= as,
()(k, i.e. B
f1 == al'
()(1-fk,
the result of
of the
the transformation
transformation is to multiply
a2lB2
up then the
the Lagrangian
Lagrangian by the
the constant
constant factor a*,
()(k, i.e. to leave the equations
equations of motion
motion
the
unaltered.
change of all the
the co-ordinates
co-ordinates of
of the particles
particles by the same
same factor correA change
sponds to the rep
replacement
of the paths of the particles
other paths,
geometrisponds
lacement of
particles by other
paths, geometribut differing
differing in size. Thus we conclude
conclude that, if the
the potential
potential energy
energy
similar but
cally similar
of the system
system is a homogeneous
homogeneous function
function of degree
degree k in the
{Cartesian) coof
the (Cartesian)
the equations
equations of
of motion
motion permit a series
series of
of geometrically
geometrically similar
similar
ordinates, t.he
ordinates,
paths, and
and the times
times of
of the
the motion
corresponding points are in the
the
motion between
between corresponding
ratio

t' /t :=
= (2'/01-Ha
(l' fl)l-lk,
r'/r

(10.2)

linear dimensions
dimensions of the two paths.
paths. Not
times
where l'fl
I'll is the ratio of linear
Not only the times
but also any mechanical
mechanical quantities
quantities at corresponding points at corresponding
but
velocities,
which is a power
For example,
example, the velocities,
times are in a ratio which
power of l'fl.
I'lI. For
energies and
and angular
angular momenta are such that
that

(10.3)
M'JM =
= (1'n)1+*'t~
(l'jl)l+lk.
v'Jv
= (/'/1)v¢,
(l'fl)lk,
E'JE
= ((l'fl)k,
w'/u =
E'lE =
w,
m'/m
following are some
some examples
examples of
of the
the foregoing.
foregoing.
The following
As we shall
shall see
see later,
later, in small
oscillations
the
potential
qua?ratic
small oscillations
potentia l energy is a quadratic
•
period of
function of
of the co-ordinates (k =
= 2). From (10.2)
(10.2) we find
find that
that the
the penod
of
function
such oscillations
oscillations is independent of
of their amplitude.
such

§10

23

similarity
Jlfechanical simiIa1'i1'y

In a uniform
e ld of
uniform fi
field
of force,
force, the potential
energy is a linear
linear function
function of the
potent al energy
tes (see (5.8)), .Le.
co-ordina
v(l'/Z).
co-ordinates
i.e. k =
= 1. From (10.2) we have r'lr
t' ft =
= vU'
fl).
Hence, for example,
example, it follows that,
that, in fall under
gravity, the time of
of fall is as
Hence,
under gravity,
the square
square root
of the initial altitude.
altitude.
the
root of
Newtonian
attraction of
of two masses
masses or the Coulomb
Coulomb interaction of
of
ewtonian attraction
In the N
charges, the potential energy is inversely
inversely proportional to the distance
two charges,
= - 1. Then r';z
t' ft
it is a homogeneous
homogeneous function
function of degree k =
apart, i.e. it
=(I'
and we can
can state, for instance, that
that the square
square of
of the time of
of revolu=
(I ' fIf[)3i2,
I)3°"2, and
orbit is as the cube
cube of
of the size of
of the orbit
orbit (Kepler
(Kepler's third
third law).
la'W).
tion in the orbit
tion
If
energy is a homogeneous
homogeneous function of
of the co-ordinates and
and
If the potential energy
the motion
motion takes
takes place
of space, there
there is a very
simple relation
relation
the
place in a finite region
region of
very simple
of the kinetic and potentia
potentiall energies,
energies, known
known
between the time average values of
between
·Dirial theorem.
as the aerial
Since the kinetic
kinetic energy
energy T
Tis
of the velocities, we have
Since
is a quadratic function of
by Euler's
putEuler's theorem
theorem on homogeneous
homogeneous functions
functions Eva»3Tl'§va
l:,va·2Tfi3va = ZT,
2T, or, putting 3oTfova
Tl3va =
pa, the momentum,
= Pa•

22T
T == 2:Pa•
P a . vVa
a
u

=

ra' p
~(LPu•
ru}- 2:ra•
Pu·
dt u
a
a

(L

go.

(10.4)
(10.4}

Let us average
average this equation
equation with
with respect
respect to time. The
The average
average value
value of
of any
any
Let
function
of time f (I)
(t) is defined
defined as
function of

f

= lim
- Jt(t)
f(z) dr.
lim~
dt .
.,. ->oo
-700 'T
1

1'

7'
T

T 00

I dz of
It
ifj(t) is the time
time derivative
derivative dF(t)
dF(t)fdt
of a bounded
bounded funcIt is easy to see that, iff(r)
I
tion
For
F(t), its mean
mean value is zero. For
tion FU),

1 dF
_ = hmF('f) F(0) = 0.
. F(T)-F(O)
Iim~f
dF dr
dt =
= lim
lim
ff =
= 0.
T

*

T-BOOT
'T->OO T

0

dt
dr

-

-'T->OO
t->
oo

1'
T

Let us assume
assume that
that the system executes
executes a motion
motion in a finite
of space
space
Let
Fruite region of
l:,pa. ra
ra is bounded, and
and the mean
mean value of
of
and with
with finite velociti
velocities.
es, Then Zpa
and
term on the right-hand
right-hand side of
of (10.4) is zero. In
In the second
second term we
the first term
replace IN
a
by
-3U/3
Pa
- oUjora
accordance with
with Newton's
equations (5.3), obtainobtainNewton's equations
to in accordance
iniT
ingt

ZT
ta' oUjvra.
viata2T == 2:ra·
a

(10.5)

If the
If
the potential energy
energy is a homogeneous
homogeneous function
function of degree k in the radius
ra, then by
by Euler's
Euler's theorem
theorem equation
equation (10.5) becomes
becomes the required
required
vectors re,
relati
relation:
on:
ZT
2T = Ku.
kO.
(10.6)
(10.6)
on the
the right of
of (10.5) is sometimes called the
the 'virz'al
virial of
of the
the system.
system.
Tt The expression on

1

24

Conse
rvation Laws
Conservation
Laws

Since
U =
Since T
T++ 0
=D
E* :::
= E, the relation
relation (10.6) can also be expressed
expressed as

U
0 = 2E,'(k+2),
2E,'(k+2),

T
T=
= kE!(k+2),
kE/(k+2),

_

§10

(10.7)
(10.i)

elms of the total energy
which
which express
express 0 and T
T in tterms
energy of
of the~system.
th~system.
T=
- In particular,
particular, for small oscillations
oscillations (k =
= 2) we have T = U,
0, i.e. the mean
mean
values
qua 1. For
values of
of the
the kinetic
kinetic and
and potential
potential energies
energies are equal.
For a Newtonian
Newtonian interaction (k =
T=
= -- 11)) 22T
= -l`7,
- 0, and E =
= -- T, in accordance with
with the fact that,
in such
such an interaction, the
the motion
motion takes pplace
lace.. in a finite region of space only
if the total energy
energy is negative
negati\re (see §15).
·
if
'N

PROBLEMS
P
R O B LE M S

PROBLEM
PROBLEM 1.
1. Find
Find the
the ratio
ratio of
of the
the times
times in
in the
the same
same path
path for
for particles
particles having
having different
different
masses
masses but
but the
the same
same potential
potential energy.
energy.
SOLUTION.
SoLUTION.

r'lt
I(m',"m).
t'/t = v(m'/m).

PROBLEM 2.
2. Find
Find the
the ratio
ratio of
of the
the times
times in
in the
the same
same path
path for
for particles
particles having
having the
the same
same mass
mass
PROBLEM
but
factor.
but potential
potential energies
energies differing
differing by
by aa constant
constant factor.

t'/t
SOLUTION. t'lt

y(U/U').
== V
(U]U').

4

CHAPTER
C
HAPTER

IIII
II

HE E
QUATIONS O
F M
OTION
THE
EQUATIONS
OF
MOTION
IINTEGRATION
N T E G R A T I O N OF T

Motion in
in one dimension
§11. Motion
THE motion of
of a system having
having one degree
degree of
of freedom
freedom is said to take place
place
in one
oue dimension. The most general
general form
form of
of the
the Lagrangian
Lagrangian of
of such a system
in fixed external conditions is
L
L

::=

!a(q)q2=%"(9)é2- U(q),
U(q)»

(11.1)
(11-1)

where a(q) is some
some function
function of
of the generalised
generalised co-ordinate
co-ordinate Q.
q. In
In particular,
particular,
Cartesian co-ordinate
co-ordinate (x,
(x, say) then
then
if q is a Cartesian

L
L =
= %m.x'
!m.i2U(x).
2- U(x).

(11.2)

The equations
equations of
of motion
motion corresponding
corresponding to these
these Lagrangian
Lagrangians can be inteThe
general form. It is not even necessary
necessary to write down
down the equation
equation
grated in a general
grated
of motion; we can start from the
the First
first integral of this equation, which
which gives
of conservation
conservation of energy.
energy. For
For the Lagrangian
Lagrangian {11.2)
{e.g.) we have
the law of
(11.2) (e.g.)
!m.X2+ U(x) =
= E. This is a first-order
first-order differential
differential equation,
equation, and can
can be inteinte%mx'-2+
/{21E
- U(x)]/m}, it follows that
grated
Since dxldt
dxfdt =
= v/{2[
Egrated immediately.
immediately. Since
=

v(!m)
1/(W)

II

rt

I

dx
do

-+
constant.
+constant.
U(x)]
1E- U(X)]

/
vi[Ew
-n

(11.3)

The
The two arbitrary
arbitrary constants
constants in the solution
solution of the equations
equations of
of motion
motion are
by the
the total
totaf energy
energy E and the constant
constant of integration.
integration.
here represented by
here
Since the kinetic
kinetic energy
energy is essentially
essentially positive,
positive, the total energy
energy always
Since
excee
ds the potential energy, i.e. the motion can take place
only in those
exceeds
place only
of space
space where
where U(x) < E. For
For example,
example, let the function U(x) be
regions of
regions
the figure a horizontal
horizontal
of the form shown in Fig. 6 (p. 26). If
If we draw in the
lin
given value of
of the total energy, we immediately
immediately find
linee corresponding to a given
regions of
of motion.
motion. In the example
example of
of Fig. 6, the motion
motion can
the possible
possible regions
the range AB
AB or in the range to the
the right of C.
occur only in the
occur
The points at which
which the potential
energy equals
equals the total energy,
The
potential energy
U(x) =
= E,

(11.4)

of the motion. They are turning
turning points, since the velocity
there
give the limits of
velocity there
is zero.
If the region of
zero. If
of the motion
motion is bounded
by
two
such
points,
then
bounded by
such
then the
motion takes place in a finite region
region of
of space,
space, and is said to be finite.
]"21zite. If
If the
motion
is
limited
region
of the motion limited on only
only one
one side, or on neither, then
then the
region of
motion
and the
the particle goes
goes to infinity.
motion is infinite and
25

26

of
of the
the Equa
Equations
of Motio
Motion
tions of
rz

Irzteg
ratiorz
Integration

§11
§11

finite motion in one dimension
dimension is oscill
oscillatory,
reA Finite
story, the particle moving r6peatedly back and forth between two points (in 1~.18.
Eig. 6, in the Pote
potential
ntial well
AB
AB between
between the points .al
x1 and kg).
x2). The
The period T of the oscillations, i.e. the
time
time during which
which the particle
particle passes
passes from x1 to
to x2
x 2 and back, is
'is twice
twice the time
time
from al
.3),
x 1 to x2
x 2 (because
(because of the reversibility
reversibility property, §5) or, by (11
(11.3),
=

/
12m
"x (2m)
( )

(11.5)

vi[E_ V'
[E- U(x)]'
um] 1
Xt(El
$C1(E)

II

11E
T(E)
( )

dx
do

XI( PE)

x2J(El

where al
x1 and xg
x2 are roots of
of equation
equation (11.4)
{11.4) for the given
given value of E. This forfoF--where
the period of
of the motion
motion as a function
function of
of the total energy
energy of
of the
the
mula gives the
particle.
U
u

A
As
~@-------»--

B

II

1-------

i

I
I

I
I

I
l
l
I
I

I
I
I

x,

1
l

I

I

X2

Xl

FIG.
FIG.

x
X

6

PROBLEMS
P
ROBLEMS

PRoDLE.\1 1. Detemmine
Determine the
the period of
of oscillations of
of a simple pendulum (a particle of
of mass
PROBLEM
by a string
string of
of length Il iin
of the
the amplitude of
of
suspended by
n a gravitational field)
Held) as a function of
the oscillations.
the
111
m

--

SOLUTION,
SOLUTIO:-!. The
The energy
energy of
of the
the pendulum
pendulum is
is E == §ml2ql2-mgl
/;:ml 2 ~ 2 -mgl COS
cos <;6==
¢-:- -· mg
mgl cos
cos 960,
r/>o, where
where

r/> is
is the
the angle
angle between
\Jetween the
the string
string and
and the
the vertical,
vertkal, and
and 960
r/>o the
the maximum
maximum value
value of
of ¢.
rf>. Calculating
Calculating
96
the
tl€.time required tO
the Qeqiod
ger,i.od aS
as· t~.time
to go
go from r/> = 0 to
to 96
r/> = 960,
r/>o, multiplied by
by four, w
v.e
find
e Find
T

=

_

4
f

j_!_z

" - 2g
2g
A

f

go

=

d
If ____:d"'~~0
¢0

0

¢»
V(cos
¢50)
v(cos 96-cos
r/>-cos
r/>o)

~0
¢0

2/Z
d¢
= 2 '~I
dr/>
2
2 /t4>)
II/ gg 0 v(sin
V(sin2/;:r/>o-sin
gf>o-sin2qf>)
0

4M

The substitution
substitution sin
sin [=sin
!r/>/sin %st>0
/;:r/>o converts
converts this
this to
to T
!r/>o), where
(I/g)K(sin %quo),
The
sin %¢>lsin
T = 4v(l/g)K(sin
§
K(k) =
K(k)

If

hr
I"
0

d[
do'
v(1-k2
sin2[)
_kg si112§)
1/(1

tiOI1S),
nall OSCilla
9150 < 11 (gf
is the
the complete
complete elliptic'
elliptic integral
integral of
of the
the first
kind. For
For sin
sin !<Po
~ !<Po~
(small
oscillations),
颻0 go
jirst kind.
is

an expansion
expansion of
of the
the function
gives
an
function K gives

+

T = 2trV(//§)(1
27Ty(l/g)(1 + 11¢¢~02+
-h</>o 2 + ...).
... ).

of

§12

27
27

Determination
Determination of the
the potential
potential energy
energy

first term corresponds to the
the familiar
familiar formula.
The First
2. Determine
Determine the
the period
period of
of oscillation,
oscillation, as
as aa function
of the
the energy,
energy, when
when aa
PROBLEM 2.
function of
of mass
mass m
m moves in
in Fields
fields for which the potential energy is
particle of

U=
= Alxl",
Alxln,
(a) U

SOLUTION.

(a):
(a):

= -U0/cosh2ax,
-Uo/cosh2 ax, -Uo < E
(b) U =

T = 2v(ZtN)

=

f

(E[A)l!N

<

0,

U = U0
Uo tanzax.
tan2 ax.
(c) U

do

1/(E -.ex")

dy
(E )1/n I v(1-yn}
dy
m) f Vu -§*5`
0

2m
E
22 .2m
.._
.../ E.
A
E

I
I

1/n

0

0

By the
the substitution
substitution y"
yn == Hz
u the
the integral
integral is
is reduced
reduced to
to aa beta
beta function,
function, which
which can
can be
be expressed
expressed
By
in
functions :
in terms
terms of
of gamma
gamma functions:
1/n
I`(1]n) .
T = ~2 / 27rm
2mn . (E )1/n _!J1/n)
T
P(%+1/n) .
nlto/ E
A
_rc!+tfn)
Z

,M

The dependence
dependence of
of T
on EE is
is in
in accordance
accordance with
with the
the law
law of
of mechanical
mechanical similarity
similarity (10.2),
(10.2),
The
T on

(10.3).
(10.3).
{b) T
T =:
= ("l(1)V(2m!IEI)~
(7TMvC2m!IEI>.
(b)
(c) T
T =
= of/aw
(7r/a)v [2f».f(E+u0)1.
[2m!(E+Uo)].
cc)

§1
§12.
Determination of
of the
the potential
potential energy
energy from
from the
the period
period of
of
2. Determination
oscillation
oscillation
Let us consider
consider to what extent
extent the form of the
the potential
energy U(x) of a
Let
potential energy
Hel
d
field in which
which a particle
oscillating can
can be deduced
deduced from a knowledge
of the
particle is oscillating
knowledge of
period
of
oscillation
T
as
a
function
of
the
energy
E.
Mathematically,
period of oscillation
energy
Mathematically, this
the solution
solution of
of the integral
integral equation
equation (11.5), in which U(x) is regarded
regarded
involves the
as unknown
unknown and T(E}
T(E) as known.
We
shall assume
assume that the required
required function
function U(x) has only
only one minimum
minimum
\'Ve shall
region of
of space
space considered,
considered, leaving aside the question
question whether
whether there
there
in the region
of the integral
integral equation
equation which
which do not
not meet
meet this condition.
condition.
exist solutions
solutions of
exist
For convenience,
convenience, we take the
the origin
origin at the position
position of minimum
minimum potential
For
potential
energy, and
and take this minimum
minimum energy
energy to be zero (Fig.
(Fig. 7).
energy,
I

u
U

I

--U=€

I l l

FIG
FIG..

7

X2

*z

XL

of

of

28
28

§12

Integration
Integration of the
the Equations
Equations of It/lotion
lVlotion

In the integral (11.5) we regard the co-ordinate
co-ordinate x as a function of U. The
The
function x( U) is two-valued:
two-valued: each value of the potential
potential energy
energy corresponds
corresponds
function
different values of
of x. Accordingly,
integral (11.5) must
must be divided
divided
to two different
Accordingly, the integral
into
U) dU: one from
x = al
into two
two parts before
before replacing
replacing do
dx by (do/d
(dxfdU)
from.'\'=
.'\'1 to x =
= 0
and the other
other from xx = 0 to xx =
= kg.
x2. We shall write the function x(
x( U) in
these two ranges as xx =
= al(
x1( U) and xx =
= kg(
xz( U) respectively.
The limits
limits of
of integration
integration with respect
respect to U are evidently
evidently E and 0, so that
we have

T(E)
T(E}

=

x/(2w)
v(2m)

~/(2%)

I

E

0

E

0

dU
/2
dx 2(U)
dU
+
m
dX2(U)
dU vi(EU) + M
'\1(2m)
dU
x/(E-U3
)
l

N2

dU

0
0 dx1(U)

E
E

ddU
U

dU
'\/(Ew/(E- U)
u)

dU

X1

l

I

d U 1/(E~ U)`

If both sides of this equation
/(@- E), where
a is a parameter,
If
equation are divided
divided by v(()(where()(
parameter,
and integrated
integrated with
with respect
respect toE
to()(,
and
to E from 0 to
a, the result
res lt is

II
a

f

0
0

T(E) dE
T(E)dE
v(()(-E)
1/(01-F)

=

vl(2m)
/'

w (2m)

IHI
a

0

f`

E

0

l 002
l al
[-dx_2
__
dx_'l]
_ _dudE
d_U_d_E_ _
ddU
U ddU
U v[(<=<,1[(()(-E)(EE)(E- U)]

un

or, chang'7
chang;ng
order of
of integration,
integration,
fig the order

1C1 d U fa
dE
T(E)dE
aT(E)
dE
fa [ dx2 dd:'-'1]
dE
1/(21")
f0 v(()(-E)
v[(()(- EXEE)(E- un
U)]'
ddUU ddU
U dU wwV01-E) = v(Zm)
0

0
0

d3C2

U
u

integral over
over E is elementary;
elementary; its value is
The integral
trivial, and
an~ we have
have
thus trivial,

fI -vc()(-E>
a

since x2(0)
x2{0) = X1(0)
x1{0)
since

0

T(E)dE
T(E) dE

1/(°( E)

'

TT.
IT.

The integral
integral over U is
The

= ='V(2"*)[~'"2((1)
TTy{2m)[x2(()()-xl(z)],
- x1(1)]»

I

= 0. Writing
()(, we obtain the final result
result::
\Vriting U in place of a,

-x U
JC2(U)
x2(U}-xl(U)
1( )

=

='.

uU
T(E) dE
11
T(E)dE
TTy(Zm)
»\/ (U- E) `
7T
"v(2m)0 '\I(U-E).

(12.1)

0

Thus the
the known
known function
can be
be used
used to
to determine
determine the
the_ difference
di~erence
Thus
function T(E)
T(E) can
11 ind ete rlyes
remain
rlse
(U) th€r
x2(U)-x1 (U). The
The functions x2({
~2(U)
x 1(U)
~he~selves
r;:: ~U tndeterx2(U)-x1(U).
I ) and .'JC1
.= U(x)
rves
cu
of
minate.
This
means
that
there
IS
not
one
but
an
tnfintty
<>f
cu
U(x)
ere
is
not
one
but
an
infinity
minate. This means that th

1

§13

29
29

The
The rejueed
reduced mass
mass

which give the
the prescribed
dependence of period
period on energy,
energy, and differ in such
which
prescribed dependence
that the difference
difference between
values of
of x corresponding to each
each
a way that
between the two values
of U is the
the same
same for every
every curve.
curve.
value of
indeterminacy of
of the solution
solution is removed
removed if we impose
impose the condition
condition
The indeterminacy
that
that the
the curve
curve U =
= U(x) must
must be symmetrical
symmetrical about
about the
the U-axis,
U-axis, i.e. that
x(U). In this case, formula (12-1)
x2{U) =
{12.1) gives for x(U) the
x2(U)
. = -x1(U)
. E
umque
expresston
unique expression

=

(U)

x(U)

_

=
'

U
u

T(E) dE
IT(E)dE
.
.
/
/
27T'\1(2m)
vi(
UE)
2 (2122) 0 w ( U - E i
11

0

(12.2)

§13.
The reduced
reduced mass
§1
3. The
complete general
general solution
solution can
can be obtained
obtained for an extremely
extremely important
important
A complete
problem, that
that of
of the
the motion of
of a system consisting
consisting of
of two interacting particles
particles
(the
{the two-body
two-body Problem).
problem).
As a first step
step towards
towards the solution
solution of
of this problem,
problem, we shall
shall show how it
can be considerably
considerably simplified
simplified by
by separating
separating the
the motion
motion of
of the
the system
system into
into
the
the motion
motion of
of the centre of
of mass
mass and
and that
that of
of the particles relative
relative to the
the centre
of mass.
The potential
potential energy
energy of
of the interaction
interaction of
of two particles
only on
The
particles depends only
distance between
between them, i.e. on the magnitude
magnitude of
of the difference
difference in their
their
the distance
radius vectors. The
The Lagrangian
Lagrangian of
of such
such a system
system is therefore
therefore
radius
2 +lm2f2
2 - U(!r1-r2!).
L == !m1f1
=§~?7'£1il12+%2'"2
i'22,(13.1)
U(l 11 r 21)~

-

Le
Lett r = 1'1
r1- rr22 be
be the
the relative
relative position
position vector,
vector, and
and let
let the
the origin
origin be
be at
at the
the
centre of
of mass, i.e.
i.e. m1r1+m=.gr2
m1r1 + m2r2 = 0. These
These two
two equations
equations give
give
m2r,-"(m1+1n2),
mzr/(ml +m2),

Substitution in (13.1) gives
Substitution
where
"·here

L

r2

-m1rr (1111-I-7722).

II

r1 =
1'1

-

(1
3.3)
(13.3)

m1m
2,"(m1+
11li11l'!./(ml
+m2)
1722)

(13.4)
(13.4)

= !mf2U(r},
J§mi'2 UU),

m
m =
=

(13.2)

called the reduced mass. The
The function
function (13.3) is formally identical
identical with
with the
the
is called
Lagrangian of a particle
of mass m
moving in an external Held
field L`(r)
C(r) which is
Lagrangian
particle of
m moving
symmetrical about aa fixed origin.
problem of
the probl~m
s the
Thu
ion of
Thus
of the mot
motion
of two interacting partic
particles
equivalent
les is equivalent
to that of
•f one par
of the motion
motton oof
particle
given external
external He
field
U(r).
From the
the
ticle in a given
ld UU). From
solution r -=
= r(t) of
of this Problem,
problem, the pa
paths
r 1 = 1'1(t)
r 1(t) and t2
r2 =
= I'2(Z)
r 2(t) of
of the
solution
ths I'1
separately,
es
relativ
ticl
e
to
par
common centre
centre of
of mass,
mass, are obtained
obtained
two particles separately, relative their common
by means of
of formulae
formulae (13.2)_
(13.2).

of

30

of

§14

Integration
Integration of the
the Equations
Equations of Notion
Jlotiou
PROBLEM
P
ROBLEM

A
A system consists of
of one particle of
of mass M
M and
and 11
11 particles with
with equal masses m. Eliminate
the
the motion of
of the
the centre of
of mass and
and so reduce the
the problem to one involving
invoh·ing n particles.
particles.

=

SOLUTION.
SOLUTION. Let R
R be
be the
the radio
radiusS vector
\'ector of
of the particle of
of mass 1\1,
(a=
... , 11)
11)
== 1, 2, ...,
JM, and Ra (a
those of
We put
of the particles of
of mass
mass m.
m. We
put re
ra E Ra
Ra-R
and take
take the
the origin
origin to
to be
be at
at the
the centre
centre
-R and
of
of mass!
mass: .7l'fR+mERa
1\1R+m~Ra = 0.
0. Hence
Hence R
R ~- -(mfp{':Era,
p.=J,f mn,
mn; Ra
Ra
R 11:ra.
R
(7f1!u) Era, where p.E.'l/i
Substitution in
in the
the Lagrangian LL = =}JIR2+§nz
~JJR2+!mLRa2U gives
gin~s
Substitution
ERa2.-- U

=

L

- fzm

Van

1(m'2]p)

°"'

aa

.

r

(Zva
a
a

+

-

U, where va

=

+

r' an

The potential
potential cenergy
depends only
only on
on the
the distances
distances between
between the
the particles,
particles, and
and so
so ca
can
be
The
n e r y depends
n be
written
as aa function
of -the
the re.
ra.
written as
function of

§14
§14.. Motion
Motion in a central Held
field

On
On reducing
reducing the two-body problem to one
one of
of the motion
motion of
of a single
single body,
arrive at the problem of
of determining the motion
motion of
of a single
single particle
we arrive
particle in an
external Held
field such
such that its potential
only on the distance
distance r
external
potential energy depends only
some fixed point.
called a central
ce11tral field. The force acting
acting on the
from some
point, This is called
T
le is F = particle
-oC(r),'cr
-(dC,dr)rrr;
(dl r dr)r/r, its magnitude is likewise a funcpartic
3U(r) /2r = tion of
of 1*r only,
only, and its direction is everywhere
everyw·here that of
of the radius vector.
tion
already been
been shown
shown in §9, the angular momentum
momentum of
of any system
system
As has already
relative to the centre of
of such
such a field is conserved.
conserved. Thc
The angular
angular momentum
momentum of
of a
relative
particle is M = r xp.
X p. Since
Since M
constancy of
single particle
M is perpendicular
perpendicular to r, the constancy
shows that, throughout
throughout the motion,
motion, the radius
radius vector
vector of
of the particle lies
M shows
in the plane perpendicular
perpendicular to M.
Thus the path of
of a particle
central field lies in one plane.
plane. Using
Using polar
Thus
particle in a central
polar
co-ordinates
co-ordinates r, ¢>
cf> in that
that plane,
plane, we
\Ye can ivrite
\\·rite the
the Lagrangian
Lagrangian as
.

L

-nz(r'2 -I- ?2¢>2) -

U(1') ;

(14.1)

II

(4.5). 'This
This function
function does not
not involve
involve the co-ordinate
co-ordinate ~, explicitly.
explicitly. Any
Any
see (4.5).
generalised co-ordinate
co-ordinate Qz:
qi which
which does
does not
not appear
appear explicitly
explicitly in the Lagrangian
Lagrangian
generalised
said to be cyclic. Tor
For such a co-ordinate we have, by
by Lagrange's
Lagrange's equation,
is said
/3
'
.
:
:
(d
do) EL
(d,dt)
cL,'ciJi
oDoqi =
= 0, so that
that the corresponding
corresponding generalised
generalised momenmomenr ii = 3L73g5
tum PLrJlcq IS
P·i-3 = 3oL,'cq.i
is an integral
integral of
of the motion.
motion. This
This leads to a considerable
considerable
simplification of
of the problem
problem of
of integrating
integrating the
the equations
equations of motion
motion when
simplification
co-ordinates.
there are cyclic co-ordinates.
generalised momentum
momentum P
p¢9 == mr2q§
mr2J is the same
same as
In the present case, the generalised
momentmn 5/28
11!~ =
= .V
Jf (see (9.6)),
(9.6)}, and we return
return to the known law
the angular ntomentnm
conservation of
of angular momentum'
momentum:
of conservation
I

;

=

1M_oq5
mr?.J = Col'lstarlt.
constant.
II

.11
in

(14.
(14.2)
2)

motion of
This law has a simple george
geometrical
interpretation in the
the plane
plane motion
of aa single
single
trical interpretation
area
of
the
Sector
field. The
The expression ».%1°
ir •• rd<,-6
rdf is the
the area of /h~ sector
particle in a central Held.
path
the
0
61'1t of
bounded
by
two
neighbouring
radius
vectors
and
an
dcn,ent
t e path
bounded by two n e i g h b o r i n g radius vectors and an 610nl

§14

31

lVIotion
Zllotion in a central field
jield

df, we can write
write the angular
angular momentum
momentum of
of the parpar(Fig. 8). Calling this area df,
tticle
i l e as

=
= Zmf,
2mj,

:U
M

f

(14.3)

where the
the derivative
derivative j is called
called the
the sectorial
velocity. Hence the
the conservation
conservation
where
sedorial velocity.
momentum implies
implies the constancy
constancy of
of the sectorial
sectorial velocity:
velocity: in equal
of angular momentum
the radius vector
vector of
of the particle sweeps
sweeps out equal areas
areas (Kepler's second
times the
times
law).T
law}.t

/0f¢

r

aw

O.

FIG.

8

The
particle in a central
The complete
complete solution
solution of
of the problem
problem of the
the motion
motion of a particle
central
most simply
simply obtained
obtained by starting
starting from the
the laws of conservation
conservation of
of
field is most
energy and
and angular momentum, without writing out
out the
the equations of
of motion
themselves.
Expressing~g in terms of M from (14.2) and substituting
substituting in the
themselves. Expressing
expression for the
the ehergy,
energy, we obtain
obtain
expression

%._m(i'3 +t2<£2) + U(r)
r

or, integrating,
is

II

t =

-=

dt

J{
JI

c/> =

II

m

+ L* to.

"-2

1\f2I'"

22
M2
112}
-[EU(r)]--·1Eum]
m
m2r2I
17221
m

I d»~l¢Li
drjj{~[E-

\-Vriting (14.2
Writing
(14.2)) as d¢>
clef>
we
we Fin
findd
*G-

dr

=

in .2-|'
7

%

II

Hence
Hence

E

M22

I

[E - U(r)]
- M }+constant.
+ constant.
U(r)]m2r2
??Z23'2

(

(14.4)

(14.5)
(14.6)

= M
111 dtlmr2,
dtjmr2, substituting
substituting do
dt from
from (14.5)
(14.5) and
and integrating,
integrating,

I

.

Mdrfr2
dolF?-a
.M
+constant.
y{2m[ EM2,'r2}
»\/{2nz[E
- C(r)]L (r)] - M2r
r2}
f

(14.7)
(14. 7)

Formulae (14.6) and (14.7) give the general
general solution
solution of the
the problem.
The
Formulae
problem. The

lation between r and
latter formula
QB, i.e. the
formula gives the
the re
relation
and cf>,
the equation
equation of
of the path.
Formula
Formula (14.6)
(14.6) gives the distance
distance 1'r from the
the centre
centre aS
as an implicit
implicit function
function of
of
The angle
angle <3-f,'>» it
it should
should be noted,
noted, always
always varies monotonically
monotonically with
with time,
time,
time. The
at c/>; ca
ws th
.2) sho
n never change
since
(14.2)
shows
that
can
change sign.
ce (14
sin

_

1'
t 'The
The law
law of
of conservation
conservation. of
of angular
angular momentum for 2a particle moving
moYing in
in a central field
field
d the
al,
called
the area integr
zntegral.
is sometimes calle

I

32

of

of

§14

Integration of the
the Equations
Equations of fllotion
1V!otion
Integration

The
The expression
expression (14.4) shows
shows that the radial part of the motion
motion can be regarded
garded as taking place in one
one dimension
dimension in a field where the "effective
"effective potenpotenenergy" is
tial energy"
U
Uerr
eff =
=

U(r)
Jl;f2,'2mr2.
+/W2,'211zr2.
I 1(r) +

(14.8)

2f2mr 2 is called
The
M2l2mr2
The quantity
quantity M
called the centrifugal energy.
energy. The
The values of
of r for which
U(r}+M2f2mr2 = E
U(r)+M2/2m1'2
E

(14.9)
(14.9)

determine the
the limits of
of the
the motion
motion as regards distance
distance from
from the centre.
determine
When equation
equation (14.9) is satisfied,
satisfied, the radial velocity
velocity r'i is zero. This does not
not
When
mean that the particle
particle comes
comes to rest
rest as in true
true one-dimensional
one-dimensional motion,
motion, since
since
mean
velocity ~g is not
not zero. The
The value r'i :=
= 0 indicates
indicates a turning point
the angular velocity
of the path,
r(t} begins
decrease instead
instead of
of increasing,
increasing, or 'vice
vice eexrsa.
versa.
of
path, where r(t)
begins to decrease
If the
>.= rain,
If
the range
range in which
which rr may vary is limited
limited only
only by the
the condition
condition rr ,~
rmtn,
the motion
motion is infinite: the particle
particle comes
comes from, and returns to, infinity.
the
If the
If
the range
range of
of rr has two limits rmln
rmtn and irmaJ
rmax, the
the motion
motion is finite and the
the
path lies entirely
entirely within
within the annulus
annulus bounded
bounded by the
the circles
circles rr = rmax
max and
= rmln.
rmtn· This does
does not
not mean,
mean, however,
however, that the path
path must be a closed
closed curve.
rr =
During
max to 7'm1n
During the
the time
time in which
which rr varies from
from rmax
rmtn and back, the
the radius
radius
vector turns
turns through
through an angle !1rf
according to (14.7), is given by
vector
All which, according

2

II

A¢>

Tmax
tmax

am! n

1li
drfr2
Ill dr/.r2

N / [2m(E-

u)- 3142/t2]`

(14.10)

The condition
condition for the
the path
closed is that this angle
angle should
should be a rational
rational
The
path to be closed
fraction
AA == 27Tmjn,
Zfrmln, where
where m and n are integers.
fraction of
of Zn,
27T, i.e. that 11cfo
integers. In that case,
radius vector
vector of
of the particle
particle will have
have made
made m complete
after.n periods, the radius
aftenn
revolutions and
and will occupy
occupy its original
original position,
position, so that
that the
the path
path is closed.
closed.
revolutions
Such cases are exceptional,
exceptional, however,
of U(r) is arbitrary
arbitrary
Such
however, and when the form of
11cfo is not
not a rational
rational fraction of
of Zn.
27T. In general,
general, therefore,
therefore, the path
the angle Alb
a
particle executing
executing a finite motion
motion is not
not closed.
closed. It
It passes
passes through the
of particle
minimum
and
maximum
distances
an
infinity
of
times,
and
infinite time
time
minimum and maximum distances
infinity of times, and after infinite
covers the entire
entire annulus
annulus between
bounding Circles.
circles. The
The path
it covers
between the two bounding
path
shown in Fig. 9 is an example.
only two
two types
types of
of central
central Held
field 'in which
which all finite motions
motions take
There are only
rlace in closed paths. They
They are those in which
potential energy
energy of
of the
place
which the potential
The former
former case is discussed
discussed in §15
§15;, the
the latter
latter
particle varies as 1/r
1,/r or as r2. The
of the space oscillator (see §23, Problem
is that of
Problem 3).
turning point
the square root
root in (14.5),
(14.5), and
and therefore
therefore the integrands
At a turning
point the
from the
the direcin (14.6) and (14.7), change sign. If
the
angle
cP
measured
angle
¢
is
measured
If
so
tion
of the
the re
radius
the turning
turning point, the
the part
partss of
of the
on each
the p_ath
path OI]
n of
dius vector to the
the
1.6.
t,
of
lue
va
side of
of that
that point
only in th
thee sign
sign of
of 4>t, for each
each valu~ of r, t.e. the path
path
point differ only
-·
' t where
e rr =
· symmetrical
· 1about
1·me 'f'
.1.. = 0.
S tartmg,
f rom aa Pow
potnt
°=- 7lIllax
so
sy, from
IS
symmetnca
ab out the
t h e line
say,
. who
.
h
7'
is
f, = 0 . Starting,
ith
w
__ rrmax
oint
wit
r
=
f ar as
the particle
particle traverses
traverses aa segment
segment of
of the
the path
path aas far
as aa P
mtn,
the
"

1.1-iotion in
in aa centaralfield
central field
Ibfotion

§14

33

then follows a symmetrically
symmetrically placed
segment to the next
= 7'max
Ymax.
then
placed segment
next point
point where r =entire path
path is obtained
obtained by repeating identical
identical segments
segments
and so on. Thus the entire
applies also to infinite
infinite paths, which
which consist
consist of
forwards and backwards.
forwards
backwards. This applies
branches extending
point (r =
rain) to
extending from the
the turning
turning point
= Ymtn)
two symmetrical branches
infinity.

or

1

I

I
I

I
I

l

I

.
I

AS

FIG.

9

The presence
energy when M ;é=ft 0, which
which becomes
The
presence of the centrifugal energy
1 fr2 when r -+
-+ 0, generally renders it impossible for the particle to
infinite as 1/v2
reac
h the centre of
of t.he
the field, even
even if the field
qne. A "fall" of
of
reach
Field is an attractive one.
centre is possible
only
potential energy
energy tends
tends suflisuffithe particle
particle to the centre
possible onl
y if the potential
oo as r ->
-+ 0. From the
the inequality
inequality
ciently rapidly
rapidly to - OO
ciently

!mr2 =
= EE- U(r)M2f2mr2 > 0,
-mr-9"
U(r)-M2/2mr2

2U(r) + 1VJ2f2m < Er2, it follows that r can take values tending
rr2U(r)-I-1W2/2m
tending to zero
only if
or
Or

[r 2 U(r)]r
...o < -]Lf2I'2m,
- M 2,'2m,
(14.
11)
(14.11)
[r2
U(r)],»_,0
i.e
i.e.. U(7')
U(r) must tend
tend to - OO
oo either as - alv2
cx/r2 with aex > .112/Zm,
Jf2j2m, or proportionally
proportionally
lltn with 11n > 2.
-ljrn
to .__
PROBLEM
PROBLEM

PROBLEMS

1.
1. Integrate
Integrate the
the equations
equations of
of motion
motion for
a spherical
spherical pendulum
pendulum (a
(a particle
particle of
of mass
mass
for a

m moving
moving on
on the
the surface
surface of
of aa sphere of
of radius
radius I in
in aa gravitational
gravitational field).
field).

SOLUTION.
In spherical
spherical co-ordinates,
co-ordinates, with
with the
the origin
origin at
at the
the centre
centre of
of the
the sphere
sphere and
and the
the
UTION. In
SOL
polar
polar axis
axis vertically
vertically downwards,
downwards, the
the Lagrangian
Lagrangian of
of the
the pendulum
pendulum is
is

iml 2(82+¢2 sin20)-l-mgl
sin•O)+mgl cos
cos 0.
8.
&"¢l2(62+d)2

34

Integration
Integration

of the
the Equations
Equations of
of Rtotfon
llfotion
of

§14

The co-ordinate
co-ordinate ¢r/> is cyclic,
cyclic, and hence
hence the
the generalised
generalised momentum P¢»
P.;,, which is
is the
the same
same as
as the
z-component of
of angular
angular momentum,
momentum, is
is conserved
conserved::
z-component

The energy is

mlf-'~;f»
mf24> sin'-39
sin2e == .Ma
Alz = constant.
constant.

e

~mF( {}"1 +¢2
+4>2 sin9-6)
sin2e) -mg!
-mgl cos 6
E = §mI2(9'-'

= ~m[202
+!J1z 2'mfg
'm/2 sin'8'6-mgfl
sin 2e -mgl cos 6,
e.
=
'nzl'29"+é_?lf.32

Hence

tt =

I

v {2 [E
[E _·----,[-!e-rr-(
il:-)c:-]'mI'3}
1/{2
[t0ff(t9)]

where the
the "effective potential
where
potential energy" is

dB
de

,:--n--:11":-:-}'9

(1)
(1)

(2)
(2)
(3)
(3)

[](~ff(6)
'mll8' sinI9-mql
Uerr(e) = =l~_-\f,~F
!1~1z 2 'mf2
sin2 e-mr:l cos 9,
e.

the angle
angle 95
rf> we find, using
using (1),
For the

55
r/> =

_H
-liz
lv(2m)
it/(zm)

I

do
ue

v

sin 2 ti\'[E-Ueu(e)]
sir.'2H
[ E - U<¢ff(9)] `·

(+)
(4)

The integrals (3) and
and (4)
(4) lead
lead to elliptic
elliptic integrals
integrals of
of the
the first
first and
and third kinds respectively.
respecti,·ely.
The

range of
of 0e in which the
the motion takes place is that where
where E > Ueff,
Uerr, and its limits
limits
The range
gh·en by the
the equation
equation E =
= UeffUcn- This is a cubic
cubic equation
equation for COS
cos 6,
e, having
hm·ing two roots
are given
between -1
+ 1;, these define
define two
two circles
circles of
of latitude on
on the
the sphere,
sphere, between which the
the
between
- 1 and +1
path lies.
path

PROBLE~I
the equations of
of motion
n>otion for a particle moving on
on the
the surface
surface of
of a
PROBLEM 2. Integrate the
(of vertical
vertical angle 22:x)
placed vertically
vertically and with ,·ertex
downwards in a gravitational
gra,·itational
cone (of
x) placed
vertex downwards
field.
Held.

SOLUTIO~. In
In spherical
spherical co-ordinates,
co-ordinates, with
with the
the origin
ongm at
at the
the vertex
vertex of
of the
the cone
cone and
and the
the
SOLUTION.
polar axis
axis vertically
vertically upwards,
upwards, the
the Lagrangian
Lagrangian is
is 1}1n(r'2+r2¢S2
!m{r2 +r 2.f,2 sitl2oc)
sin 2cc) --mgr
-mg1· cos
cos of,
cc. The
The cocopolar
ordinate £6
rf> is
is cyclic,
cyclic, and
and Ill.:
]11, ==
= mr2§£
mr2 4> sin'3a
sin 2o: is
is again
again conserved.
consen·ed. The
The energy
energy is
is
ordinate

E
E

=

°-=

2,'mr 2 sm0-x--I-mgr
!mr2+!-W,
sin2:x+mgr cos
cos a,
a.
§mrl9'+§~.'W;2,'mt8

lf '

By the
the same method
method as
as in Problem 1, we End
find
By

dr
dr
'
~ '{2[i€-L*m(t))
{2[£- Uerr(r)] ~»}'
111}
dr
m.,
J,],
dr
v
~1(2m) sings:
sin2:x. r2\
'[E-L'cff(r)]'
1/(Zm)
VI \ '[E-U.:fr(r)]

t =
r = •
U

co
Corr(r)
Uerflf)

o

111,2

lf

s

--+mgr COS
cos):.
'x.
- _ --- -I-nwr
2mr2 sln2a
sin 2cc
2mr2
as

conditionE=
L'err(r) is (if
J,1, -:.
or= 0) a cubic
cubic equation
equation for r, having
ha,·ing two
two positive
positi,·e roots
roots;;
The condition
E = LTf.*f[(Y)
(if If;
two horizontal circles on
on the
the cone, between which the
the path lies.
these define two

PROBLE~I
the equations of
of motion for a pendulum of
of mass
mass mg,
m2, with
with a mass 1111
PROBLEM 3. Integrate the
m1
the point of
of support which can
can move
mo,·e on
on a horizontal
horizontal line
line lying
lying in the
the plane in which 7712
mz
at the
mO\'eS (Fig. 2, §5).
moves

SOLlTIO:-:. IIn
the Lagrangian derived
deri\'ed in §5, Problem 2, the
the co-ordinate .>:
cyclic. The
SOLUTION.
n the
x is cyclic.
Px, which is the
the horizontal component of
of the
the total momentum of
of the
the
generalised momentum Pr,
therefore conserved
conser,·ed:I
system, is therefore
Pr = (am +m2).\?+m2fgl' cos ¢ = constant.

(1)
(1)

. ( 1) is. zero
nstant on
The system
system may
may always
always be
be taken
taken to
to be
be at
at rest
rest as
as aa whole.
whole. Then
Then the
the co
constant
in (l) is zero
The
and integration
integration gives
gi,·es
and
(2)
(m 1 +m2).\:+m2l sin
sin ¢>
rf> =-'
= constant,
constant,
.
(2)
(rm-I-m2)x-I-mzl
not
zontal
not grove
mm·e hori
honzontall
which expresses
expresses the
the fact
that the
centre of
of mass
mass of
of the
the system
system does
docs
y.
which
fact that
the centre

15
§15

Kepler's
Kepler problem

w e End
(1), we
find the
~he energy in
in the form
form
Using (I),

é"H2/3¢°2(1

E =

Hence
I-[ence
t

t

=

J

z
l

m

2

cos29b)

fJ
.Ll

ml +7712

m2
nz2

2(m1 +m2) •
2(m1-I-m2)

35

-meg! cos

(3)
(3)

m1
+m2
1111
+m2. sin2';l>
sin 2<f>..d¢.

E +n12gl cos
E-I-m2gl

goj, drf>.

Expressing the co-ordinates
co-ordinates x2
-"2 = x
x +l
r/>, y = l COS
cos go
r/> of the particle mg
m2 in
in terms
terms of
of go
r/>
Expressing
+ l sin of),
by means
means of
of (2),
(2), we
that its
its path
path is
is an
an arc
arc of
of an
an ellipse
ellipse with
with horizontal
horizontal semisemiby
we find
find that
axis
As ml
axis lmll(m1
lm1/(111I +1rr2)
+1112) and
and vertical
vertical semi-axis
semi-axis Z.
l. As
nz 1 -->
~ co
oo we return to the familiar
simple penfamiliar simple
dulum, which
which moves
mo,·es in
in an
an arc
arc of
of aa circle.
circle.
dulum,

§15. Kepler's problem

An important
which the
potenimportant class of
of central
central fields is formed
formed by those
those in which
the poteninversely proportional to
tor,
and the
the force accordingly
accordingly inversely
inversely
tial energy is inversely
r, and
They
the fields of
of Newtonian
gravitational
attracproportional
,.2._ The
Newtonian g
ravitational attracy include the
pr 0p ortional to 1·2.
tion and
and of
of Coulomb electrostatic interaction,
interaction; the
the latter
latter may be either attracattraction
repulsive.
tive or repulsive.
Let us First
first consider
consider an attractive
attractive field, where
Let

U=
= -a/r
-CI./r
C/. a positive constant.
constant. The
The "effective" potential
potential energy
with O13
C/.
1lf2
U
on
MY
Ueflfr =
= »
·--+---+-

r

e

(15.2)

2mr'2.2
Zm:

1'

(15.1)

shown in Fig. 10. As r ->»
-+ 0, Ueff
Uerr tends to +
-+ oo
is of the form shown
+ oo, and as r ->~
it tends to zero from
from negative
negative values
values; for r =
= 1Vf2jm7.
minimum value
value
lt
]l~I2/nza it has a minimum
Uefl.
min
Ue
in
fi, m

J

=

:'.

(15.3)

2.'lf2.

21
Tll'Y.
-- m
y 2 »'2m0_.

L/df.»

r

FIG.
FIG.

10

It is seen at once
once from Fig. 10 that
that the motion
motion is finite
finite for
forE
and infinite
infinite
E << 0 and
fforE
o r E > 0.

of

36

of

§15

Integration of the
the Equations
Equations of 3-lotion
!Vlotion
Integration

The
tiThe shape of
of the path is obtained from
from the
the general
general formula
(14.7). Subs
Substiformula (I4.7).
tuting there ['
I' =
= - O!
o: rr and effecting
effecting the el
elementary
integration,
we
haYe
emcntary intmigration, we have
(J
I/r) - (1711/JI)
(Jlfr)(nw}1i)
cp = cos"1--~--cm;-l____
-·+constant.
9+ coxlstaut.
m~1- .
2mE+
Z1nE
+ -m2:~.2·)
II

J(

0

Iii:!
1112

,
/

)

Taking
putting
Taking the origin of
of ¢cp such
such that the constant
constant is zero,
zero, and P';ltting
f
= .U-'-frm.,
Jf'!·/mCJ.,
/

ee =

c

(2EJJ!!:m:x2)],
vy[l[1 ++ (21_a.112,,m;
l,

II

p

we
can write
equation of
of the path
path 8S
as
we can
write the equation

pjr = 1 +
co~ QS.
cp.
-I-ee CON

(15.-t)
(15.4)

(15.5)

equation of
of a comic
conic section
section with
with one focus at the origin
origin;, ZP
2p is called
This is the equation
latus rectum of
of the orbit
orbit and e the
the eceentrifity.
eccentricity. Our choice of
of the origin
origin of 96
cp
the lotus
seen from (15.5) to be such
such that the point
point where quo
cp =
= 0 is the point
nearest
is seen
point nearest
origin (called
(called the perihelion).
to the origin
equivalent problem
problem of
of two particles
particles interacting
interacting according to the
the law
In the equivalent
(15.1 ), the orbit
orbit of each
each particle
particle is a conic section,
section, with one focus at the centre
centre
(15,1),
of mass of
of the
the two particles.
of
It
seen from (15.4) that, if E < 0, then the eccentricity
eccentricity e < 1,
the
It is seen
l , i.e. the
orbit is an ellipse (Fig. 11
11)) and the motion is finite, in accordance
accordance with what
\Vhat
orbit
has been said
said earlier
earlier in this section.
section. According
According to the formulae
formulae of
of analytical
analytical
has
geometry, the major
major and minor
minor semi-axes
semi-axes of
of the ellipse
ellipse arc
geometry,
2
p/(1-e
- 62))
aa= 1),'(1

=

o¢.»'2.]EI,
:x/ZJEJ,

6 = I).'*~ / (1 - 62) = Ill, \ '(2m:EI).

1*'

j_j_______

2b
2b

!

.

Ir4Ir<--202o ._

FIG.

(15.6)

JF'

I
I
I

*I

11

least possible
value of
of the
the energy is (15.3),
(15.3), and
and then
then e =
= 0, i.e.
the ellipse
ellipse
The least
possible value
~i.e. the
ds
depen
becomes
major axis of
of the ellipse
ellipse depends
becomes a circle. It
It may be noted that the major

of the particle,
particle, and not
not on its angular
angular momentum.
momentum. The
only on the energy of
least and greatest
greatest distances from the centre of
of the field (the focus of
of the
least
arc
ellipse) are
Ymin ':= P,:(1
p/(1 +e)
= a(1
a(le),
l'max
= 1).-'(1
P:'(la(l +e). (15.7)
(15./)
flax =
-8)e) = a(1-l-6)train
+e) =:
--e),
can,
5.4), can, of
These expressions, with a and e given
given by (1
(15.6)
(15.4),
of Course,
course,
5. 6) and (1
E.
also be
obtained directly
directly as
as the
the roots
roots of
of the
the equation
equation Ueff(")
Uerr(r) = E.
also
be obtained
.--.'=

.--'.=

§15

Kepler's
problem
K€plet'5 Problem

37

The
The period
period T of
of revolution in an elliptical orbit is conveniently
conveniently found by
using the
the law of conservation
conservation of
of angular momentum
momentum in the form
form of'the
of the area
using
from zero to
integral (14.3).
(14.3). Integrating
Integrating this equation
equation with respect to time from
integral
T, we have Znzf
For an ellipse
2mf =
= TM, where ff is the area of
of the orbit. For
ellipse
ff == tab,
7Tab, and by using the formulae (15.6) we Find
find

T

=

==

27Ta312y'(mfC1.)
21-ra3/2V(m/a)

(15.8)
7TC1.'\ 1(m/21Ej3).
I2IEi3).
The
proportionality between
between the square
period and the cube
The proportionality
square of
of the period
cube of
of the
been demonstrated
linearr dimension
dimension of the orbit has already been
demonstrated in §10. It may
linea
also be noted
noted that the period
period depends
depends only on the energy
energy of
of the particle.
particle.
1 i t . If
ForE
;:;;. U0 the motion is info
infinite.
For
E ,>
If E > 0, the eccentricity e > 1, i.e. the
hyperbola with the origin as internal
internal focus (Fig. 12). The
The disthe path is a hyperbola
tance of
of the perihelion
perihelion from the focus is
=

P (@+ 1)
P/(e+
l

I

I

a(e1),
a(e-1),

Z
=

-x.!2E is the "semi-axis"
"semi-axis" of
of the hyperbola.
hyperbola.
'x,f'2E

(15.9)

II

where a =
1)
= p/(€2pf(e2-1)

Tlnin
1
Tmin =

'.7rOt1/(771

*-ole-I) -

FIG.
FIG.

I

l-

i

X
x

12

If E =
If
= 0, the eccentricity e
e =
= 1, and the particle
particle moves
moves in a parabola
parabola with
perihelion
p. This case
Ymtn =
= fp.
case occurs
occurs if the particle starts
starts from rest
perihelion distance
distance Armin
at in
finity.
infinity.
Th
e 1co-ordinates
The
of the particle
particle as functions
functions of
of time in the orbit may be
ordinates of
fo
und by
found
by means
means of
of the general
general formula
formula (14.6). They
They may be represented
represented in a
.
conv
enient parametric
convenient
follows.
parametric form as foll
ows.
Le
Lett us first consider
consider elliptical orbits.
orbits. With a and e given by (15.6) and (15.4)
we
\\·e can
can write the integral
integral (14.6)
(14.6) for the time
time as
=

II

pa,

1

II

=

I
J2IEI
ZIEI J \/[
V[
J
__
J
' 171
m

'm
a
ma

a
-;-

rrdr
dr
2 +(C1./IEI)r-(M
2 /2miEI}]
- rr2+
(a/|E|)r- (m2/2m|E:)]
rrdr
dr
'\/[a2e2- (r(r- a)2]
a)2] .·
1/[£I2€2-

"

Integration
Integration of
of the
the Equations
Equations of
of Ilfotfon
111otion

38

I
J I

cos §g converts the integral to
to
may
.
Jma3
sm g)
( 1- e cos
cost)
dg =
=
-;Ctt) +
+constant.
constant.
(1
E) do?
(§- enin

The obvious substitution
r - aa =
substitution r=
II

r.*

t =

J

ma3
ma3
--;a

§15

-- bae
e

a

If time is measured
measured in such a way that the constant is zero, we have the
following parametric
parametric dependence
dependence of r on tt::

3
(15.10)
= ,'(ma
,':x}(t-e~in.t},
x '(1»»@3,')(§0 simi).
the
the particle being at perihelion
perihelion at t =
= ().
0. The
The Cartesian co-ordinates
co-ordinates

tt

II

I - e cos f).
r=
= a(
a(1-ecost),

y axes being
IIe l to the
= r cos ¢,,
cfo, y =
= r sin cPt, (the x and
andy
being respectively
respectively para
parallel
x=

major and minor
minor axes of the ellipse)
ellipse) can
can likewise be expressed
expressed in terms of
of
major
g. From
From (15.5) and
and (15.10) we have
the parameter
parameter f.
ex
= P
-r =
= a('(cos§-e),
ex=
p-r
= a(1-el3)-a(1-e
a(1-e3 )-a(1-e cost)
cost)=
ae(cost-e);

equal to y(r2x3). Thus
y is equal
I(t2-.~,;2).

= He
a\ (1-62)
(1- e2 ) sing.
y =

x =
= a(cos§-e),
a(cost-e),

(15.11)

complete passage
passage round
round the
the ellipse
ellipse corresponds
corresponds to an increase
increase of
of gg from 0
A complete
tO
to 21-r.
27T.
Entirely similar
similar calculations
calculations for the hyperbolic
hyperbolic orbits
orbits give
Entirely

r =
1),
= a(e cosh§cosht-1},

tr

x =
= a(ea( e- cosh§),
cosh t},

y

==

~\/(ma3/a)(e
inhf-f),
\l(ma3fa:}(e ssinht-t},
do '(e2- 1) sinh§,

where the parameter
g varies
varies from
from - oo
oo to +
+ OO.
oo.
where
parameter g
Let us now consider
consider motion
motion in a repulsive
repulsive field, where
where
Let

U = air
a:/r

effective potential
potential energy is
Here the effective

(a
(ex > 0).

(15.12)

(15.13)

M2
a:
Uerr =
= --I-+-Uefi'
rr

2mr2
2mr2

and decreases
decreases monotonically
monotonically from
from + OO
oo to zero as r varies
varies from zero to
The energy
energy of
of the particle
particle must
must be positive,
positive, and the
the motion
alwa) s
infinity. The
infinity.
motion is alway
The calculations
calculations are exactly
exactly similar to those
those for the attractive Held.
field.
infinite. The
infinite.
The path is a hyperbola:
The
hyperbola:

p/r
pfr =
= -1-I-ecos¢>,
-1 +e coscfo,

(15.14)

and e are again given
given by
(15.4). The path passes
the centre of
of the
where Pp and
by (15,-4).
passes the
field in the manner
manner shown in Fig. 13. The
The perihelion
Field
perihelion distance is
(15
.15)
Ymin = pj(e-1)
a(e+ 1).
(15.15)
Vmln
p/(e- 1) = a(e+1).
time dependence
dependence is given
given by
by the
the parametric
parametric equations
equations
The time

=

rr =

a(e oos
cosht+
1),
h§+ 1).
a(e
xx =
osh§-1-e),
= a(c
a(cosht+e),

= v(ma3fcx)(e
v(»~w@)(e ssinht+~).
ing*-Q'
tt =
k £a,V'(82._._
1)
yy :~*
sin
=-· av'(e2-l)sinhg.

(15.16)
(1
16)

§15

39

Kepler's
Ixepler"s problem
rI'

To
To conclude
conclude this section, we shall show that there is an integral
integral of
of the mowhich exists
exists only
only in fields U ::::
= air
cxfr (with either
either sign
sign of
of of).
o:). It
tion which
It is easy to
direct calculation
calculation that the quantity
quantity
verify by direct

VXM
v xM+cxr/r
+ url

(15.17)

rs constant.
constant. For its total time derivative is v'
v xM
M+a
cxvfrcxr(v -· r)/13
r)fr3 or,
or,
v / r - ar(v
is
= mr xv,
xv,
since M =
mv
Putting my'
vanishes.
vanishes.

mr(v· v°)
v}-mv(r·
v)+cxv/r-cxr(v·
mdv- mv(r - v•)
+ avl7' - ar(v - rr),'r3.
),'t=*.

= arl13
o:rfr3 from
from the
the equation
equation of
of motion,
motion, "·e
that this
this expression
expression
we find
find that
yy

O
0

x
X

I

o (I +e)

F1G.
FIG.

1133

Th
e direction
The
direction of
of the conserved
conserved vector
vector (15.17)
(15.17) is along the major
major axis from

and its magnitude
magnitude is 0.€
o:e.• This is most simply
simply
focus to the perihelion, and
the focus
seen by considering
considering its value at perihelion.
seen
perihelion.
It should
should be emphasised
emphasised that the integral (15.17) of
of the motion, like M and
one-valued function
function of
of the state (position
(position and velocity) of the particle.
E, is a one-valued
We
that the existence
existence of such
such a further
further one-valued
one-valued integral
integral
\Ve shall see in §50 that
is due to the degeneracy of
of the
the motion.
motion.
PROBLEMS
P
ROBLEMS

PROBLEM 1. Find the time
time dependence of the co-ordinates
co-ordinates of aa particle
particle with energy
energy E
E = 0
PROBLEM
mov
ing in
moving
in a parabola
parabola in
in aa Field
field U
U = -air.
-rt.fr.

SOLUTION.

In the
the integral
intell'ral
In

t

We substitute r = M2(l 'l""02)/2m01
the required dependence :
r

x

ml

J

-I-172),

%P(1 -122),

rrdr
dr

1

1/[(2 a/m)r-(I\4'2,'m2)]

%P(1 +"'2), obtaining the following parametric form of

z
:v

V("¢P3/(1)-£°2(1

P'2-

+

§'22).

40
40

Integration
Integration

of
Motion
of the
the Equations
Equations of
of illation

§15

+

The parameter
parameter 12
'1 varies
varies from
from -oo
- oo to
to -l-oo.
oo.
The

PROBLEM
PROBLEM 2.
2. Integrate
Integrate the
the equations
equations of
of motion
motion for
for aa particle
particle in
in aa central
central Field
field
U =

(or
(rx

-a/r2

> 0).
0).

M (1- )]»
MJ(( -)1»

SOLUTION.
SOLUTION. From formulae
formulae (14.6)
(1+.6) and (l4.7)
(14.7) we have,
have, if
if go
r/> and
and ttare
appropriately measured,
measured,
are appropriately

JII/f
' ~/J
~J/_

_
(a) forE>
fur E > 0 and
and .M'2/2m
l'vf2/2m >
> a,
rx, 1 =

1'vf2 -2m :x
M2-Znzx

(b) for
forE>
M 2/2m <
< a,
rx, -1 =
(b)
E > 0 and M2l2m
(C) or

an

r

cases
In all three cases

1
r= -

2mE

Zma
2
cos
cos[r!>J ( 1- m")],
M2
M2

_

h

o /J(( ~')]~
-2mrx
- - 1 )] ,
M2
M2

2mlEI
2miEI cosh [ r/>
2m :x- M" 2
2m-m2

I

( ET.»2

M2
2m

Zmac
-2m:x
- - 1 )] .
1'vf2
MY

)]~

-I-0x

-

Qu:

E

2mm

2mE
in [ r/>
sinh
2mrx-l'.f2 s
Zma-Ilfz

r =

1
2 f2m <
f E
d ."fj,f
Il/I2l2M
(c) for
E .:::
< oz,
oc, -1 =
_i 0 and

2
2mE
-- mE

-

~rr

In
In cases (b)
(b) and
and (c)
(c) the particle "falls" to
the centre
centre along
along aa path
path which
which approaches
approaches the
the
to the

origin
as go
rf> -»
->- oo. The
The fall
from aa given
given value
value of
of rr takes
takes place
place in
in aa Finite
finite time,
time, namely
namely
fall from
n as
origi

: l ( ~2m- - )
M2

-"v(im)

J(

a_

M2
.

2m

)}.

PROBLEM
PROBLEM 3. When a small correction
correction 8U(r)
8U(r) is
is added to
to the
the potential energy
energy U
U == -a;'r,
-rx/r,
the paths
paths of
of Finite
finite motion
motion are
are no
no longer
longer closed,
closed, and
and at
at each
each revolution
revolution the
the perihelion
perihelion is
is disdisthe
a small angle 89b.
8¢. Find 896
8rf> when (a) 8U
8U = Blr2,
f3/r 2 , (b) 8U
8U = 'y/r3.
yfr3.
placed through a
SOLUTION. When
\Vhen rr varies
varies from
from Vmin
rmin to
to lrmax
rmax and
and back,
back, the
the angle
angle ¢>
r/> varies
varies by
by an
an amount
amount
SOLUTION.

(14.10), which we write as
as
(l4.10),

3

A¢,

tmax

QM

tmln

/[2m(E-U)-

M2

r2

]

dr,

to avoid the
the occurrence of
of spurious divergences. We
We put
put U
U = -rx/r~l~8U,
-rxfr+8U, and
and
iin
n order to
expand
the
integrand
in
powers
of
8U;
the
zero-order
term
in
the
expansion
gives
27T,
and
expand the integrand in powers of SU, the zero-order term in the expansion gives 2-rr, and
the Erst-order
first-order term
term gives
gives the
the required
required change
change 866:
8rf>:
the

a r

am

max

2m'8U dr

f ¢l2»»(E+~)

tmln

a

r

-

'

M2
r

2

II

8¢,

a

( M l,.28Ud
2m

-an-

71

1

(1)
(1)

where we
we have
have changed
changed from
the integration
integration over
over rr to
to one
one over
over ¢,
rp, along
along the
the path
path of
of the
the "un"unwhere
from the

perturbed'' motion.
motion.
perturbed"

In case
case (a),
(a), the
the integration
integration in
in (1)
(1) is
is trivial:
trivial: 856
8rf> = -2m8m[1M2
-27Tf3m/l'vf2 = -27rB/aP,
-27Tf3/rxp, where 2P
2p (15.4)
In
lrlgiven by
U =
the Iatus
latus rectum
rectum of
of the
the unperturbed
unperturbed ellipse.
ellipse. IIn
case (b) t=s
r 2 8U
= 'ylr
y/r and,
and, with
with 11/r~given
by
n case
is the
(15.5), we have
have 8g6
8rf> = -6'rraym2/.M'4
-61rrxym2/M4 == -6w/(1P2»
-6wy/rxp2.
(15.5),

CHAPTER
C
H A P T E R IIV
v

COLLISIONS B
BETWEEN
PARTICLES
COLLISIONS
E T W E E N PARTICLES

of particles
§16. Disintegration of

In
momentum and energy alone can
IN many
many cases the
the laws of
of conservation
conservation of momentum
used to obtain
obtain important results concerning
concerning the properties of
of various
various mechmechbe used
It should
should be noted
noted that these properties are independent
independent of
anical
cal processes. It
ani
the particular type of interaction between the particles involved.
Let us consider
consider a "spontaneous" disintegration
disintegration (that is, one not due to
Let
external
external forces)
forces) of
of a particle
particle into two "constituent parts", i.e. into two other
particles
which move
move independently
independently after the
the disintegration.
disintegration.
les which
partic
This process
process is most
most simply
simply described
described in a frame of
of reference
reference in which
which the
particle is at rest
rest before
before the disintegration.
disintegration. The law of conservation
conservation of
of momenmomenparticle
tum
the two particles
tum shows
shows that
that the
the sum
sum of
of the
the momenta
momenta of
of the
particles formed
formed in the
disintegration
disintegration is then
then zero;
zero; that is, the particles
particles move
move apart
apart with equal and
opposite
opposite momenta.
momenta. The magnitude
magnitude Pg
Po of
of either
either momentum
momentum is given by the
of conservation
conservation of
of energy:
law of
2
Po
Po 2
P02
P02
Ei
=
E1,+-+E2i+-;
Et = E1VI'
+E2i+
1
27721
27722
Zm1
Zm2

here 1711
Et and E
E21
m1 and mg
m2 are the masses of the particles, EH
2 t their internal
energies, and
and ET
Et the internal
internal energy
energy of
of the original particle.
particle. If
the "disenergies,
If eE: is the
integration energy", i.e. the difference
difference
Integration
c

(16.1)

= Ea - Eli Ear,

whi
ch mus
which
mustt obviously
obviously be positive,
then
positive, then

(- +2_)) -

1
2 1
1
P02
=
f.Po2(~
P2mo2,
(16.2)
= aP0 7721 +1722
2m
m1 m2
which determines
determines Po,
Po; here
here m
reduced mass of
of the two particles. The
The
which
m is the reduced
velocities
Po/"2b u20
velocities are U10
v1o =
= Po/m1,
v2o =
· Po/m2.
P0/m2.
Let us now change
change to a frame
frame of
of reference
reference in which
which the
the primary
primary particle
Let
particle
mov
es
with
moves
velocity V
before the
the break-up. This frame is usually
usually called
called the
the
velocity
V before
labo
ratory system, or L system,
laboratory
system, in
in contradistinction
contradistinction to the centre-of-mass
in which
the total momentum
momentum is zero. Let us consider
consider
system, or C system, in
which the
one
of
the
resulting
particles,
one of the
particles, and
and let v and
and vo
vo be its velocities
the L and
velocities in the
the C system·respectively. Evidently v = V +vo, or v - V = vo, and so

II

E:
G

V2-2~L*V COS 9

J

=

(16.3)
(16.3)
le at which this particle moves relative
angle
relative to the
the direction of
of
where
re 6e is the ang
whe
equation
is
gi
ves the
the
the vel
velocity
This equation gives
the velocity of
of the
the particle as a function
ocity V. Th
'z,=2-|-

=.~02,

42

§16

Collisions
Collisions Between
Betwem Partieles
Particles

of its direction of motion in the L system.
system. In Fig. 14 the velocity
velocity vvis
repreis represented
vector drawn
drawn to any point
point on a circles'
circlet of radius
radius 'to
vo from a point
sented by a vector
A at a distance
distance V from the centre. The
The cases V < 'to
v 0 and V > 'to
v0 are
are shown
shown
in Figs. 14a, b respectively.
respectively. In the former
former case 6e can have any value, but
but in
the latter
particle can
can move only forwards,
forwards, at an angle 6ewhich
which does
does
latter case the particle
not
not exceed
exceed Gray,
emax• given
given by
sin
sin Gray
emax =

'Col
V,
'l.,'o/V;

(16.4)

this is the direction of
of the
the tangent
tangent from
from the point
point A to the
the circle.
circle.

. e
gm.!//

A

v

A

/f

59

/'

/'R

\

\

C'

v

\

\

\
\

v

\

\

.; w

x

J

*~._

_

_.

.

_.,.

"\

..I

(of V<v0

FIG.
FIG.

1»

""\.L
`*».

(b)V>Vo
V )V0
(b)

14

The
The relation
relation between
between the angles 6e and 60
eo in the
the L and C systems
systems is evidently (Fig. 14)
tan
sin 90/(v00 COS
60 + V).
tane6* =
= 100
vosineof(v
coseo+V).

equation is solved for cos 90,
eo, we obtain
If this equation

eo

cos 60 =
COS

-

V .
V s1n26*
sin2e i± cos 6e
vo
Oo

j(1 -V
vo

2
sin2e).
2

(16.5)

6.6)
(1
(16.6)

For 'zz0
vo > V the relation between
between 60
eo and 6e is one-to-one
one-to-one (Fig. 14a). The plus
plus
For
ver,
eo =
= 0 when 6
e=
= 0. If
however,
sign must be taken in (16.6), so that 60
If 'VO
to < V, howe
each value
value of 6e there
there are two values of
of 60,
eo,
the relation is not one-to-one: for each
which
correspond to vectors
vectors vo drawn
drawn from
from the
the centre
centre of
of the
the circle
circle to the
the
which correspond
Band
given by the two signs in (16.6).
points B
and C (Fig. 14b), and are given
tegration
physical applications we are usually concerned
concerned with the
the disin
disintegration
In physical
of not one but
but many similar
similar particles, and
a-nd this raises
raises the
the problem
of the
the
problem of
of the
the resulting
resulting particles in direction,
dire<;tion, energy, CtC.
etc. ~
distribution of
distribution
Wee shall
in
are randomly
randomly oriented
oriented
I.e. isospace, 1.e.
assume that the primary particles are
tropically on average.
tropically

.
.
f h'ich
h Fig. 14 shows
shows 3a diametral
diam t 1
More precisely,
precisely, to
to any
any point
pomt on
on aa sphere
sphere of
of radius
radms to,
vo, of
o wh
w 1c Fige ra
Tt . More

16
§§16

of patticles
particles
Disintegration of

43.
43

system, this problem
very easily solved:
solved: every
every resulting particle
In the C system,
problem is very
same energy,
energy, and their
their directions
directions of motion
motion are
are
(of a given kind) has the same
isotropically
isotropically distributed.
distributed. The
The latter fact depends on the assumption
assumption that
that the
primary particles
primary
particles are randomly
randomly oriented,
oriented, and can
can be expressed
expressed by saying
that
particles entering
that the fraction
fraction of particles
entering a solid angle element
element doo is proportional
proportional
to do0,
do 0 , i.e. equal
equal to do0/47-r.
doo/47T. The
The distribution
distribution with respect
respect to the angle 90
eo is
obtained by putting do
doo =
= 27-r
27T sin 60
eo d60,
deo, i.e. the corresponding
corresponding fraction is

!2

eo

1. sin 60 too.
deo.

(16.7)

(1j2m"L·oV) dT.
(1/2?.*zt':0V)

(16.8)

The
The corresponding
corresponding distributions
distributions in the L system
system are obtained
obtained by an
appropriate transformation.
transformation. For
appropriate
For example,
example, let us calculate
calculate the kinetic energy
distribution
in
the
L
system,
Squaring
the
equation
v0-I-V,
system. Squaring
equation v = vo
+ V, we have
distribution
2:2
V2+2Z'0V cos 60,
= d(o2)/2o0V.
v2 =
= 1702+
v02+ V2+2"L·oV
eo, whence d(cos 90)
eo)=
d(v2)/2voV. Using the
the
2
ml or m2
mg depending on which kind of
energy T =
= -21mz:2,
fmv , where m is m1
kinetic energy
particle is under
particle
under consideration,
consideration, and substituting
substituting in (16.'7),
(16.7), we find tlte
the redistribution:'
quired distribution

The
m(o0The kinetic energy
energy can take values between
between Train
T min =
= tm(
vo- V)2 and
Tmax
= §m(z20+
T max=
~m(vo + V):2._
V)2. The
The particles are, according
according to (16.8), distributed
distributed
uniformly over
over this range.
range.
uniformly
\\"hen a particle disintegrates
disintegrates into more than two parts, the
the laws of conWhen
servation of
of energy and momentum
momentum naturally allow considerably
considerably more freeservation
dom as regards
regards the velocities
velocities and
and directions
directions of motion
motion of the resulting
resulting particles.
dom
particles.
particular, the energies
energies of these particles
system do
go not have
In particular,
particles in the C system
determinate values. There is, however,
however, an upper limit to the kinetic energy
determinate
To determine
determine the limit, we consider
consider
resulting particles.
particles. To
of any one of the resulting
system formed
formed by all these particles
particles except
except the one concerned
concerned (whose
(whose
the system
denote the "internal
"internal energy" of that system
system by El.
mass is m
ml,1 , say), and denote
Et'»
energy of the particle
particle ml
m1 is, by (16.1) and (16.2),
Then the kinetic energy
2
T10
Tw =
= P02}211z1
Po f2ml =
= (M-1rz1)(Ei-E15-E¢')/M,
(M-ml)(Ei-Eli-Et')JM, where M is the mass of the
evident that
that T10
T1 o has its greatest
greatest possible
primary particle.
particle. It
primary
It is evident
possible value
For this to be so, all the resulting
resulting particles
particles except
except 7721
m1
when E/
Et' is least. For
moving with the same
same velocity. Then Et'
E/ is simply
simply the sum
sum of their
must be moving
Eli- Ei' is the disintegration
disintegration
internal energies,
energies, and the difference EiInternal
Et-E11-Es'
ene
rgy
energy e.
E. Th
Thus
us
T[0IM3X =-"

""

11l1)€l11/I.

(16.9)

P
ROBLEMS
PROBLEMS

• PROBLEM
PR_<JBLEM 1. Find
Fin? the
the relation between the
the angles
angles 61,
OI, 62
0 2 (in
(in the
the LL system)
system) after a disintegratio
tiOn
mto
two particles.
particles.
n int
o two

SOLUTION.
So~uriON. In
In the
the C
C system,
~ystem, the
the corresponding angles are related
related by
by B10
010 == 7vr-920.
7T-020 , Calling
Calling
formula (16.5) for
910
for each o
010 simply (90
Oo and using
us~ng formula
off the
the two
two particles,
particles, we
can put
put
we can
V-I-'<'J1o
o cos
V
+v1o cos (90
Oo ==
= U10
VIo gin
sin 80
fkr cot
cot 01,
01, V-'I.'2
V -v2o
cos Bo
Oo == U20
V2o sin
sin 60
Oo cot
cot 62.
02. From
From these
these two
two
equations
we must
equations we
must eliminate
eliminate 90.
Oo. To
To do
do so,
so, we
we First
first solve
solve for
cos 6!0
Oo and
and sin
sin 60,
Oo, and
and then
then
for cos

44

§17

Collisions Between
Between Particles
Particles
Collisions

form the
the sum of
of their squares, which is unity. Since O10,/u20
v1o/v2 0 =
= m2/rm,
m2/m 1 , we have
have Finally,
finally, using
(l6.2),
(16.2),
(m2/m1) sin2l92
+(m1[m2) siN291--2
sin202+(m1/m2)
sin201-2 sin 91
01 sin 92
02 COS(lO1
cos(01-I-92)
+02)
=
=

. I>(0 +02
si
sin-2(01I +02).
.
2
(m1+m2)V
(ml
V2 n
)
2E
2e

-I-7/12)

PROBLEM 2. Find the
the angular distribution of
of the
the resulting particles in
in the
the

system.
L system.

SOLUTION. When to
vo >
> V, we substitute
substitute (l6.6),
(16.6), with the plus sign of the radical,
radical, in
in (l6.'7),
(16.7),

do[

obtaining
obtaining

.n 60 d o[22-V
!é si
sm
V

vo
U0

11 -l~(V2/'Uo9'")
+(V2/vo2) cos 29
20 ]
v[1-(V2/vo2)
sin20)
x/[1 -(V2/1002) sin29]

cos 6+
0+--,--'----'--'----cos

0s .;;; ii).
rr).
(0 i.;;; 6

When 'UO
V, both possible relations
vo < V,
relations between
between 90
Oo and
and 9
0 must
must be taken
taken into account.
account. Since,
Since,
when 90 increases,
increases, one
one 'value
·value of
of 90
Oo increases
increases and
and the
the other decreases,
decreases, the
the difference (not the
the
of the
the expressions d cos 90
Oo with the two
two signs of
of the
the radical in
in (16.6) must be taken.
taken.
sum) of
result is
is
The result

sin 0 do

1 -l-(V2/vJo2) ms 26

(0 S.
.;;; 19
0 S.
.;;; 0m8
Omax).
X)-

x/[1 -(V2!V02) sin26]

Determine the
the range of
of possible values
of the angle 60 between the directions
PROBLEM 3. Determine
values of
of motion
motion of
of the
the two
two resulting
resulting particles
particles in
in the
the LL system.
system.
of

SOLUTION. The angle
angle 6
0 = 61-l~62,
01 +02, where 61
01 and
and 62
02 are the
the angles
angles defined by
by formula (16.5)
(16.5)
Problem 1),
1), and
and itit is
is simplest to
to calculate the
the tangent of
of 6.
0. A
of the
the extrema
extrema
(see Problem
A consideration of
of the
the resulting expression gives the
the following
following ranges of
of 6,
0, depending on
on the relative magniof
tudes of
of V,
vw and
and U20
v2o (for definiteness,
definiteness, we
we assume
assume v2o
v1o):
0 < 'or
7T if
if U10
vw < V < 2120,
v2o,
i): 0 < 6
no > v
tudes
V, '010
n--60
rr-Oo

<
< 60 <<

7r
7T

ifif V

<

Vin, 0
v1o,

<

6
0

<

60
V > 020.
Oo ifif V
v2o. The value of
of 60
Oo is given
given by
by

sin 60
Oo == V(vw+V2o)f(V2
+vwv2o).
sin
V('vlo -l~O20)/( V2 +'U1
0020).

§17. Elastic collisions
A collision between
involves no change
change
between two particles
particles is said to be elastic if it involves
in their
when the law of conservation
conservation of energy
energy
their internal state. Accordingly,
Accordingly, when
is applied
applied to such
such a collision, the internal energy
energy of the particles
particles may be
neglected.
neglected.
The collision is most simply described
described in a frame of reference in which
which the
The
(the C system).
system). As in §16, we
centre of
of mass of the two particles
centre
particles is at rest (the
distinguish by the suffix 0 the values of
of quantities
quantities in that
that system.
system. The
The velodistinguish
of the particles before the
the collision are related
cities of
related to their velocities
velocities v1
VI and
va
v 2 in the laboratory
laboratory system
system by V10
VlO =
= m2v/(m1+m2),
m2v/(m1 + m2), V20
v2o =
= -m1v/(m1-I-m2),
- m1v/(m1 + m2).
= v1-v2;
see (13.2).
where v ==
V 1 - V 2 , S€€
of the law of conservation
conservation of momentum,
momentum, the
the momenta
momenta of the two
Because of
particles remain
remain equal
equal and
and opposite after
after the collision,
collision, and
and are also unchanged
unchanged
particles
conservation of
of energy.
energy. Thus, in the C system
system
magnitude, by the law of conservation
in magnitude,
the collision
collision simply rotates
rotates the
the velocities,
velocities, which
which remain
remain opposite
opposite in direction
direction
the
denote by 110
no a unit
unit vector
vector in the direcdirecand unchanged in magnitude.
magnitude. If we denote
and
of the velocity
velocity of th
thee particle ml
m 1 after the collision, then the velocities
tion of
velocities
by priIne5)
primes) are
are
of the
the two
two particles
particles after
after the
the collision
collision (distinguished
(distinguished by
of
v10' =
= m2'z'n0i'(m1+m2),
m:tt•no,'(mi + m2),
vm'

V2o'
v20'

=

+

2

-mrlJnof(mi m ).
-ml'vI10,/("'1+M2)'

(17
.1)
(17.1)

§17

Elastic collisions
collisions
Elastic

45

In
In order
order to
to return
return to
to the
the LL system,
system, we
we must
must add
add to
to these
these expressions
expressions the
the
velocity V
V of
of the
the centre
centre of
of mass.
mass. The
The velocities
velocities in
in the
the LL system
system aft€~
after the
the
velocity
collision are
are therefore
therefore
collision

n 0 (7721 -|7712) + (7?21V1 +
??"£2V2ll(7IZ1 -|v1' = 17221)
m2vno/(m1
+m2)+(m1v1
+m2v2)/(m1
+
vi
I

vv2'
2
f

=
=

7722),
m2),

- ttlloflo
m1vnoj((Ml
m1 +
+ 1712)
m2) +
+ ("21v1
(m1v1 +7t22V2)/(Ml
+ m2v2)/( m1 +
+ Mg).
m2)·
-

(17.2)

No further information
information about
about the
the collision
collision can be obtained
obtained from the laws
conservation of momentum
momentum and energy.
energy. The
The direction
direction of the vector
vector no
of conservation
no
depends on the
the law of
of interaction
interaction of
of the particles
particles and
and on their
their relative
relative position
position
depends
during the collision.
The
The results
results obtained
obtained above may be interpreted
interpreted geometrically.
geometrically. Here
Here it is
convenient to use momenta
momenta instead
instead of velocities,
velocities. Multiplying
Multiplying equations
equations
more convenient
(17.2) by ml
m1 and
and mg
m2 respectively, we obtain

P1'
PI'
P2'
2I
P

+

+P2)/(ml +7722),
+ m2),
+P2)/(m1
-mvno+m2(p1
+m2),
-?iz'8n0+m2(
P1++p2)/(m1
p 2)/(Mr-I-1MH2),

= ming
mvno + 7721(p1
m1(P1
=
=
=

(17.3)

where
where m == ming/(ml-l~mg)
m1m2/(m1 +m2) is the reduced
reduced mass. We
We draw aa circle of
of radius
mv and
and use
use the
the construction
construction shown
shown in
in Fig.
Fig. 15.
15. IfIf the
the unit
unit vector
vector 110
no is along
mo
OC, the vectors
vectors AC
AC and CB
CB give the momenta
momenta pa'
p1' and
and pp2'
respectively.
2'

When
points A and B
When pr
p1 and pp22 are given, the radius of th
thee circ
circlee and the
the points

here on
are fixed,
but the
the point
point C
C may
may be
be any
anywhere
on the
the circ
circle.
are
fixed, but
e.
e

"E

PI

O

A-5

5§§=

tn"-1

2

oc~ m
mv
y
d"c=
Ml

'**z

MI+M2

( pl-. Pa)

( p I + P2)

FIG.
FIG.

15

Let
Let us
us consider
consider in more
more detail the case where
where one of
of the particles ('*1:Li)
(m 2 , say) is

at
ore the collision.
~t rest
rest bef
before
c?llisi?n. In that
t.hat case the distance
distance OB =
= 1112191/(1121
m2p 1f(m 1 +t712)
+m 2) =
= my
mv
is
IS equal
equal to the
the radius,
radms, i.e.
I.e. B lies
hes on the circle.
circle. The vector AB
the
AB is equal to the
momentum P1
Pl of
of the particle
particle ml
m1 before
before the
the collision.
collision. The point
point A lies inside
inside
or outside
<m
g or ml
outside the
the circle,
circle, according
according as 7721
m1 <
m2
m1 > mg.
m2. The corresponding
corresponding

46

§17

Collisions Be
Bttween
Particles
Collisions
tween Particles

diagrams
diagrams are shown in Figs. 16a, b. The
The angles 61
01 and 69
02 in these diag
diagrams
rams
are the angles
of motion
motion after
after the
the collision
collision and
and the
the
angles between
between the
the directions
directions of
direction
of impact
impact (i.e. of. P1).
Pl)· The
The angl
anglee at th
thee centre,
centre, denoted
denoted by X,
x, whi
which
direction of
ch
gives the direction
direction of no,
no, isis the
the angl
anglee thron
through
direction of
of motion
motion
which'the direction
g h which·the
of mi
ml is turned
turned in the C
c system. It is evident
evident from th
thee Figure
figure that 91
el and 92
e2
can be expressed
X by
expressed in terms of x
tan
el
n 61
ta

---r

. _ .

1112

(17.4)

::

C

am,s
*'

>' 5

0I

A

92

J

C

.P

-

s_ i_ x
m2smx
,
??21'l"??12
m1+m2 COSX
cosx

=

m 1 <mz
((o)
o ) ,f'NI<f.772

A-B=
XBopp.1

A

3x

15

61

(bl m1>f"2
fTIJ>mz
(b)

A0/OB : ml/m2
AO/OBo
fTIJ/mz

•;
|

FIG
FIG..

16

_

may give also the
the formulae
formulae for the
the magnitudes
magnitudes of the velocities
velocities of the
We may
x:
particles after
after the collision,
collision, likewise expressed
expressed in terms
terms of XZ
two particles
'Vl
01

If

=

2 +m2 2 +2mlm2 COSX)
(m12+m22+21t21m2
v(ml
cosx)
1721-I-H12

v,

102'

2m1'z'

N11+1Wa

1

sm EX'

(17.5)
(17.5)

The sum
sum 61+
81 + 62
82 is the angle between the directions
directions of motion of the
The
particles after the collision. Evidently 91
is-r if 7121
81 + H2
82 '3> irr
m1 < mg,
m2, and 91
Bt .|.+ 62
82 < !rrr
if 1721
ig.
m1 > Wm2.
When the two particles are moving
moving afterwards
afterwards in the
the same
same or in opposite
opposite
When
= w,
rr, i.e. the point
point C lies on the
(head-on collision),
collision), we have x
directions (head-on
X =
16b; p1'
p 2'
same direcdirecdiameter through
through_ A, and is on OA (Fig. 16b,
diameter
1 ' and P
2 ' in the same
16a; PI'
p1' and PA
p2' in opposite
opposite directions).
directions).
tion) or on OA produced
produced (Fig. 16a,
this case the velocities
velocities after
after the collision
collision are
In this

1?

Vt' =
V1

m1-m2

1121-M2

V,
V,

f

27721

Vo

(17
(17.6)
.6)

1721-l-N22
M1+M2
m1+m2
the maximum
.
•
d the
·• d e, an
and
maximum
This value
of v2'
v 2 ' has
has the
the greatest
greatest posslble
possib 1e magnitude,
magmtu
Thls
value of
f

n

'

Elastic
Elastic collisions
collisions

§17

47

energy which
which can be acquired
acquired in the collision by a particle
energy
particle originally at rest
therefore
is therefore
_
.r
_ 1
2
2
max - 1
2m2v2 ma
max
x E2 max
2772272

477217722
4mlm2

I

_

1

1

E1,

2E1,

(m1 +m2)
(1?21+1722)2

(17.7)

where
m1'zJ12 is the initial energy
where E1
£1 =
= la
}m1v12
energy of
of the incident
incident particle.
If
If 7721
m 1 < mg,
m 2, the velocity
velocity of
of 7/21
m1 after
after the collision
collision can have any direction.
direction.
If ml
through an angle
If
m1 > mg,
m2 , however, this particle can be deflected only'
only·through
nott exceeding
exceeding Hyrax
Bmax from its original
original direction,
direction; this maximum
maximum value of 61
01
no
corresponds to the position
position of
of C for which AC
AC is a tangent to the circle
corresponds
16b ). Evidently
(Fig. 16b).

sin Qomx = OC/OA = m2/m1.

(17.8)

at
The collision of two particles of equal mass, of which one is initially at
rest, is especially simple. In this case both
and A lie on the circle (Fig. 17)
both B
Band
17)..

p.
Vt

Y

A1

I

19l

O

I

I

I

I

I

1

I

e1 ==
61

lx.
;`X»

131'
'L'l = 'u
v
1

c:

p2

x

19

'\.._

Then

/

I

)

B

'

FIG.
F1G.

17

e2
62

cos tx.
f,

'UQ'
v2

1

l(rr-x).
l2(w-x)»
=
iX.
= 'u
v sin
sinh.

=

After
collision the particles
particles move at right angles to each other.
After the collision

(17.9)

(17.10)

PROBLEM
PROBLEM

Express the
the velocity
velocity of
of each
each panicle
particle after
after aa collision
collision between
between aa moving
moving particle
particle (mi)
(mt) and
and
Express
another at
at rest
rest (ma)
(m2) in
in terms
terms of
of their
their directions
directions of
of motion
motion in
in the
the LL system.
another
system.

1
1
SOLUTlON.
SOLUTION. From
From Fig. 16 we have P2'
P2 =
= 2OB
20B cos 62
02 or U2'
V2 =
= 21v(m/m2)
2v(m/m2) cos 62.
02. The momenmomentum 191'
Pt :=:
= AC
AC is given by OCR
OC2 == I102-l~P1'2-2AO
A0 2+pt'2-2AO.. al'
Pt' cos 91
Ot or
tum
to'1
M1-M2
Vt' ) 2 2m
2m Vt
mt -m2
61
- - - cos
COSOt+
= 0.
(- U
m2
m1-l~m2
v
m2 Uv
m1+m2
Hence

('u1')2

1

'U1I

v
v

_

M1
M1

+m2

+

cos 61i

ml

1

-l~m2

V(M22 -m12 sin291) ;

for
for m1
mt >> m2
m2 the
the radical may have either sign,
sign, but
but for
for M2
m2

> m1
mt it
it must
must be
be taken
taken positive.
positive.
>

48
48

Collisions Between
Between Particles
Particles
Collisions

§18

§18. Scattering
As already mentioned in §l'7,
§17, a complete calculation of the result of a
sion between
collision
between two particles
determination of
of the
the angle x)
x) requires
requires
particles (i.e. the determination
colli
the solution
particular law of interaction
of motion
motion for the particular
interaction
solution of the equations
equations of
involved.
involved.
We shall
shall First
first consider
consider the equivalent
equivalent problem
problem of the deHection
deflection of a single
particle of
particle
of mass m moving
moving in a Field
field U(r) whose centre
centre is at rest
rest (and is at
the centre
centre of
of mass of
of the
the two particles in the original problem),
problem).
shown in §14, the path
particle in a central
central field
As has been
been shown
path of a particle
Field is symmetrical
point in the
metrical about
about a line from the
the centre
centre to the nearest
nearest point
the orbit
orbit (OA
in Fig.
Fig. 18). Hence the two asymptotes
asymptotes to the orbit make equal angles (c/>o,
with this line. The
The angle X
x through
through which the
the particle
particle is deflected
deflected as it
say) with
passes the centre is seen from Fig.
Fig. 18 to be
passes
(18.1)
lrr- 2<;60IZr/>ol·
xX = I"
(¢)09

1

1

1
1
1

1

1
I
I
1

1I

1I

1I

1I

1I

1I

1
I

1I

1I
1I

iI

a
a

I
I

X
X

-_ AA

\

I

I

I

1

,f

/

I

_,.

_,.

/'

---------~~~---------1~-------------

i

________
-

0

-

v

-

-

-

#

-

\ /

C/1

I

.__-l¢_...__

.

/

n

-

-

-

..__._____Q"_\_¢b___-_______
e:_~~----------FIG.

18

-------------

The
The angle ¢>0
cf>o itself
itself is given, according
according to (14.7),
(14. 7), by
II

.3

c/>o=

I

oo
00

Tmtn
tmln

(Mfr2)dr
(Il///r2)d1r

'\/{2m[E- m2/t2}'
v{2m[E- U(t)]
U(r)]-M2fr2}'

(1
8.2)
(18.2)

taken between the nearest approach
approach to the centre and infinity. It should
should be
taken
recalled that rain
rmtn is a zero of the radicand.
radicand.
recalled
For an infinite
infinite motion, such as that
that considered
considered here,
here, it is convenient
convenient to
For
use instead
instead of
of the constants
constants E and JW
]1,1 the velocity 7300
'L'oo of the particle
particle at infinity
and the impact
impact parameter
The latter
latter is the
the length
length of
of the
the perpendicular
and
parameter p.
p. The
perpendicular
the direction of
of voo,
distance at which
which the
the particle
from the
the centre
centre 0O to the
from
Woo, i.e. the distance
particle
The
gy
1
if there
there were no Held
field of force (Fig. 18)
energy
would pass the
the centre
centre if
would
8)... !he ener
and the
the angular
angular momentum
momentum are
are given
given in
in terms
terms of
of these
these quantities
quanttttes by
and
n
1;

= '§mz1002,
!nn>oo2,
=

jlJ = Mpeoo,
mp<t'ro,
IW

(18.3))
(18.3

§18

49

Scattering
Scattering

formula (18.2) becomes
and formula
becomes
=

f

II

4>o
950

OO
00

r

I
Tmtn

(pfr2)dr
(p/t2
) dr
..
y[l(p2fr2)- (2U/mvoo2)]
V [1 -(p2lt2)-(ZU/"ww2)]

(1
8.4)
(18.4)

Together
X as a function of p.
p.
Together with ((18.1),
18.1), this gives x
In physical applications
applications we are usually concerned
concerned not
not with the deflection
deflection
of a single particle
but with the scattering of a beam of
of identical
identical particles
particles
particle but
voo
on
the
scattering
centre.
The
different
incident
with
uniform
velocity
incident
uniform velocity Voo
scattering centre. The
particles
impact parameters
therefore
particles in the beam have different impact
parameters and are therefore
scattered
scattered through
through different angles X.
X· Let
Let dN
dN be the number of particles
particles
x and x+dx.
x+dx. This number
number
scattered per
per unit time through
between X
scattered
through angles between
proporitself is not
not suitable
suitable for describing
describing the scattering
scattering process, since it is proporitself
of the incident
incident beam.
beam. We therefore
therefore use the ratio
tional to the density
density of
tional

do'
dO" = dNln,
dNfn,

(18.5)

do'
dO" = 2».rrp
27Tp do.
dp.

(18.6)

number of particles
through unit area of
where itn is the number
particles passing in unit time through
cross-section (the beam being assumed
assumed uniform over its crossthe beam cross-section
section).
section). This ratio has the
the dimensions
dimensions of
of area and is called
called the effective
entirely determined
determined by the form of the scattering
scattering
scattering cross-section.
cross-section. It is entirely
the most important
important characteristic
characteristic of the scattering
scattering process.
and is the
field and
process.
We shall
shall suppose
suppose that
that the relation
relation between
between X
x and Pp is one-to-one,
one-to-one; this is
scattering is a monotonically
monotonically decreasing
decreasing function
function of
of the
so if the angle of scattering
impact parameter.
parameter. In
In that
that case, only those
those particle's
impact parameters
parameters
impact
particles whose impact
lie between
p(x) and p(x)
p(X)+dp(X)
X and
+ dp(x) are scattered
scattered at angles between
between x
between p(x)
x+dx. The
The number
number of
of such
such particles
particles is equal
equal to the product
the
x+dx.
product of n and the
between two circles
circles of radii p and
and p-l~dp,
p + dp, i.e.
t.e. dN
dN = 211-p
2TTp dp .. n. The
The
area between
cross-section is
i~ therefore
therefore
effective cross-section

order to find the dependence
dependence of do'
dO" on the angle of scattering,
scattering, we need
need
In order
only rewri
rewrite
(18.6)) as
te (18.6

(18.7)
dO" == 2"p(x)ldp(x)/dxl
27Tp(x)ldp(x)/dxl do
dx.
do'
(18-7)
Here we use the modulus
modulus of
of the derivative
derivative op/dx,
dpfdx, since the derivative
derivative may
may
Here
Often do'
dO" is referred
referred to
to the
the solid
solid angle
angle element
element
usually is) negative.T
negative. t Often
be (and usually
of the plane angle element dxdx. The
The solid angle between zones
~ones
do instead of
with vertical
vertical angles
angles X
x and
and x+dx
is do
do=
27T s
sinx
dx. Hence
Hence we
we have
have from
with
X+ dX is
= 2».rr
i x dxfrom
(18.7)
dO"
do

ldpl

= p(x)
P_(x) ldp d
do.
six d
0.
stnxdx

(18.8)

p(X) is
ny-valued w
-|ress'o
t IfIf the
the function
funcltlionh p(bx}
is ma
mh anyf-val~d,
we mustt obviously
the sum off such
b •
ch expressions
ly ttake
k th
the
as (18
(18.7}
overr all
a t e branc
ranches
es of
o this
thiS funct
function.
as
ions o nous
.7) ove
a e
e sum o su
mm 1 ns

so
50

§18

Collisions
Collisions Between
Between Particles
Particles

Returning now to the problem
problem of the scattering
scattering of a beam of
of particles,
no t
particles, not
by a Fixed
fixed centre of
of force, but by other
other particles
particles initially at
at rest, we call
can say
that (18.7)
( 18. 7) gives the effective cross-section
cross-section as aa function of
of the angle of
of
that
scattering
scattering in the centre-of-mass
centre-of-mass system.
system. To
To Find
find the
the corre
corresponding
expression
sponding exp
ression
as a function of
of the scattering
scattering angle 6e in the laboratory
laboratory system, we must
must
express
express X
x in (18.7) in terms
terms of
of 6e by means of formulae (l'7.4).
(17.4). This
This gives
scattering cross-section
cross-section for the incident
incident beam of
expressions for both
expressions
both the scattering
particles
particles initially
(x in terms of
of 61)
81) and that for the particles
initially at rest
rest (X
(x in terms
terms
particles (X
of 62).
82)PROBLEMS

the effective
effective cross-section for scattering of
of particles from a perfectly
PROBLEM 1.
l . Determine the
of radius a (i.e.
(i.e. when the
the interaction
interaction is such that
that U =
= oo for r <
< a and U =
= 0
rigid sphere of
for r >a).
> a).
particle moves freely outside the
the sphere and cannot penetrate
penetrate into
into it,
SOLUTION. Since a particle
path consists of
of two
two straight
straight lines symmetrical
symmetrical about
about the radius
radius to
to the
the point
point where
where the
the
the path
particle strikes. the sphere
sphere (Fig. 19). It is evident
evident from
from Fig.
Fig. 19 that
panicle
r/>o = a sin !{1r-x)
= a cos
p =a
= a sin 560
§(1r-X) =

/'

1/1

g

ix.
!x-

FIG.
FIG.

V

§L¢°__-- ____-L________ _
I/

/-.,V

19

Substituting in
in (18.7)
(18.7) or
or (1S.8),
(18.8), we have
have
Substituting

do
da = re"
!1ra 2 sin
sin X do
dx = 1102
!a2 do,

(1)
(1`)

i.e. the
the scattering is
is isotropic in
in the
the C system.
system. On
On integrating do'
da over all
all angles,
angles, we find
find that
i.e.

2, in
which the
was,
the total
total cross-section
cross-section o'a == 1ra
in accordance
accordance with
with the
the fact
fact that
that the
the "impact
"impact area"
area" which
the
the
particle must
must strike
strike in
in order
order to
to be
be scattered
scattered is simply
simply the
the cross-sectional
cross-sectional area
area of
of the
the sphere.
sphere.
particle
In order to
to change to
to the
the LL system,
system, x
X must
must be expressed
expressed in
in terms
terms of
of 61
Ot by
by (l7.4).
(17.4). The
In
mbcalculations are
are entirely
entirely similar
similar to
to those
those of
of §l6,
§16, Problem
Problem 2,
2, on
on account
account of
of the
the formal
formal rese
resembcalculations
ticle
(16.5). For
For m1
mt < ma
m2 (where ml
mt is
is the mass
mass of
of the
the par
particle
lance between formulae (17.4) and (l6.5).
and ma
m2 that of
of the
the sphere)
sphere) we have
have
·
and

{/?,>§';2§11

1

2 cos 20t ]
2
+(mt/m2)
(
2(7i11lM2) COS 01+ 1 +(
( / )• . 20 ] do1,
dat == i"02[
!a2 [ · 2(1izt/m2) cos Ot +. /[
'[
dot,
V 1mt 1n2 - Sill I

d0'1

where do1
dot == 2-rr
27T sin 01
Ot d61.
dOt. If,
If, on
on the other hand, m2
m2
'

_
do'l
in"
dat =!a

< m1,
mt, then

1 -l~(m1/m2)2
+(mt/m2) 2 cos 201
20t d
do1.
ot.
V[1
-(mi/m2)22 SiI]26
V[1-(mt/m2)
sin 2 0]]

bt ·ned directly by subFor
we have
For m1
mt :=:
= ma,
m2, we
have day
dot == a2lcos
a 2 icos 011
Otl do1,
dot, which can
can also
also be
be obtain
o 31 ed directly by substituting x
= 201
201 from
from (17.9)
(17.9) in
in (1).
(1).
stituting
X =

§18

Scattering
Scattering

For
For a sphere
sphere originally at
at rest,
rest, X
x = 7r-262
TT-202 in
in all
all cases, and
and substntution
substitution in
in (1)
(1) gives

51

do'2 = aglcos 921 d o .

PROBLEM 2.
2. Express
Express
PROBLEM

effective cross-section
cross-section (Problem
(Problem 1)
1) as
as aa function
function of
of the
the energy
energy e€
the effective

lost by
by a scattered particle.

by aa particle of
of mass
mass Ml
m1 is
is equal to
to that gained by
by the
the sphere
sphere of
of
SOLUTION. The energy lost by
2 ) 10002
mass ma.
-l~m2)2]
V 0} sin2§x
m2. From (17.5)
(17.5) and
and (17.'7),
(17.7), e€ = Ez'
E2' = [2m12m2/(mi
[2mi2m2/(m1+m2)
sin 2!x = emu
€max sin2§x,
sin 2!x.
e =
nu. The
whence
m y sin X
whence de
d€ = ~€max
x do:
dx; substitntirig
substituting in
in (II,
(1), Prnhlem
Problem 1,
1, we
we have
have ddo
= 1rq2
Trq 2 dele
d€/€maxThe
scattered
scattered particles
particles are
are uniformly
uniformly distributed
distributed with
with respect
respect to
to e€ in
in the
the range
range from
from zero
zero to
to
Emax.
'max-

Find the
the effective cross-section as a function
function of
of the
the velocity 'Uno
Voo for
PROBLEM 3. Find
for particles
in a Held
field U
U ,._ r"".
rn.
scattered in

to (lO.3),
(1 0.3), ifif the
the potential energy is
is a homogeneous
homogeneous function
function of
of order
SOLUTION. According to
·k == -n,
-n, then
then similar
similar paths
paths are
are such
such that
that pp ,._ v"2I",
v- 2 /n, or
or pp == ve()-2/"f(X),
voo-2 fnJ(x), the
the angles
angles of
of deiiecdeflec'k
4 fn do.
_tion X being equal for
for similar paths.
paths. Substitution in
in (18.6)
(18.6) gives do'
da ,._ voo.tion
v¢0'4/"

PROBLEM 4. Determine
Determine the
the effective
effective cross-section
cross-section for
for aa particle
particle to
to "fall"
"fall'' tto
the centre
centre of
of
PROBLEM
o the
a field
U == -oc/r2.
-cr./r 2•
field U
a

to the
the centre are those for
for which 22cr.
> 771P2'U002
mp2 voo2 (see
SOLUTION. The particles which "fall" to
a >
2 ). The
(14.11)), i.e. for
for which the impact
impact parameter does not
not exceed Prix
pmax == y(2cr.fmvoo
(l4.1l)),
I(2e/me-..2).
2
2
effective cross-section
cross-section is
is therefore
therefore oa == 7Tpmax2
7rpmax = 2na,'mvw2.
2TTcr.,'mvoo •
effective
PROBLEM 5. The same as Problem 4, but for
field U
U = -air'
-cr.{rn (n
(n >
> 2, aa>
for a Held
> 0).

SOLUTION. The
The effective
effective potential
potential energy
energy Uerr
Uerr == mpl3v¢02/2r2-<1/y"
mp2voo 2 /2r 2 -cr./rn depends
depends on
on rr in
in the
the
SOLUTION.

= Uo =

in Fig.
Fig. 20. Its
Its maximum value is
manner shown in
Uerr,max E
Ueff,max

£{, ..
U"

2voo 2/rm)"/(n-2l.
!(n-2)cr.(mp
n-2) a(mp2'v¢<)8/atz)"/("'2).

T"

r

FIG. 20
FIG.

The
The particles
particles which
which "fall"
"fall'' to the
the centre are those for which
which Uo < E. The
The condition
condition Uo = E
gives pray,
Pmax, whence
whence
gives

o
o

_

2)(2-n) in( a/mUoO2)2/nI
= *rM(71
Trn(n-2)!2-n)fn(cr./mvoo2)2/n.

•

PROBLEM 6.
6. Determine the
the effective
effective cross-section for
for particles of
of mass
mass m1
m1 to
to strike
strike aa sphere
sphere
of
ss mg
of ma
mass
m2 and
and radius
radius R
R to
to which
which they
they are attrac
attracted
in accordance
accordance with
Newton's law.
law.
ted in
with Newton's

SOLUTION.
SoLuTION. The
The condition
condition for
for aa particle
particle to
to reach
reach the
the sphere
sphere is
is that
that ram
rmtn <
< R,
R where
where arm
r 1
is
the
h is
is the point
point on
on the
the path
path whic
which
is nearest
nearest to
to the
the centre
centre of
of the
the sphere.
sphere. The
The greatest
grea'test possible
posstbl:
value
; this
value of
of pp is
is given
given by
by fmln
rmtn = RR;
~his is
is equivalent
equivalent to
to Ueff(R)
Uerr(R) == E
E or
or Qm1voo2pmaxz
lm 1v 002Pmax2fR2cr.JR
R2_a/R
12 ('y
=}m1v.,.,2,
g the
=lm1voo2, where
w~ere a"' == 7'177
ym1m2
(y bein
beu~g
the gravitational
gravitational constant)
constant} and
and we
have put
put m
m z~ Ml
m1 on
on
we have
2, we
the
lving for
the assumption
assumption that
that ma
ma ~ m1.
m1. So
Solvtng
for Prna
Pmaxx2.I
fin.ally obtain
obtain Cr
a=
TTR2(1 +2'rmz/Rv=»")+2,m2/Rvoo2).
we Finally
= ..rrR2(1

>

52

§18

Collisions Between
Between Particles
Particles
Collisions

\Vhen Uoo
voo ->->- oo the
the effective cross-section
cross-section tends, of
of course
course,, to
to the
the geometrical
cross-section
-section
geometrical cross
of the
the sphere.
sphere.
of

PROBLEM 7. Deduce
Deduce the form
form of
of a scattering
scattering Held
field U(r),
U(r), given the
the effective
effective cross-section
cross-section
as a function
function of
of the
the angle
angle of
of scattering
scattering for a given
given energy
ener~ E.
E. It
It is assumed
assumed that
that U(r)
U(r) decrea
decreases
ses
monotonically
monotonically with r (a repulsive
rep•.llsive Held),
field), with U(0)
U(O) > E and U(
U( oo) = 0 (O.
(0. B. Fms
Fmsov
1953).
).
ov 1953
SOLUTION. Integration of
to the
of do'
da with respect
respe-ct to
to the
the scattering
scattering angle
angle gives, according
according to
the
formula
formula

f_l(du/dX)
(do/dx) dx
dx =
7T

x
X

7Tp 2,

(1)
(1)

the square of
of the
the impact parameter,
parameter, so that p(x)
P(X) (and therefore x(p)) is
is known.
known.
We put
put
eVe

s =
= llr,
1/r,

formulae (18.l),
(18.1), (18.2) become
become
Then formulae

II

%[*of-x(x)]

vr1[1 -(U/E)].
-<UJE)J.

zu =

x = 1/pn-...

1/(xzv2 -52)

(3)
(3)

a

where s0(x)
so(x) is
is the
the root
root of
of the
the equation :>:rv2(s0)
:~:ru 2 (so)-so
= 0.
0.
-8022 =
Equation (3)
(3) is
is an integral equation
equation for
for the
the function
and
function w(s),
zu(s), and
similar to
to that used
used in
in §l2.
§12. Dividing both sides of
of (3) by
by V
v(
a-x)
(a-x)
to xx from
from zero to
to a,
a:, we Find
find
to

JJ
II I

(2)
(z)

may be
be solved by
by a method
and integrating
integrating with respect
respect

a s0(x)
dx
. 7r-X(x)
TT-x(x) .
dx
dsdx
j"
ds dx
. so(x)
=
v[(xzu
-s2)(a-x)]]
J
J 2 . V(£x
1/[(xzv2-s2)(a-sr)
-x)
v(o:-;")
a

a
<X

ij
6

0

2

0

so(o:)
s'0(a) a
0

=

'or

7T

by parts on the
the left-Hand
left-hand side,
or, integrating by

0

x/ [(J\"*u2 - s 2 ) ( a - x ) ]

x(so)
A¢(S'o)

J --;--·
nu

so(o:)
so
•(a)
•

0

ds

-'-'s

. (a--36) ddxX dodx =
7T\1a- \l(a-x)7r\/a-J
dx
dx
a
a

dxds
dx
ds

7r
7T

Jj'

so(a)

f o r ) ds'
ds

•

.
--.--'_

ru
to

0

This relation is differentiated
differentiated with
with respect to a,
a:, and then
then s0(a)
so( a) is replaced
replaced by Ss siinply
simply;;
2
2
accordingly
a
is
replaced
by
s
fzu
and
the
result
is,
in
differential
form,
,
accoMingly
replaced
s2]zv3,
the result
differer rial
7T
*or

or
or

d(s/w)
-!d.(s2fw2)
a
w ) -%<1(#'2/2112)

--:rd
-TT d log
log ivzu =

f

nu?
s"
s12w2
0

Jj

s2fzc2
82/z1.:2

d(s,12v)
d(s/rv)

o0
-.

X'(x) do
x'(x)
dx

(TT/zu) ds
(for/zu)

_,.::....~-- =

v[(s2/zc2)-x]
1/[(s"/2¢2)-rv]
X'(x) dx
x'(.\:)
dx
V[(52/262)

-=*¢]

.

v

.
ation
the tight-hand
right-hand
0 n the
'on QD
-.
-.
.
.
.
This equation
equation can
can be
be nntegrated
integrated immediately
nnmedJately If
1f the
the order
order of
of lntegiatl
mte~ C e. U = 0)»
0), we ha
hav
Thns
ve,
(1'¢'
side is
is inverted.
inverted. Since
Since for
for s == 00 (i.e.
(i.e. rr--+
co) we
we must
must have
have nu
w =
e,
->~ OO)
side
r

=

'·

§19

53

Rutheftordk
/V,ltheiford's formula
formula

on
and p, the
following two
on returning
returning to
to the
the original
original variables
variables rrand
the following
two equivalent
equivalent forms
forms of
of the
the final
final

result:•
result

~ Jcosh- (p/rw) (dx/dp) do}
dp}
pH
rw

nu
w = exp{

1

:r

ao
00

1

cosh'.(p/rw) (dx/dp)

expo I M002
re

1 ao
1 Ioo

- exp (- -

7T
71'

rw
re

x(p) dp
dp

l.
}

;V (p2- 788202)
r2w2 )

(4)
(4)

•

This formula determines implicitly the
function w(r) (and therefore U(r))
for all
the function
U(r)} for
all rr

> ram,
rmtn,

in the
the range
range of
of r which
which can
can be
be reached
reached by
by aa scattered
scattered panicle
particle of
of given
energy E.
E.
i.e. i11
i.e.
given energy

§l9. Rutherford's formula
§i9.
One of the most
most important
important applications
applications of the formulae derived above is
scattering of charged
charged particles
particles in a Coulomb
Coulomb Held.
field. Putting
Putting in (18.4)
to the scattering
U ===
= a/r
cxfr and
and effecting the elementary
elementary integration, we obtain
obtain

-

c/>o ...
= cos
cos--11
Q50

cxfmvoo2p
P
,,
2
[1 +(°=/1'W<18p)2]
v[1
+(o:/mvoo p)2]
x/
(X/77172002

- x)
X) from (18.1),
p2 =
= (a2/m2-0004)
(cx2fm2v004) tan2¢>0,
tan2cf> 0 , or, putting
putting 960
c/>o =
= Hrrwhence per
p2

=
=

(0t2/m2°z:oo4)
(CJ.2jm2~'oo4) cot?-éx.
cot~x-

Differentiating this
this expression
expression with
with respect
Differentiating
respect to
or (18.8) gives
or

-

do -

"r

' OC

r

(19.1)

substituting in (18.7)
x and substituting

Ra' 2 2

ddar =
do/'sin4~§X.
= (0C/21i1'2,'c=o2).?'
('Y./2m'Doo2)2do/sin4ix·

(1
9.2)
(19.2)
(19.3)

This is Rutherford's formula.
formula. It
It may be noted that the effective cross-section
is
is independent
independent of the sign of
of of,
o:, so that
that the result
result is equally
equally valid for repulsive
repulsive
and attr
active Coulomb
attractive
Coulomb Fields,
fields.
•
reference
Formula (19.3) gives the effective cross-section
cross-section in the frame of reference
Formula
of mass of
of the
the colliding
colliding particles
particles is at rest,
rest. The
The transin which
which the centre
centre of
In
transform
ation to the laboratory
of formulae
formulae (1'7,4).
(17.4).
formation
laboratory system
system is effected by means of
Fo
particles initially
Forr particles
initially at rest we substitute
substitute X
x == 7-r-262
rr- 202 in (19.2) and obtain
obtain
dog
= Z7-r(o:,'mi°002)2
2rr(o:,'mi·oo2)2 sin 62
02 d02,'cos362
d02/cm;302
da2 =
=
(19.4)
= (cn/m'vo¢8)2
( o:fmvoo2)2d02lCOS302.
do2jcos302.

The same transformation
transformation for the incident
incident particles
particles leads, in general, to a very
very
The
complex formula, and we shall
shall merely
merely note two particular
complex
particular cases.
If
If the
the mass
mass mg
m2 of
of the scattering
scattering particle
particle is large compared
compared with the mass
m1
scattered particle,
particle, then x
X r-:c
~ 61
01 and m z~ #11,
tn1, so that
7721 of the scattered
dO'l
gl,
= (Q/4E1)2
(o:/4EI)2 do1sil'14do1/sin4J;.OI,
(19.5)
da1 =
(1
9.5)

2 iS
Where
m1ww2
where E1
£1=
= !m1voo
is the energy
energy of
of the incident
incident particle.
particle.

54

Collisions
Collisions Between
Between Particles
Particles

§19

= (a/E1)'2 cos 61 do1/sin461.

(19.6)

Jéml)
by
If the masses of the two particles
If
particles are equzd
equal (ml
(m1 =
= 7722
m2,, m
m =
= !m
then by
1), then
(17.9) X
x == 261,
201, and substitution
substitution in (19.2) gives
3 01
61d61/sin361
cos81d81/sin
= 21r(a/E
2r.(cx./El)
da1 =
1)'-82 cos
dol

If the particles
particles are entirely
rest cannot
If
entirely identical, that which was initially
initially at rest
c.annot
distinguished after the collision. The
The total effective cross-section
cross-section for all
be distinguished
and 62
particles is obtained
obtained by adding
adding d61
dol and dog,
dcr2, and replacing 61
eland
e2 by their
common value 6:
e:
common
1
1
do
E1)- . 4 +---- ) cos 6Odo.
(19.7)
do.
da = (cx/El)2(
sm 0
cos40
s1n49
c0549
II

c)

..

Let
Let us return
return to the general
general formul
formulaa (19.2) and
and use it to determine
determine the
distribution
distribution of the scattered
scattered particles
particles with respect
respect to the energy
energy lost in the
collision. When
When the masses of the scattered
scattered (ml)
(m1) and scattering
scattering (mg)
(m 2) particles
particles
latter is given in terms
terms of the angle
arbitrary, the velocity
velocity acquired
acquired by the latter
are arbitrary,
[2"'
1/(1
of scattering
Too sin 'l£X,
scattering in the C system
system by et'
v2' =
= [2ml/(ml
tx; see (1'7.5).
(17.5).
121 + m 2 )]~·oo
I
The
~§1t"2'i'12'2
The energy
energy acquired
acquired by mg
m2 and lost bby
m1 is therefore eE =
= }m
2v2'2
>€ Ml
2
- (2m2i'11z2)'v0<8
(2m2fm2)voo2 sin2-éx,
sin h. Expressing
Expressing sin lax
lx inin terms of
of eE and
and substituting
substituting
=
(19.2), we obtain
obtain
in (192),

mm

do

(19.8)

2s7'(0¢2 1?l2'z'0€2) d6/62.

required formula: it gives the effective cross-section
cross-section as a function
function
This is the required
of
max =
E, which
which takes values from zero
zero to Emax
= 2m2'voO22m2.
2m2voo2/m2.
of the energy loss e,
PROBLEMS
PROBLEM
S

=

PRODLE~l 1.
1. Find
Find the
the effective
effectiYe cross-section
cross-section for
for scattering
scattering in aa field
field U
U =
PROBLEM

SOLUTL0::-1. The
The angle
angle of
of detection
deflection is
is
SOLUTION.

The effective
effecti,·e cross-section
cross-section is
is
The

x

I

77

'\

2TT2J..
2-n"1

da = - - do
11lt':x:2
7?ZZ'sc2

aft?
afr 2

((a
a

0).
> 0).

1

{l -1- 2 OL 'mp2'v_1:F-'}

do
··-x2(2Tr-x)2
X°'(21r-~X)°- sin
Sill x
X
1rr-x
7"(-x

PROBLEM 2.
2. Find
Find the
the effective
effectiYe cross-section
cross-section for
for scattering
scattering by
by aa spherical
spherical "potential
"potential Well"
well"
PROBLEM

of radius
radius a and
and "depth"
"depth" Uo
Uo (i.e.
(i.e. aa field
with U
U = O
0 for
for r >
> aa and
and U
U == -U0
-Uo for
for rr <
<a).
of
field with
a).

SOLUTIO:-\. The
The particle
particle maxes
mo\'es in
in aa straight
line which
is "refracted"
"refracted" on
on entering
entering and
and leavleavSOLUTION.
straight line
which is
ing
the
well.
According
to
§7,
Problem,
the
angle
of
incidence
a
and
the
angle
of
refraction
ing the well. According to §7, Problem, the angle of incidence a and the angle of refraction
2 ). The angle of
(Fig. 21)
21) are
are such
such that
that sin
sin a'sin
a'sin B
{3 =
= n,
11, where
where 12
11 :=:
= "\1(1
+2Uo/mvoo
of deflection
deflection
Bf3 (Fig.
x/(1 +2
U0,1'm-0002).
X=
2(a-{3). Hence
Hence
is X
= 2(a--B).
sin(a-h)
in(a-x)
s

.

.

sm aa
sm
A

. . . - . _

=
= cos
cos $3-cot
!x-cot

.
sin ix
"'sm
!x

I

11

= -.
11
12

om the
ident fr
Elimin
Eliminating
from this
this equation
equation and
and the
the relation
relation aa sin
sin aa = p, which
which is
is ev
evident
from
the
ating a from
diagram, we
find the
the relation
relation between
between pp and
and x:
diagram,
we find
X'

#Lx
lx

n0°n 2 Sir12
sin 2
p'
p2o :== 02
a2'--=---:--c-=--.=.:.:..--:-- _ ,
l-2n
cos Q(
-2n COS

.,2+
n'-3-I-l

lx

§20

Small-angle scattering
scattering
Small-angle

55

Finally, differentiating,
differentiating, we
have the
the effective
effecti,·e cross-section
cross-section::
Finally,
we have

a 2n 2
!x-l)(n-cos ix)
!x)
a2n0~
(n cos by-l)(n-cos
d
do.
o' = - - - da=
2
--2n cos %x)2
4cos!x
(n +1
+1-2ncos!x)2
4
cos lx
(n°~

The angle x
from zero
a), where
where COS
X varies from
zero (for pp = 0)
0) to
to Xmas
Xma" (for pp =
=a),
cos ixrnax
!xmax = 1
1/n.
in.
The total
total eFfective
effective cross-section,
cross-section, obtained
obtained by
by integrating
integrating do'
da over
over all
all angles
angles within
within the
the cone
cone
The
X << Xmas,
xma", is,
is, of
of course,
course, equal
equal to
to the
the geometrical
geometrical cross-section
cross-section aTTam2..
X

r/

J/6'

°-..

"x

\

'He

';

/`

/

/

I

I

<21

1

,<@§»f"

I

1

/

I

1
\ /

/I

I

,140

T

'I

FIG,
FIG. 21

§20. Small-angle scattering
The calculation
calculation of the effective cross-section
cross-section is much simplified if only
The
those collisions are considered for \Yhich
which the impact parameter
parameter is large, so
and the angles of deflection are small. The
The calculation
calculation
that the field U is weak and
carried out in the laboratory system,
system, and the centre-of-mass
centre-of-mass system
system
can be carried
not be used.
need not
need
.
direction of the initial momentum of the scattered
scattered
We take the x-axis in the direction
particle ml,
by-plane in the plane
plane of
of scattering.
scattering. Denoting
Denoting by pa'
p1' the
particle
m1 , and the xy-plane
momentum
momentum of the particle
particle after scattering,
scattering, we evidently
evidently have sin 61
e1 == Plv'/Pl'·
Ply'/P7'.
For small
small deflections,
deflections, sin 61
e1 may be approximately
approximately replaced
replaced by 61,
e1, and Pl'
For
Pl' in
the denominator by the initial momentum P1
PI = 1lll'l'cc:
rfzltso :

e1
61

p

(20.1)
(20.1)

' / ~·
'of 1'z.'0@.
ly I:II
!LCC•
P1!/ff

Nex
t, sinc
Next,
sincee p,
py = Fy,
they-direction
y-direction is
F21, the total increment of momentum in the
~
~

"\./
'\./

F_y do.
If Fydl.

II

Plv'
Ply

=

QQ

(20.2)

-00
-oo

The
The force Fy
= -2U/331
-cUfoy =
= --(dUfdr)ctjoy
= -(dU/dr)_yi'r.
-(dUfdr)yfr.
Fe, =
( d Uldr)E'1'j3y =
Sin
ce the integral (20.2) already
Since
already contains the small quantity U, it can be
calculated, in the same
same approximation,
approximation, by assuming
assuming that the particle is not
calculated,
deflected
path, i.e. that it moves in a straight line by
deflected at
at all from its initial patl1,
y =
= pp
r,
with uniform
uniform velocity
velocity '<'oo.
Thus we
we put in (20.2) Fy =
= -(dU/dr)p,
-(dU/dr)p,r,
with
Thus
'1'00.
do
dt =
= dx!'?:0@.
dxfvcc. The result is
Ply'

p

'nu

oo

-of

d U do
dr

1' •

56

§20

Collisions Between
Between Particles
Particles
Collisions

Finally,
Finally, we change the integration
integration over x to one over
over r. Since,
Since, for a straight
straight
path,
= x2+p2.
x2+p2, when x varies
varies from
from - oo to + oo, r varies
varies from
from oo to p
pat
h, r2
V2 =
The integral
integral over
over x therefore
therefore becomes
becomes twice
twice the integral
integral over
over r
and back. The
2
from p to oo, and
and do
dx =
= r dtlV(""'"-p2).
dr/y(r -p2). The
The angle of
of scattering
SGattering 61
el is thus
given byt
by
oo

2p

61

9

7Tl1'U00"'

dU
dr

p

dr
w2-pQ)

1

(20.3)

and this is the form of
of the function
function 6*1(p)
OI{p) for small
small deflections.
deflections. The
The effective
cross-section for scattering
scattering (in the L system)
system) is obtained
obtained from (18.8) with
81
cross-section
with 61
instead
instead of
of X,
x. where sin (91
el may now be replaced
replaced by 91
el::
da
d
O'

|~~~ Pam
p(Ol) do1
do1..

=

dde1
6*1

(20.4)

61
e1

PROBLEMS

1. Derive formula (20.3)
(20.3) from
from (l8.4).
(18.4).
PROBLEM l.

I II/'[1f~/[1

SoLUTION. In
In order
order to
to avoid
avoid spurious divergences,
divergences, we
we write (18.4) in
in the
the form
SOLUTION.
form

</>o=
Q60

a
_!.._
op
up

R

92

p2-

72
r2

rmln
'min

2U ]dr,
]dr,

2U

mvoo 2
7710002

and take as the
the upper limit some
some large finite quantity R,
R, afterwards taking
taking the
the value
value as R
R ->-->- oo.
and
Since U
U is
is small,
small, we
we expand
expand the
the square
square root
root in
in powers
powers of
of U,
U, and
and approximately
approximately replace
replace
Since
7'm
rmtn
p:
ln by PZ
1560

f

R

p

p dr
r2v(l _p2/?'2)

+9

up

oo

U(r) dr

M'iJoo2v(1 -p2/12)

D

integral tends to
to !1r
giving
The first integral
Qfr as R-->R ->- oo. The second integral is integrated by
by parts, giving
17

2¢0

23

up

=r-

equivalent to
to (20.3).
(20.3).
This is equivalent

f

oo

p

V02-P2)

fI

77121002

2p
2p
mvoo 2
7730002

oo
00

p
p

dU

dr

dr

dU
dr
dU
dr
dr v(r2-p2)
dr
V( r2 -p2)

PROBLEM 2.
2. Determine
Determine the
the effective
effective cross-section
cross-section for
small-angle scattering
scattering in
in a Held
field
PROBLEM
for small-angle
U == air"
n > 0).
a./r" ((n
•

X is the
above derivation
derivation is
is applied
applied in
in the
the C system,
system, the
the expression
expression obtained
obtained for ~is
the
Tt IfIf the above
B1
and x are
8
same
with
m
in
place
of
mt,
in
accordance
with
the
fact
that
the
small
angles
an
X are
•
same with m in place of mi, in accordance with the fact that the small angles
related
= m2x/(m1+m2)ll
related by
by (see
(see (17.4))
(17.4)) 91
e.=
m2x/(ml +m2).

§20

57

Small-angle scattering
scattering
Small-angle

e.=----'---If

SOLUTION. From
From (20.3)
(20.3) we have
have
SOLUTION.

61 =

.

oo
00

Z
pxn
2pxn

mlv,2
7711U13"
9

p

dr
dr

,»n+1v(r2_p2)
,.n+Iy'(r2-p2)

•

The substitution
substitution p2]r2
p2/r2 = uu converts
com·erts the
the integral
integral to
to aa beta
beta function,
can be
be expressed
e"-"Pressed
The
function, which
which can
in terms
terms of
of gamma
gamma functions
functions:
in

o. =
61

2'XVTT
2w=»f

m1'v102p"'
7rllVoo 2pn

r(!nH)
.. r@»~»+5)
.

_

1"(,,,)
r(~n)

1

•

Expressing pp in
in terms
terms of
of 61
81 and
and substituting
substituting in
in (20.4),
(20.4), we
we obtain
obtain
Expressing
do

= 1- [
n

2 v'1rf(!n
\.',,F(&n +§)
- +!)
F({»n)
f(!n)

.

2!1f1~
91_2_2/n
"'Q; ]2/n
o~-2-2/n do1
dol..
2
m1voom1voo
a)

CHAPTER v
V
CHAPTER

SMALL
OSCILLATIONS
S
MALL O
SCILLATIONS

in one dimension
§21. Free oscillations in
A VERY common
common form of
of motion
motion of mechanical
mechanical systems
systems is what are called
called
small oscillations of
of a system
system about
about a position
position of
of stable
stable equilibrium.
equilibrium. We shall
of all the simplest
simplest case, that of
of a system
system with only one degree
degree
consider first
consider
First of
of
of freedom.
Stable equilibrium
equilibrium corresponds
corresponds to a position
position of the system
system in which
Stable
which its
potential
energy U(q) is a minimum.
minimum. A movement
movement away from this position
position
potential energy
results in the setting
setting up of
of a force -- dU/dq
dUfdq which
which tends to return
return the system
system
results
Let the equilibrium
equilibrium value
value of
of the generalised
generalised co-ordinate
co-ordinate
equilibrium. Let
to equilibrium.
qo. For
For small deviations
deviations from the equilibrium position,
q be go.
position, it is sufficient
expansion of the difference
retain the First
first non-vanishing
non-vanishing term in the expansion
to retain
U(q)- U(q0)
U(qo) in powers of
of qq- go.
qo. IInn genera
generall this is the second-order
second-order term
term::
U(q)U(qo) ;::: §1k(q-q0)2,
}k(q- qo)2, where k is a positive
positive coefficient, the va
value
of the
lue of
U(Q) U(Q0)
second derivative U"(q) for q = QUO»
q0 • We shall
shall measure
measure the potential
energy
potential energy
second
from
from its minimum
minimum value,
yalue, i.e. put
U(qo) =
= 0, and
and use the symbol
symbol
put U(g0)
" '

(21.1)
xX=
= Qq-{jo
QUO
for
the deviation
for-the
deviation of
of the co-ordinate
co-ordinate from its equilibrium
equilibrium value. Thus
'

U(x)

=
...=
'°

}kx2.
-kx2.

(21.2)

The
The kinetic energy of
of a system
system with
with one degree
degree of
of freedom
freedom is in general
general
of
=}a(q)xl2. In the
of the form -a(q)q
}a(q)q2
= !a(q)x2.
the same
same approximation,
approximation, it is sufficient to
°2 =
replace
= nu.
qo. Putting
Putting for brevity
brevityt a(q0)
a(q0 ) =
= m,
replace the function a(q) by its value at q =
expression for the Lagrangian
Lagrangian of
of a system
system executing
executing
we have the following expression
small oscillations in one dimension
dimension :t
zt

L

= J2mxl2
tmx2-!kx2.
-- lakx2.

The corresponding
corresponding equation
equation of
of motion
motion is
The
or
where

(21.4)

.if-I-cu2x = 0,

(21.5))
(21.5

/
w = v-vl(kfm).
(k/"2)-

(21.6)

II

m:2E+kx
mx+kx =
= 0,

(JJ

-ordinat€~
tT It
It should
should be
be noticed
noticed that
that m
m is
is the
the mass
mass only
only ifif xx is
is the
the Cartesian
Cartesian co
co-ordinate.

1!

(21.3)

Such a system
system is
is often
often called
called aa one-dimensional
011e-dimensional oscillator.
oscillator.
Such
58

§21
§21

Free
Free oscillations
oscillations in
in one
one dimension
dimension

59

Two independent
independent solutions of the linear differential equation (21.5) are
Two
and its general
general solution
solution is therefore
cos wt and sin wt, and
therefore

x

+

(21
.7)
(21.7)

= et
c1 cos
cos w
wtt -|- cg
c2 sin
sin wt.
wt.

This expression
expression can also be written
written

x = a cos(wt-l-a).
cos( wt + ()().

(21
.8)
(21.8)

II

Since
Since cos(wt+a)
cos( wt + ()() =
= cos wt cos a-sin
()(-sin wt sin a,
()(, a comparison
comparison with (21.7)
(21. 7)
shows
shOWS that the
the arbitrary constants
constants a and a()( are related
related to 61
Cl and
and C2
C2 by
1/(e12+I2l2),

II

a

tana

=

-

C`2!I(l1.

(21.9)

near a position
position of
of stable equilibrium,
equilibrium, a system
system executes
executes harmonic
harmonic
Thus, near
The coefficient
~oefficient a of the periodic
factor in (21.8) is called the
oscillations. The
oscillations.
periodic factor
oscillations, and the argument
argument of the cosine
cosine is their
their phase;
amplitude of the oscillations,
phase ,
initial value of
of the
the phase,
evidently depends
depends on the choice
choice of
a()( is the initial
phase, and evidently
the origin of time. The
frequency of the oscilThe quantity w is called the angular frequency
lations; in theoretical
theoretical physics, however, it is usually called
called simply
simply the frelations,
frequency, and we shall
shall use this
this name henceforward.
henceforward.
The frequency
frequency is a fundamental
fundamental characteristic
characteristic of
of the oscillations,
oscillations, and is
The
independent of
of the initial
initial conditions
conditions of
of the
the motion.
motion. According
independent
According to formula
entirely determined
determined by the properties
properties of the mechanical
mechanical system
system
(21.6) it is entirely
itself. It
It should
should be emphasised,
emphasised, however,
however, that
that this property of the frequency
assumption that the oscillations
oscillations are small,
small, and ceases to hold
depends on the assumption
higher approximations.
approximations. Mathematically,
Mathematically, it depends
depends on the
the fact that
that the
in higher
potential
energy is a quadratic
quadratic function of
of the co-ordinate.T
co-ordinate. t
potential energy
The energy
energy of
of a system
system executing
executing small oscillations
oscillations is E =
= !m.X2
The
2+
+ k!kx2
x2
=
!m(x2 + w2x2) or, substituting
substituting (21.8),
=~n1(:»E°2+w2xl3)
E = -émw2a2.

(21.10j
(21.1
0:

It
square of
of the amplitude.
It is proportional to the square
The time
time dependence
dependence of
of the co-ordinate
co-ordinate of
of an oscillating
oscillating system
system is often
The
conveniently represented
represented as the
the real part
complex expression
expression::
conveniently
part of a complex
II

exp(iwt)],
x = re [A exp(z•wt)],

complex constant,
constant; putting
where A is a complex

----

A === a exp(zl:x),
exp(i7.),

111
(21,
(21.11)

(21.12)
(21.12]

we return to the expression
expression (21.8). The
The constant
constant A is called
called the complex
ordinary amplitude, and its argument
argument is the
amplitude; its modulus is the ordinary
amplitude,
initial phase.
initial
phase,
use .of
exponential factors is mathematically
mathematically simpler
simpler than
than that of
.The
he use
of exponential
tr1g0I*01u@tr1cal
es be
they arc
are unchanged
differentiation.
trigonometrical on
ones
because
cause they
unchanged in form by differentiation.
t It
It

1
er, i.e.
· her Ord

highcr
)llg

. d good
U(x)
has at =v
+
function U
( x ) has
d 1f the functiOn
''"ttl1 goo if
.,..
" :>
> 22;, sec §11, Problem
Problcn1 2(a).
H

therefore do
es no
does
nott ho
hoild

u

~

..n

...

.

.

= 0 a minimum of
of

60

Small Oscillations
Oscillations

§21
§21

So long as all the operations
operations concerned
concerned are linear
linear (addition,
(addition, multiplication
multiplication
by
by constants,
constants, differentiation,
differentiation, integration),
integration), we may omit
omit the
th~ sign
sign re throughthroughand take the
the read
real part
part of
of the Final
final result.
result.
out and
PROBLEMS
P
ROBLEMS

Express the
the amplitude and
and initial
initial phase of
of the
the oscillations in terms of
of the
the
PROBLEM 1. Express
initial
initial co-ordinate xo and
and velocity
velocity to.
vo.

=

/w

SOLUTION.
SoLuTION. . aa= V(X02+'J02lW2)9
y(xo 2 +vo 2 2),

tan aa=
= -ool¢V-€*€0.
-vo/wxo.

PROBLEM 2.
2. Find
Find the
the ratio
ratio of
of frequencies
frequencies cu
w and
and w'
w' of
of the
the oscillations
oscillations of
of two
two diatomic
diatomic
PROBLEM
molecules consisting
consisting of
of atoms
atoms of
of different
different isotopes,
isotopes, the
the masses
masses of
of the
the atoms
atoms being
being m1,
m1, me
m2 and
and
molecules

' m2'.
'
m1',
m1,
m2.

SOLUTION. Since
Sipce the
the atoms
atoms of
of the
the isotopes
isotopes interact
interact in
in the
the same
same way,
way, we
we have
have kk = k'.
k'.
SOLUTION.
m in
in the kinetic energies of
of the molecules are
are their
their reduced
reduced masses.
masses. AccordAccordThe coefficients m
ing to
to (21.6)
(21.6) we
we therefore
therefon: have
have
ing

J

wr
w'

N'11tN2(tH1I-*-f]12I)
•mm2(m1' +m2')

rn1'nt2'(•m ~l-m2)
+m2) .·
m1'm2'(m1

-;:;
cu =

the frequency of
of oscillations of
of a particle of
of mass
mass m which
free -to
to
PROBLEM 3. Find the
which is free
along a line
line and
and is attached to a spring
spring whose
whose Other
other end
end is fixed at
at a point
move along
point A (Fig. 22)
at a distance Zl from the
the line. A force
the spring to length Z.
l.
at
force F is required to extend the

SA

S

S`

§5:L
I

I

m

X

F
IG • 2
FIG.
222

SoLuTION. The
The potential
potential energy
energy of
of the
the spring
spring is
is (to
(to within
within higher-order
higher-order terms)
terms) equal
equal to
to
SOLUTION.
2
)
the
force
F
multiplied
by
the
extension
8[
of
the
spring.
For
X
<·l
we
have
8[
=
v(i2+x
-l
the force F multiplied by the extension '61 of the spring. For x <i~l we have Sl = d(F+x2) -I
= x2]2l,
x 2/2l, so
so that
that U
U ==
= Fx2[2Z.
Fx2/2l. Since the kinetic energy is
is §mxl2,
~mx 2 , we have w =
= 1/(Flmf).
v(F/ml).
=

PROBLEM 4. The same as Problem 3, but
for a
but for
a particle of
of mass
mass m
m moving on
on a circle
circle of
of
radius rr (Fig. 23).

,,,,--t--- - ,
'

I

I

I

I

/f

in

4-r*I

/Q'

"s

J'I.vJ/

Ftc.
FIG.

23

\

§22
SoLuTio:-;.
SOLUTION.

Forced oscillations
oscillations
Forced
In this
this case
case the
the extension
extension of
of the
the spring
spring is
is (if
(if ¢</>
In

< 11))

61
61

Bl
= \"[t8+(l+r)2-2r(l+r)
Ill=
,'[r2 +(l+r) 2-2r(l+r) cos ¢>]-!
t/>]-l z~ r(l+r)g62[2l.
r(l+r)t/> 2/21.

V [F(r-l-l)]mrl].
is T == !mr2,p,
and the
the frequency is
is therefore w ==
= v[F(r+l)fmrl].
The kinetic energy is
l-rnr2.£'-3, and

PROBLEM 5.
5. Find
Find the
the frequency
of oscillations
oscillations of
of the
the pendulum
pendulum shown
shown in
in Fig.
Fig. 22 (§5),
( §5),
PROBLEM
frequency of
whose point
point of
of support
support carries
carries aa mass
mass m1 and
and is
is free
to move
move horizontally.
horizontally.
whose
free to
SOLUTIO:-/. For
For ¢,
</>
SOLUTION.

Hence

< 1 the formula derived in §14,
§l4, Problem 3, gives
s.

T =
= §1t211n2l2£2/(m1
~•mm2l2,f,2/(m1 -I-mz),
+m2),

U
U == émzglsbz!rmglt/> 2.

w = 1/ [g(m1 +ff12)/m1l]-

PROBLEM 6.
6. Determine
Determine the
the form
of aa curve
curve such
such that
that the
the frequency
frequency of
of oscillations
oscillations of
of aa
PROBLEM
form of
particle on
on it
it under
under the
the force
of gravity
gravity is
is independent
independent of
of the
the amplitude.
amplitude.
particle
force of

SoLuTIO:'<". The
The curve
curve satisfying
satisfying the
the given
given condition
condition is
is one
one for
the potential
potential energy
energy
SOLUTION.
for which
which the

of
of aa particle moving on
on itit is U
U = }ks
!ks 2,, where
where ss is
is the
the length
length of
of the
the arc
arc from
from the
the position
position of
of
equilibrium. The
The kinetic
kinetic energy
energy T ==
= *}m.§2,
!m.i 2, where
m is
is the
the mass
mass of
of the
the particle,
particle, and
and the
the frefreequilibrium.
where m
quency is
is then
then w = V
v(kfm)
whatever the
the initial
initial value
value of
of s.
s.
quency
(him) whatever
In aa gravitational
gravitational field
field U
U ==
= may,
mgy, where
where yy is
is the
the vertical
vertical co-ordinate.
co-ordinate. Hence
Hence we
we have
have
In
!ks 2 == mgy
mgy or
or J'
y = w~s
llut d.92
ds 2 == dx2+dy2
d.x2 +dy 21, whence
whence
M32
w2s22 '2g. But

xX=
1

f, [(d5/dy)2-1]
[(ds/dy)L1) do
dy= Jv[(g/2,;,
y)-1) do.
dy.
Iv'
I x/[(§/2w2J')-1]
1

:=

2

The integration
integration is
is conveniently
conveniently effected
effected by
by means
means of
of the
the substitution
substitution yy :=: : g(1-cos t)/4'*'2»
~)f4w 2,
The
1
2
which yields x'."C = g(
g(~
+sin
t), 4w
These two
two equations
equations give,
give, in
in parametric
parametric form,
the equation
equation
which
w'3.• These
form, the
sin §),'4
£ -Iof the
the required
required cun'e,
cun·e, which
is aa cycloid.
cycloid.
of
which is

§22. Forced oscillations
Let us now consider
consider oscillations
oscillations of a system
system on which
which a variable
variable external
external
Let
called forced
oscillations, whereas
those discussed
discussed in
force acts. These arc called
force
forced oscillations,
whereas those
§21 are free
oscillations. Since
Since the oscillations
oscillations are again
again supposed
supposed small, it
free oscillations.
that the external
external field is weak, because
because otherwise it could
could cause
cause the
implied that
is implied
take too large
values.
displacement x to take
displacement
large values.
The system
system now has, besides
potential energy
energy !kx2,
additional
The
besides the potential
-ékxg, the additional
t) resulting
resulting from the external
external field.
Expanding this
potential energy
energy Ue(x,
potential
U€(x, Z)
Held, Expanding
term as a series of
of powers
powers of
of the small quantity
quantity x, we have
additional term
addition:-d
U€(x,
U6(0, z)+
x é'U¢i3x x.-L10- The
Ue(x, t) ~ Ue(O,
t)+x[oUe/ox]x=O·
The first term is a function of time
time only,
and may therefore
therefore be omitted
omitted from the
the Lagrangian,
Lagrangian, as being
being the total time
time
of another function of time. In the second
second term
term --[3
- [o Uefox]x=o
derivative of
derivative
U6/3x]g;=0 is
external "force" acting
acting on the system
system in the
the equilibrium
equilibrium position,
and
the external
position, and
of time, which we denote
denote by F(t).
F(t). Thus the potential
is a given function of
potential
~nergy involves
involves a further term --xF(t),
-xF(t), and
ap.d the Lagrangian
Lagrangian of the system
system
energy
lS
is
L
L =
= !~rna2:2-9,-kx2+xF(t).
~m;~2-~·kx 2 +xF(t).

The corresponding
Corresponding equation
equation of motion
mxtkx =
= F(z)
F(t) or
motion is m5é+kx

.1)
(22.1)
(22

(22.2}
£+
F(z)ln,
(22.2)
x+ w'2x
w 2x =
= F(t}fm,
have
e \\`€
wher
where
we haYe again
introduced the
the frequency w of
of the free oscillations.
ag ai n introduced
ral soluti
The gene
general
f
h"
"nh
1"
d"rr
· 1 equation
•
on of
o this
t 1s 1inhomogeneous
omogeneous linear
tnear differential
tuerentla
equatton
solution
*FF1Cie11Nts
"th co
constan
coefficie
t .·IS
.
.
nstantt COC
with
1
W
s •s A:
x = x0+x1,
xo + x1 , where xo
x 0 1s
the general solution of
of
is the

62

Small Oscillations
Oscillations
Small

§22

corresponding homogeneous
homogeneous equation
equation and al
X1 is a particular
integral of
of
the corresponding
particular integral
inhomogeneous equation.
equation. In the present
case kg
xo represents
the inhomogeneous
present case
represents the free
oscillations 'discussed
"discussed in §21.
oscillations
Let
Let us consider
consider a case of
of especial
especial interest,
interest, where the
the external
external force is itself
itself
a simple
y:
simple periodic
periodic function of
of time, of
of some
some frequency
frequency -y:

F(r)
cos('yt -I-18).
F(t) = ffcos(yt+f3).

(22.3)

We seek a particular
particular integral
integral of equation
equation (22.2) in the form xi
x1 =
= b cos(-yt
cos(yt -l-B),
+ {3),
with the same
same periodic
factor. Substitution
Substitution in that
that equation
equation gives
with
periodic factor.
= f/m(w2-y2);
adding the solution
solution of
of the homogeneous
homogeneous equation,
equation, we
b=
f / M ( w 2 - ' ) / 2 ) , adding
general integral
integral in the form
obtain the general
obtain
z
a cos(wt
-I-B).
+ a) -I- [f/m(w2 - y`2)] cos('yt
x =a
cos(wt+ex)+[f/m(w2-y2)]
cos(yt+f3).

(22.4
(22.4))

The arbitrary
arbitrary constants
constants a and aex are found from the
the initial
initial conditions.
conditions.
The
Thus ·a system under
action of
of a periodic force
force executes
executes a motion
motion which
which
Thus'a
under the action
of two oscillations,
oscillations, one with the
the intrinsic
intrinsic frequency
frequency w of
combination of
is a combination
the
the system
system and one with
with the frequency
frequency y of
of the force.
force.
The solution (22.4) is not valid when resonance occurs, i.e. when the freThe
of the external
external force is equal to the intrinsic
intrinsic frequency
frequency cu
w of
of the
quency y of
quency
To find the general
general solution
solution of
of the equation
equation of motion in this .case,
case,
system. To
we rewrite
rewrite (22.4) as

x

=
- cos(wt +18)],
= a cos(w
cos( wtt -|+ex)+
[f/m( w2- 'y2)][cos(yt
y2)][cos(yt -I-B)
+ {3)+/3)],
a) + [f/m(w2-

where a now has a different
different value. As 'yy ->
-+ w,
second term
term is indeterminindeterminwhere
w, the second
of the form 0/0.
OfO. Resolving
Resolving the indeterminacy
indeterminacy by L'HospitaTs
L'Hospital's rule, we
ate, of
have
x =a
= a cos(wt
-l- fx)-I-(f/Zmw) zt sin(wt
+18).
cos(wt+ex)+(ff2mw)
sin(wt+f3).

(22.5)
(22.5)

amplitude of oscillations
oscillations in resonance
increases linearly
linearly with the
Thus the amplitude
resonance increases
oscillations are no longer
longer small and the whole theory
theory given
time (until the
the oscillations
above becomes
becomes invalid).
above
Let us also ascertain the nature of
of small oscillations
oscillations near
near resonance,
resonance, when
when
Let
y =
= 0u-I-€
w + E with eE a small
small quantity.
quantity. We
We put
general solution
solution in the com'y
put the general
plex
plex form
x = A exp(zCwt)
€)t] == [A
-|- B exp(z•d)]
exp(iwt) +
+ B exp[i(w
exp[i( w +
+ E)t]
[A+
exp(id)] exp(iwt).

(22.6)

quantity A
A+
exp(iEt} varies
varies only
only slightly over
over the
the period 21-r/w
27Tjw
Since the quantity
+ BB exp(z'et)
of the
the factor
factor exp(ioJt),
exp(iwt), the motion
motion near
near resonance
resonance may be regarded
regarded as small
small
of
of variable
variable amplitude.t
amplitude. t Denoting
amplitude by C, we have
oscillations of
oscillations
Denoting this amplitude
C =
= [A-l-B
lA + B exp(zld)
exp(iEt} 1.I. Writing A
A and B in the form a exp(z•a)
exp(iex) and b exp(iB)
exp(i/3)
respectively, we obtain
obtain
respectively,
C2 = a2.-|a2.+b2+2ab
cos(d+f3-ex).
CO
b2 + Zab cos(et+
,B -01).

The "constant"
"constant" term
terrn in
in the
the phase
phase of
of the
the oscillation
oscillation also
also varies.
varies.
tt The

(22.
2.7)
(2

63

Forced
Farced oscillations
oscillations

§22

Thus the amplitude
amplitude varies periodically
periodically with frequency eE between
between the limits
Thus
la-bl
Ia- b I S~ C S.
~ a-I-b.
a+ b. This phenomenon
phenomenon is called beats.*beats. 1- 1/ 'JL:
The equation of motion
motion (22.2) can be integrated
integrated in a general
general form for an
The
F(t). This is easily done by rewriting
rewriting the equation
equation
arbitrary extern:-d
external force F(r).
arbitrary
as

d
-(JE
+ i x ) - z•w(.:i:+ i x )
-(x+iwx)-iw(x+iwx)
dt
do

or
,where
~where

=

1
-F(t)
-FU)
m
m

d§ldf--iwf
dgjdt-iwg == F(t)/'m,
F(t)Jm,

(22.8
(22.8))

= .1*§+&uX
x+iwx
5g =

(22.9)

(22.8) is of
of the First
first order,
order. Its solution
solution when
complex quantity.
quantity. Equation
Equation (22.8)
is a complex
the right-hand
right-hand side is replaced
replaced by zero is 5
g=
= A
A exp(iwt) with constant
constant A.
&
before, we seek a solution
solution of
of the inhomogeneous
inhomogeneous equation
equation in the form
As before,
§g =
A(z)
exp(z•oJZ),
obtaining
for
the
function
.4(t)
= A(t) exp(iwt), obtaining
A(t) the equation A(z)
A(t)
=
= F(t)
F( t) exp(-iwt)/m.
exp(- iwt)fm. Integration
Integration gives the solution of
of (22.9)
(22. 9)::
=

of

1
exp(1•wt)
exp( - i t ) df +50 lI'
exp(iwt){f -F(t)
~F(t)exp(-iwt)dt+go
}•

II
~w»

g

t

m

0

(22.10)

where the constant
constant of
of integration
integration £0
go is the value of
of 5
g at the instant
instant tr =
= 0.
where
required general
general solution; the function x(t) is given by the imaginimaginThis is the required
of (22,10),
(22.10), divided
divided by oJ.T
w. t
ary part
part of
The energy
energy of
of a system
system executing
executing forced
forced oscillations
oscillations is naturally
naturally not conThe
source of
of the external
external field.
served, since
since the system
system gains energy
energy from the source
served,
Let us determine
determine the total energy transmitted to the system during
during all time,
Let
time,
According to formula (2210),
(22.10), with
assuming its initial
initial energy
energy to be zero. According
assuming
lower limit
limit of
of integration
integration - oo instead
instead of
of zero and with §(g(- oo) = 0,
the lower
-+ oo
co
we have for zt ->

lg(co):2
\§(00)£2

=

The energy
energy of
of the system
system is
The

E
E

~21

m2

I

00
oo

1

-00
-oo

F(r)
- i t ) dt
F(t} exp(
exp(-iwt)dt

r.

= %t~(x°2+<»2x2)
!m(x2 + w2x2) == %~t~!§%2.
~-m!~i2.

2

9
. obtain the energy transferred
Sub
stituting 1It(
Substituting
co) i-,
12,
we
transferred:I
g=( OO)
b e obtain

E
:::
E=

.t

1

~t5§
00

2~1 1J

2m

"

F(z)
exp(-ioJt)dt
F(t)exp(-iwt)dt~~;

-----co
e F(r)
'fhe forc
force
F(t) m
must
1t The
ust, o
0 ff cours
e, be
tten id
•
course,
be wri
written
in" real form.
forum.

(22.11)
(22.11)

(22.12}
2)
(22.1

64

§22

Small Oscz°IIatz•o11s
Oscillations
Small

it is determined
determined by the squared
squared modulus of
of the Fourier
Fourier component
component of
of the
force F(z)
F(t) whose frequency
frequency is the intrinsic
intrinsic frequency of the
the system.
system.
In particular,
particular, if the external
external force
force acts only
only during a time short
short in
m comparison with
exp( -iwt) ~ 1. Then
Then
parison
with 11fw,
W e we can put exp(-iwt)

E

=

f

t

_!__((f 1~Iz)dr
F(t)dt
,

1
--2m

OO
00

-oo
-oo

result is obvious: it expresses the fact that
that a force of
of short duration
duration
This result
gives
bringing about
gives the system
system a momentum
momentum ¢ff F dr
dt without bringing
about a perceptible
displacement.
PROBLEMS
P
R oB L E M S

PR6BLEM l.
1. Determine the forced
of a system
system under a force F(t)
F(t) of
of the
the followPROBLEM
forced oscillations of

x

ing tforms,
at time tt = 0 the
the system
system is
is at rest
rest in
in equilibrium (x
(x = x' == 0):
0): (a) F
F = Fo,
Fo,
ing
o , ifif at
a constant,
constant, (b) F
F = at,
at, (c)
(c) F
F =
= Fo exp(-at),
exp(- at), (d)
(d) F
F = Fo
Fo exp(-at)
exp(- at) cos
cos Bt.
{3t.
a

SOLUTION.
-cos wt).
wt). The action of
SoLUTION. (a)
(a) xx = (F0,'mw'3)(1
(Fo/mw2)(1-cos
of the constant force results in
in a dis-

placement of
of the
the position
position of
of equilibrium
equilibrium about
about which
which the
the oscillations
oscillations take
take place.
place.
placement
(b) xx == (a,-'m¢u3)(¢ut-sin
(a;mw3)(wt-sin wt).
wt).
(b)

(c) xx =
= [F0g'm(w2-I-a3)][exp(-at)-cos
[F0 fm(w 2 +o:2)][exp(-at)-cos wt+(a,'w)
wt+(o:,'w) sin
sin or].
wt].
(c)

(d)
(d)

= Fof-(w2+°¢2-52)
Fo{-(w2 +o:2-(32) cos w¢+(<==/<»)(w2+a'2+B2)
wt+(rx/w)(w2+rx2+f32) sin
sin wt+
xx =
wt-p

.

+

+ m2-B3)2
+exp(at)[(
w2+ a2-52)
o: 2-(32) cos Bt{3t-2o:f3
sin B/]};'m[(w2
{3t]}/m[( w 2 -|o:2-(32)2 +4a2B2]
+4o:2{32).
-lexp(- at)
[(w'-2
2:15 sin

This last case is
is conveniently treated by
by writing the force id
in' the complex form
form
= Fo exp[(-a+iB)t].
exp[(- ex +i{3)t].
F =

the final amplitude for
for the
the oscillations
oscillations of
of a system under a force
PROBLEM 2. Determine the

which is zero for t < 0, Fot/T
> T (Fig. 24), if
if up to
to time
which
F0t]T for 0 < t < T, and Fo for tF :>
= 0 the
the system is at rest in equilibrium.
t =
F

I

I

/1

*B

I

I

I
I
I

r
l

FIG.
FIG.

f

24

SOLUTio!'!. During the interval O
0 < tt < T
by the initial
SOLUTION.
T the oscillations are determined by
(Fo/mTwB)( wt-sin wt).
wt). For tt > T
T we
in the
the form
condition as xx = (FolmTw3)(wt-sin
we seek a solution in
form
xx =
= e1
c1 cos w(t--T)+e2
w(t-1)+c2 sin
sin w(t-T)-}-F0;'mw2.
w(t-1)+Fo/mw2.

x

of x and
and x' at
at tt = T gives
gives et
Cl = -(F0,'mTw3)
-(FtJ/mTwB) sin
sin wT,
c2 == (Fo/mTw°)
(Fo/mTw8 ) X
The continuity of
wT, et
2
(1-cos
w1). The amplitude is aa=
v(c1 -I-e22)
(2Fo/mTw3) sin
sin }wT.
iwT. This is the smaller,
smaller,
+c22) = (2170/mTw3)
X (1
-cos wT).
= x/(612
the more
more slowly
slowly the
the force
force Fo
Fo is
is applied
applied (i.e.
(i.e. the
the greater
greater T).
the
T).
{or aa finite
finite

force Fo
PRoBLEM 3. The same as Problem
Problem 2, but
but for
for aa constant
constant force
Fo which
wh1ch acts
ac for
PROBLEM
(Fig. 25).
25).
time T (Fig.
•

ts

§23

Oscillations of
of systems
systems with
with more
more than
than one
one degree
degree of
offreedom
freedom
Oscillatzbns

65

SoLUTioN. As
As in
in Problem
Problem 2,
2, or
or more
more simply
simply by
by using
using formula
formula (22.10).
(22.10). For
Fort>
have
SOLUTION.
t > T
T we
we have

free oscillations
oscillations about
about xx == O,
0, and
and
free
c.__

s _=

~

=

FO

F~
iN
1~l

T

exp(iwt)J eexp(
dt
x p (-iwt)
- i w t ) dt
cap(iwt)l
o0

.F''
[1-exp(
-iwT)] exp(iw!).
e:\:p(iwt).
_
[1
-exp(-iwT)]
Fri

:cum
IW111

F
L

FO

f
------L---4T----------t
7'
I

FIG.

The
The squared
squared modulus
modulus of
of

25
25'

Q*g gives
gives the
the amplitude
amplitude from
from the
the relation
relation l£l2
lgl 2 == a'B8w2.
a 2 w2 • The
The result
result is
is
{2Fo.'mw2 ) sin
sin %wT.
!wT.
a = (2Fo./mw2)

PRoBLEM 4.
4. The same as Problem 2,
2, but for
a force F0t,'T
Fot/T which acts
acts between
betwee_n tt == 00 and
PROBLEM
for a
tt = T (Fig.
(Fig. 26).
F

f
------~--~---------,
rT

FIG.
FIG.

26

SOLUTI0::\1, By
By the
the same
same method
method we
obtain
SOLUTION.
we obtain

a == (F0,*lTmw3)v[w2T2--2wT
fFo!Tmw 3 )\ 1 [ w 2 T~-2wT sin
sin wT-I-2(1
wT+2(1-cos
a
-cos w1)].
wT)].

PROBLEM
2, but
for a
wt which
which acts
PRoBLEM 5.
5. The
The same
same as
as Problem
Problem 2,
but for
a force
force Fo
Fo sin
sin wt
acts between
between tt

and rt == T

= 27r]w
27Tf w (Fig.
(Fig. 27).

E

=0

F
F

r

SoLUTION,
SOLUTION,

FIG.

27

Substituting in
in (22.10)
(22.10) F(t)
F{t) == Fo
Fo sin
sin wt
Substituting
wt

integrating
integrating from 00 to
to T,
T, we
obtain aa == F0W.'MW2.
FoTT.'mw 2 •
we obtain

=

Fo[exp(iwt)-exp(
-iwt)]!2i and
and
Fo
[¢xp(iwt) -exp( --£wt)]/2i

scillations of
§23.
§23. O
Oscillations
of systems
systems with
with more
more than
than one
one degree
degree of
of freedom
freedom
'fhe theory of
of free
.illat
Iatmns
.
'th s degrees
d egrees of
f reedom is
.
The
free osc
ions
osctl
o f systems
systems WI
o f freedom
IS
of
with
anal
OgouS to
analogous
to tha
thatt given
given in
in §21
§21 for
the
case
s
=
1.
for the case 5 = 1.

66

§23

Small Oscillations
Oscillations
Small

Let
Let the
the potential
potential energy
energy of
of the
the system
system U
U as
as aa function
function of
of the
the generalised
generalised
co-ordznates
qi (i
(i == 1, 2, ...,
... , s)
s) have aa minimum for
for Qi
qi == Qin.
qw. Putting
co-ordinates Qs

qi-qiO
Qi
9:10

(23
.1)
(23.1)

=8»- Z ki:.:-*Ci-*FA-,

(23.2)

E

(23.3)

Xi=
Xi

small displacements
displacements from equilibrium
equilibrium and
and expanding
expanding U as a function
for the small
of the Xi as far as the quadratic
quadratic rems,
terms, we obtain
obtain the potential
potential energy
energy as a
of
positive
definite quadratic
quadratic form
positive definite

U

=

Le

where we again take
take the minimum
minimum value
value of
of the potential
potential energy
energy as zero.
Since the coefficients lack
XfXkI
!?ik and km
!?ki in (23.2) multiply the same quantity
quantity xix,,,
it is clear
k!.:i~
clear that
that they may always be considered
considered equal:
equal: km
kik =
= kki·
In the kinetic energy,
(see (5.5)),
energy, which has the general
general form 212"ik(Q)Qdik
!~aik(q)tiitik(see
we put ii
qi =
= Qin
qiO in the coeHicients
coefficients au.;
aik an
and,
denoting ¢1w(Q0)
aik(qo) by mu.,
mtk, obtain
obtain
d, denoting
definite quadratic
quadratic form
form
positive definite
the kinetic energy as a positive
-1
2

i,I:

Mike.

coefficients Mu.:
11lik also may
may always be regarded as symmetrical:
symmetrical: M
mik.
Tllkt·
The coefficients
i l l=: him.
Lagrangian of
of a system
system execut
executing
small free oscillations
oscillations is
Thus the Lagrangian
ing small

L
L

=

:::

l2(mikXiXk-kikXiXk)·
§(t12zk£z1'51.:- kfkxi-Tk )_

(23.4)

f,k
i,/8

now derive
derive the equations
equations of
of motion.
motion. To
To determine
determine the derivatives
derivatives
Let us now
involved, we write the total differential
differential of
of the Lagrangian
Lagrangian::
involved,

dL

t

+

- ki1.-1lkdNi).
- kfkxi
= 7;1 Z(??2£ki'5
2( miki.·i dig
dxk + M5ki/6
mikx,, dig
dxikik:l.'i dank
dxkkikxkdxi)·
as

i,k
i,k

Since the value of
of the sum is obviously
obviously independent
independent of
of the naming
naming of
of the
Since
can interchange
interchange iz' and
and k in the first
terms in the parensuffixes, we can
First and third terms
theses. Using
Using the symmetry
symmetry of
of mik
kik• we have
theses.
Min and kilt3

d JCi).
dL
dL =
= Z("¢iA:J?kdi°i-kiz.:91'k
2(mikXkdxi-ku,x~.:dxi)·

Hence

k

are therefore
therefore
Lagrange's equations are

2 "1i1.:X°n
mikXk + Z
2 kikxk
kikXk =
Z
k

cLjcxi

2 fmikx,,,
nikfi3

=

II

3Ll3xi
oLfoxi

k
L:

0

=

-

22ki1,xk.
kiI.:xk:»
k

... ,s);
(i == 1, 2, ...,s),

(23.5)
.5)
(23

they form
a set
set of
of ss linear
linear homogeneous
homogeneous differential
differential equations
equations with
constant
they
form a
with constant
coefficients.
coefficients.
As usual,
usual, we
we seek
seek the
the ss unknown
unknown functions
functions xk(t)
Xk(t) in
in the
the form
As
form

Xk == AA:
Ak
xi

.

exp(iwt),
€xP(1•c,o1f),
. d.
d etermme
b e determined.
where
Al.: are
where Ak
are some
some constants
constants to
to be

.. .- g ((23.6)
23.6 in
s u b stttutlfl
In the
the
Substituting
)

(23.6)
_{23.6)

§23

Oscillations of
of so/stems
systems with
with more
more than
than one
one degree
degree of
of fzreedom
freedom
Oscillations

67

(23.5) and
and cancelling
cancelling exp(z•wt),
exp(iwt), we obtain
obtain a set
set of
of linear
linear homohomoequations (23.5)
geneous algebraic equations to be satisfied by the Ak:
As:

2,(w2mik+
kik)AkJ.: =
§(
- wgmzk
+ keA:)A
k

0.

(23.7)

If this system
If
system has non-zero
non-zero solutions,
solutions, the determinant
determinant of
of the coefficients
coefficients
must vanish
vanish::
lkzk-w2f'Hu.:l = 0.
(23.8)

clzaracten"stic equation and is of degree s in 0,2_
w2. In
In general, it has
This is the e/zaracteristic
different real positive roots wox2
(e<. =
= 1, 2, ...,
... , s),
s); in particular
particular cases, some
some of
of
s different
w.,,2 (Rx
these roots
roots may coincide.
coincide. The quantities Wa
wa thus
thus determined
determined are the characclzaracthese
ten"stic frequencies or eigerq'?equencz•es
eigenfrequencies ollf
o_f the system.
system.
teristie
It is evident
evident from
from physical
that the roots of equation (23.8)
(23.8) are
It
physical arguments that
and positive.
positive. For
For the existence
existence of an
an imaginary
imaginary part
part of
of w would
would mean
mean
real and
presence, in the time dependence
dependence of
of the co-ordinates
co-ordinates we
Xk (23.6), and so
the presence,
of
the
velocities
Xk,
of
an
exponentially
decreasing
or
increasing
factor. Such
of
velocities in, of
exponentially decreasing
increasing factor.
factor is inadmissible,
inadmissible, since
since it would
lead to a time
time variation
variation of
of the
the total
a factor
would lead
energy E =
= U
would therefore not be conserved.
conserved.
energy
U++ T of the system, which would
The same result may
SO be derived mathematically.
may al
also
mathematically. Multiplying
Multiplying equation
(23.7) by As*
Ai* and summing over i, we have 2(-w21"@k+kik)Ai*/lk
~( -w2mik+kik)Ai*Ak = 0,
whence
whence 0,2
w2 =
= 2kze/1z*AL=l2"'lzkAi*/lk.
~kikAi*Ak/~mikAi* Ak. The
The quadratic
quadratic forms
forms in the numerator
numerator
denominator of this expression are real, since the coefficients
coefficients km
kik and
and
and denominator
Min
2kkfAzAk*
mtk are real and symmetrical: (ZkzA:Az*AA:)*
(~kikAi*Ak)* =
= 2kz1UA41¢*
~kikAiAk* =
= ~kkiAtAk*
= ~kikAkAi*·
Ek,,,,/1,,A,*. They
positive, and therefore
They are also positive,
therefore w2 is positive.t
positive. t
The frequencies
frequencies wa
wa having been
been found, we substitute
substitute each
each of
of them
them in
The
equations (23.7) and find the corresponding
corresponding coefficients Arr.
Ak. If
equations
If all the roots
of the
the characteristic
characteristic equation
equation are
are different,
different, the coefficients
coefficients As
Ak are prowa of
portional to the minors
determinant (23.8) with w
portional
minors of the determinant
w =
= Wa'
wa. Let
Let these
minors be 11ka·
solution of the differential
differential equations
equations (23.5) is
minors
A k a - A particular
particular solution
therefore Xl.:
Xk = 11,.aCa
exp(iwat), where
where Ca is an arbitrary
arbitrary complex
complex constant.
therefore
A;,¢aCa exp(zlwat),
The general
general solution
solution is the sum
sum of
of s particular
The
particular solutions. Taking the real
part,
part, we write

wh
ere
where

Xk
re.;

= re

Z
2: AA,flCu
/).kaCa exp(z•wat)
exp(iwat)
s

tz=l
a=1

0a
E-)a

=

-

2_Ll1A·a0a,
Aka®¢p

re[Ca exp(z`wat)].
exp(iwat)].

a
a

(23.9)

(23.10)

Thus the
the time
time variation
variation of
of each
each co-ordinate
co-ordinate of
of the
the system
system is
is aa supersuperThus

po
sition of s simpl
position
simplee periodic oscillations (91,
01> 92,
0 2, ...,
... , (98
0 8 with arbitrary amplitudes and
and phases
phases but
but definite
definite frequencies.
tudes
frequencies.

f
.. .. d fin.. ..
,• h h
..
The
fact
that
k tk .is
.Is posntnve
firom
h t. Th
de
Ku.:
definite
flni
f t th(2t3a2;;1uadratic
form Wit
with
the
definitio
( a quad ratic form
t e coeflicnents
coeffi c1ents
pos1t1ve
e 1te is
IS seen from
2 2 for real val
the
variables.
If
the
complex
quantities
As
are
written
the'[. :hty
as a~+ ~b ) for real values
If
A~:
ues of
. of•
,,
€Xpli¢h1 y as a»+ib,, we ha
eXP( ""+ihk)
again using
using the
the symmetry
symmetry of
of km
ktk,, Z`»k4/¢Ai
:Ekt~:At*Ax
AT< .=
= 2k¢k(a¢-abs)
:Ekt~:(at -lbt) xX
1, ) = Eh 11,";,' we;~ve,
ve, again
:><
ak
h<kb,b,., which
which is
is the
the sum
sum of
of two
two positive
positive definnte
definite forms.
forms.
x(-<1'=+'
ac
(k'a""'*+2""k6fb»,

r.; w,.

68
68

Small Oscillations
Oscillations
Small

§23

The question naturally
naturally arises
arises whether
whether the
the generalised
generalised co-ordinates can
can be
The
chosen
chosen in such a way that
that each
each of
of them
them executes
executes only one simple
simple oscillation.
o3cillation.
The
points to the answer. For,
The form of
of the general
general integral
integral (23.9) points
For, regarding
regarding
the s equations
equations (23.9) as a set
set of
of equations
equations for s unknowns ®.,,
0,., we can
express
express (91,
81, (92,
82, ...,
... , 98
0 8 in terms
terms of the co-ordinates
co-ordinates al,
.'1:1, xg,
x2, ...,
... , Xs~
x 8 • The
quantities @1
0 a may
may therefore
therefore be regarded as new generalised co-ordinates,
called normal co-ordinates, and they execute
execute simple
simple periodic oscillations,
oscillations,
called
called normal oscillations of the system.
called
The normal
normal co-ordinates
co-ordinates 9a
0 a are seen
seen from their definition
definition to satisfy the
The
equations

€~)a+ Wa2®a = 0.

(23.11)

%ma(®a22 -_ Wa2®a2)1
L =
= E
,Lima(0a
wa20a2),
L

(23.12)

means that
that in normal
normal co-ordinates the equations
equations of
of motion
motion become
This means
become s
independent equations. The acceleration
acceleration in each
each normal
normal co-ordinate
co-ordinate depends
depends
independent
e pendence is entirely
only on the value of that
that co-ordinate,
co-ordinate, and
and its time ddependence
entirely
only
determined by the initial values
\'alues of
of the co-ordinate
co-ordinate and
and of
of the corresponding
corresponding
determined
velocity. In
In other
other words, the normal
oscillations of
of the
the system
system are
are completely
completely
velocity.
normal oscillations
independent.
independent.
It
evident that the Lagrangian
Lagrangian expressed
expressed in terms of
of normal
normal co-ordinates
It is evident
sum of
of expressions
expressions each
each of
of which corresponds
corrLsponds to oscillation
oscillation in one dimendimenis a sum
sion with
with one of
of the frequencies
frequencies wa,
form
sion
wa, i.e. it is of the form
a
a

ma are positive
positive constants.
constants. ll\/Iathematically,
Mathematically, this means
means that the
where the Ma
transformation (23.9) simultaneously
simultaneously puts
puts both
both quadratic forms-the kinetic
transformation
potential energy
energy (23.2)--in
(23.2)-in diagonal
diagonal form.
energy (23.3) and the potential
energy
The normal co-ordinates are usually
usually chosen
chosen so as to make
make the coefficients
coefficients
The
of the squared velocities
velocities in the Lagrangian
Lagrangian equal
equal to one-half. This
This can be
of
achieved by simply defining new normal co-ordinates
co-ordinates Qx
Q x by

Then
Then

Qa

L

ma

to

(23.13)

1 ""(Q 2_ w 20 2).
= 1;
wa'-4
=2
HQ?
- L...;
a
a ,_,a
a
a
9

The above
above discussion
discussion needs
needs little alteration
alteration when
when some
some roots of
of the characThe
equation coincide.
coincide. The
The general
general form (23.9), (23.10)-of
(23.10)·of the integral
integral of
teristic equation
of motion
motion remains
number s of
of
the equations of
remains unchanged,
unchanged, with the same number
and the only difference
difference is that the coefl»-\-Qicicnts
coefficients !1k:x
corresponding to
terms, and
As, corresponding
multiple roots
roots are not
not the minors
minors of
of the determinant,
determinant, which in this case
case
multiple
vanish.t
vanish. t
·tmposst'b·thty
· of
f
Th e impossibility
Tt The
. o

the
· the
·
h'1ch contain
- po,~-c
rs of
of
the time
timeth$18
as
1w
terms 1n
in
which
pow€f5
t h e general
genera1 integral
mtegra
con tam
h 0 ws
- 8she
w s that
th at th
.
e
which
as the
the exponential
exponential factors
is seen
seen from
from the
the same
same argument
argument as
as that
that whtch
e
factors is
well as
·
h
ld
·
I
h
I
f
t'
f
energY¢nergY
of
frequencies are
are real:
real: such
sue terms
terms ·wou
v1o ate the
t e law
aw of
o conservation
conserva 1on o
frequencies
would violate

§23

Oscillations
Oscillations of
of systems
systems with
with more
more than
than one
one degree
degree of
offreedom
freedom

69

Each
Each multiple
multiple (or, as we say, degenerate) frequency
frequency corresponds
corresponds to a number
number
of
normal co-ordinates equal to its multiplicity,
but the choice
of normal
multiplicity, but
choice of these
these coThe normal
normal co-ordinates
co-ordinates with equal
equal w:<
enter the
ordinates is not unique.
ordinates
unique. The
Wa enter
and potential
potential energies
energies as sums 2Q,8
~Qa2 and
and 2Qa2
~Qa 2 which
which are transformed
transformed
kinetic and
same way, and they can
can be linearly transformed
transfo~:med in any manner
in the same
manner which
does not alter these sums of squares.
The normal
normal co-ordinates
co-ordinates are very
very easily found
found for three-dimensional
three-dimensional oscilThe
lations of
of a single
single particle
particle in a constant
constant external
external field. Taking the origin
origin of
of
Cartesian co-ordinates
co-ordinates at the point
energy U(x, y, z) is
Cartesian
point where the potential
potential energy
minimum, we obtain this
this energy
energy as a quadratic
quadratic form in the variables
x, y, z,
z,
a minimum,
variables x,
and the kinetic energy
m(i2+j'2+22) (where
energy T =
= lm(x2+j2+z2)
(where m is the mass of
of the
particle) does not depend
depend on the
th~ orientation
orientation of the co-ordinate
co-ordinate axes. We
therefore have only
only to reduce the potential
potential energy to diagonal
diagonal form
form by an
therefore
appropriate choice
choice of axes. Then
appropriate

L = -%m(s&2+Y2+22)-i§(k1x2+k2y2+k3z2),

(23.14)

L =.= Lo
LE + 2:Fk(t)xk,
§jF,,,(£)x,,
L

(23.15)

and the normal
normal oscillations
oscillations take place in the x, y and z directions
directions with freand
quencies
UJ1 = v(kl/m),
V(k1/m), We
we = v(kafm).
V023/m1. In
quencies Wl
w2 = 1/(k2/'"))
v(k2/m), wa
In the particular
2
case of a central
field
(k1
=
kg
=
kg
E
k,
U
=
§kr2)
the
three frequencies
(k 1
k2 = ka =
lkr )
frequencies
central
equal (see Problem
Problem 3).
are equal
The use of
of normal
normal co-ordinates
co-ordinates makes possible
reduction of a problem
problem
The
possible the reduction
of
than one degree
degree of freedom
freedom to a
of forced
forced oscillations
oscillations of a system
system with more than
series of problems
problems of forced oscillation
oscillation in one dimension.
dimension. The
The Lagrangian
Lagrangian of
of
series
including the variable
variable external forces,
force,s, is
the system, including
k

where L0
Lo is the Lagrangian
Lagrangian for free oscillations.
oscillations. Replacing
Replacing the co-ordinates
co-ordinates
where
Xk by normal
normal co-ordinates, we have
al:

where
put
where we have put

é2(Q
a

2»-

II

L

=

EMoQ ,
a

(23.16)

2:Fk(t)t1ka/vma.
;F,,,(¢)A,,,//m.

II

fa(t)
f(i)

~»2Q2) +

k

The corresponding
corresponding equations
equations of
of motion
motion
The

2Qa =
Qa+ Wa
= Mr)
fa(t)
Q-|w 2Q

each
each involve
involve only one unknown
unknown function Qa(t).
PROBLEMS

(23.17)

.
.. •. of
PROBLEM
PJ<OBLEM 1.
1. D
·• h two
L Detennin
degrees
of
whose
angian is
is
=€t€n"""1¢
l( . 2 e.. 2the
t h e oscnllatnons
oscdlatlons
o f a system
system •wlth
Wit
degrees
o f • ffreedom
reedo m whose
an
. twoone-dlmenslonal
•
2
gqaugncy
we
}(x2+y2)
*}wo2(x2
-I-y2)-I-ocxy
(two
identical
~a~freQUency wo co~ 1;;~ )-lwo."(x +y,+axy
one-dimensional systems of
f
el
C0\.1pl¢d
by
e~ge
P
by an
an interaction-acxy).
lnteractton-axy).
o

Lzglfr

C

70

§24

Small Oscillations
Oscillatiotts
.Small

2
SOLUTION.
w0O~'x
SoLuTioN. The
The equations
equations of
of motion
motion are
are 56+
.~+wo
." =
= ay,
ay, ;v"+w02y
.Y+wo 2y == ax.
ax. The
The substitution
substitution

(23.6) gives

Ax(w02-w2) = alqy3
Ay(¢"02""2) = URL
{1)
2 == w02-a,
The characteristic
characteristic equation
equation is
is (w02-w2)2
(wo2-w2)2 == as,
a 2, when
whence
wo2-a, w
wo 2+a. For
For
w12
w22
The
ce w1
22 == 0002-1-0.
cu
i , the
w == iwt,
the equations
equations (1)
{1) give
give Ax
Ax=
for w == we,
w2, Ax=
:=.- Ay, and for
Ax = -Ay. Hence :1e
x =
(Qi-I-Q2)[V2,
from the
(Qt+02)/v2, y == (Q1-Q*.2)W2.
(Ql-Q2)/v2, the
the ooefHcients
coefficients 1]V2
1/v2 resulting from
the normalisation
of the normal
normal co~ordinates
co-ordinates as
as in
in equation
equation (23.13).
(23.13).
of
For aex <<
~ 0002
w0 2 (weak
(weak coupling)
coupling) we:
we: have
have w
w0 |-j - ia/w0,
/;:cx/w 0 , W
w22 '*
~ to;
w 0 ; + ia[w0.
/;:cxfw 0 • The
The
For
w,1 ~ to
variation of
x
and
y
is
in
this
case
a
superposition
of
two
oscillations
with
a
of andy
in
of
:with almost
equal
most equal
frequencies,
i.e. beats
beats of frequency
w2 w 1 = a[w0
cxfw0 (see §22). The amplitude of
of y i sis a
cu,
frequencies, i.e.
frequency W2
minimum when
when that
that of
of'x
is aa maximum,
maximum, and
and vice
vice ·versa.
minimum
'x is
versa.

< 1, 962r/>2 < 1),1), the
the Lagrangian derived
derived in
in §5, Problem

PROBLEM 2.
2. Determine
Determine the
the small
small oscillations
oscillations of
o~ a
a coplanar
coplanar double
double pendulum(Fig.
pendulum (Fig. 1,
1, §5).
§5).
PROBLEM
SOLUTION.
SoLUTION. For small oscillations (951
{r/>1

1, becomes
becomes
1.

!{mt +m2)l12d~l2
+m2)1Nt 2 -I-%m2/22d22
+/tm212~2 2 +m2l1l2qI»IJ>2-=.l;(ml
+m21tl2</>t~2-Hmt +1122)g/1';f)12
+nz2)gltr/>t 2-§M2§l2qS22.
-}m2g12rp22•
L == %('*21

The equations
equations of
of motion
motion are
are
The

lm +/2't;2+§¢'2 =

+m2)grp1 =
= 0,

(nz1

+m2)11~1 +m2l2qii2+(m1
+m212~2+(m1 +M2)§5t>1
(011-I-1'112)/1$1

Substitution of
of (23.6)
{23.6) gives
gives
Substitution

2m212 = 0,
At(m1+m2){g-hw2)-A2w
A
I(M1-I-m2)(g -l1w'*) -A2W2m2l2

The roots
roots of
of the
the characteristic
characteristic equation
equation are
The
2 ==

Wt,2
w1,2-*
0

g

g

212211112
2mtltl2

2

0.

2 = 0.

-A11tw +A2(g-12w ) =
-A1z1w2+A2(g-l2w2)

{(ml
+nz2){h -I-12)
+12) ±
v(mt +mw[(m1
+nl2)V[(nzl -I-m2)(lI
+m2){1t -I-l2)2-4mIlIl2]}.
+12) 2-4mtltl2]}.
{(m1-I-m2)(l1
_+ M014I

As m1
->- oo the
to the values v(g/l|)
As
m1->the frequencies tend
tend to
y{g/1t) and
and x/(8/I2)9
y{g/12), corresponding to
to independent oscillations
osci!Iations of
of the
the two
two pendulums.
pendulums.
den~t

PROBLEM 3. Find
Find the
the path of
of a particle in
in a central Held
field U
U = §kr2
lkr 2 (called
{called a space oscillator).
oscillator).

in any
any central Held,
field, the
the path lies in
in a plane,
plane, which we take
take as the
the by-plane.
:11y-plane.
SOLUTION. As
As in
The
x, y is
The variation
variation of
of each
each co-ordinate
co-ordinate ·"•
is aa simple
simple oscillation
oscillation with the
the same
same frequency
W =
= Vy(k/m):
a cos(w1!-I-a),
cos(wt+a), y == b cos(wt-I-B),
cos(wt+tl), or
or ,'\:=a
COS 96,
rp, y == b cos(~;f)-I-8)
Cos{rp+8)
x = a cos
cu
( k [ m ) : xX=
= a
Solving for COS go
and
8cos
r/>-bsin
8sin 66,
r/>, where
where go
r/> == it-I-a,
wt+a, 8 == B-a.
fl-a.Solvingforcos
r/> and
and sin
sin ¢,
r/>and
= b cos 8
cos 96-b
sin Ssin
equating the
the sum
sum of
of their
their squares
squares to
to unity,
unity, we
the equation
equation of
of the
the path:
path:
equating
we find
find the
.
-

x2

a2
as

+ +
b2
b2
y2

2xy

2x;v
cos
- cos 88 = sin28.
sin28.
ab
as
1

This
with its
This is
is an
an ellipse
ellipse with
its centre
centre at
at the
the origin.t
origin.t When
When 8 = 00 OI'
or 1-r,
TT, the
the path
path degenerates
degenerates to
to aa
segment of
of aa straight
straight line.
line.
segment

Vibrations of
of molecules
§24. Vibrations
If
have
system of
of interacting
interacting particles
particles not
not in an external field, not all
If we hat
Fe a system
oscillations. A typical example
example is that of
of its degrees of freedom relate to oscillations.
molecules. Besides motions
motions in which the atoms
atoms oscillate
oscillate about
about their positions
molecules.
positions
equilibrium in the
the molecule,
molecule can
can execute
execute translational
translational
of equilibrium
molecule, the whole molecule
and rotational motions.
and
Three degrees
degrees of freedom
freedom correspond
correspond to translational
translational motion,
motion, and
and in general
general
the same
same number to rotation,
rotation, so that, of
of the
the 3n degrees
degrees of freedom
freedom of a molemolermed
atoms, 372
3nAn exception
exception is fo
formed
cule containing
containing n atoms,
- 6 correspond to vibration. An
cule
. energy
ld iiwith potential
' a H
h in
The
thatt the
the pat
path
field
fact tha
e fact
Tt Th
ial energy
nt
te
Po
th

already be
been
mentioned in
in §14T`
§14.
en mentioned
already

e

ed curve
U == }kr2ikr" is
is aa clos
closed
curve has
has
U

of molecules
Vibrations of

§24
§24

71
71

by molecules
molecules in which the atoms
atoms are collinear, for which there
there are only two
rotational
degrees
of
freedom
(since
rotation
about
the
line
of atoms is of no
rotational degrees of
rotation about
of
signiiicance), and
and therefore
therefore 312-5
3n- 5 vibrational
vibrational degrees
degrees of freedom.
freedom.
significance),
In
In solving
solving a mechanical
mechanical problem
problem of
of molecular
molecular oscillations,
oscillations, it is convenient
convenient
eliminate immediately
immediately the
the translational
translational and
and rotational
rotational degrees of freedom.
freedom.
to eliminate
The
momentum of
The former
former can be removed
removed by equating
equating to zero the total
total momentum
of the
molecule.
molecule. Since
Since this condition
condition implies
implies that
that the centre of mass
mass of the molecule
molecule
can be expressed
expressed by saying
saying that the three
three co-ordinates of the
is at rest, it can
ra =
= ra0'1'Ua1
rao + Ua, where
where Ra()
rao is the radius
of mass
mass are constant. Putting
Putting to
centre of
vector of
of the equilibrium
equilibrium po.sition
of the ath atom,
atom, and
and Ua
deviation from
vector
position of
ua its deviation
this position,
position, we have the condition
~mara = constant
constant = 2mata0
~marao or
condition Emara

Emaua = 0.

(24.1)

To eliminate
eliminate the rotation of the molecule,
molecule, its total
total angular momentum
To
momentum
must be equated to zero. Since
Since the
~he angular
angular momentum is not
not the total
total time
derivative
function of
of the
the co-ordinates,
co-ordinates, the condition
condition that it is zero
zero canderivative of
of a function
general be expressed
expressed by saying that some such function
function is zero. For
not in general
For
small oscillations,
oscillations, however,
however, this can
can in fact be done.
done. Putting
Putting again
small
ra =
= rao+Ua
rao + Ua and
and neglecting
neglecting small
small quantities
quantities of the second
second order
order in the
to
ua, we can write the angular
angular momentum
momentum of the molecule
molecule as
displacements up,

a

M
M

1
=

.Q
L"later
mara Xva
'Xva

=
~

2L Mario
a =
marao xX iita
1

f
(djdt)
1

EL marao X Ua.
Mara() X Up.

condition for this to be zero is therefore,
therefore, in the same
same approximation,
approximation,
The condition

2

N2araf) X

ua

0,

(2
4.2)
(24.2)

in which the origin may be chosen arbitrarily.
The normal
normal vibrations
vibrations of the molecule
molecule may
may be classified
classified according
according to the
The
corresponding motion
motion of the atoms
atoms on the basis
basis of a consideration
consideration of the symsymcorresponding
metry of the equilibrium
equilibrium positions
atoms in the molecule.
molecule. There is
metry
positions of the atoms
general method
method of doing
doing so, based
the use of group
group theory, which we
a general
based on the
discuss elsewherefi'
elsewhere.t Here we shall consider
consider only some
some elementary
elementary examples.
examples.
discuss
If
distinguish· normal
If all n atoms in a molecule lie in one plane, we can distinguish.»normal
vibrations in which
which the atoms
atoms remain
remain in that
that plane
those where they
vibrations
plane from those
The number
number of
of each
each kind is readily determined.
determined. Since,
Since, for motion
motion
do not. The
in a plane,
there
are
2n
degrees
of
freedom,
of
which
two
are
translational
plane,
Zn
freedom,
which
are translation2ll
and one rotational,
rotational, the number
number of
of normal vibrations
vibrations which leave
leave the atoms
atoms
and
vibrational
2n- 3. The
The remaining (3n-6)-(2n(3n- 6)- (2n- 3) =
= n
nin the plane is 2n-3.
- 33 vibrational
degrees of
of freedom
freedom correspond
correspond to vibrations
vibrations in which the atoms move out
out
degrees
of the
the plane.
of
Fo
ich
~orr 3.a linear
linear mo
molecule
we can
can distinguish
distinguish longitudinal
longitudinal vi
vibrations,
which
brations, wh
lecule we
I-naIntain
o:amtain the linear form, from vibrations
vibrations which bring the ato
atoms
out
of
line.
ms olla of line,
ce a
Sin
Stnceh.ahinotion
of
n particles
in
a
line
corresponds
to
n
degrees
freedom
n
parti
cles
n
in a line corresponds to degrees of fi€€d0m,
of
which
of
W IC One
one is
is t
I .
.
.
.
th
~ _
rans auonal, the
the number
number of
of vibrations
vLbrations which
which leave
leave th ee 3I0m$
atoms,
tl'aIlslational,
Sec
Qunntr.n,
tum
Quan
1t See
..

'lb

•~
•
anzcs7 §l00,
§100, Pergamon
Pergamon Press,
Press, Oxford
OxforU 1970.
I97b.
xWechanics

-l.t<~ech

72

§24

Small Oscillations
Oscillations
Small

in line is n -1.
number of vibrational
vibrational degrees of freedom
freedom of a
1. Since the total number
3n- 5, there
there are Zn-4
2n- 4 which bring
bring the atoms out of line.
molecule is 372-5,
linear molecule
These Z
2n-4
vibrations, however, correspond
correspond to only
only 7n-2
different freThese
n - 4 vibrations,
2 - 2 different
quencies, since
since each such vibration
vibration can occur
occur in two mutually
quencies,
mutually perp~cular
perp/endiCular
planes
through the axis of the molecule.
molecule. It
It is evident
evident from symmetry
syn;nnetry that
planes through
each
each such
such pair
pair of
of normal
normal vibrations have equal
equal frequencies.
frequencies.
PROBLEMST
PROBLEMSt

PROBLEM: 1.
1. Determine
Determine the
the frequencies
frequencies of
of vibrations
vibrations of
of aa symmetrical
symmetrical linear
linear triatomic
triatomic
PROBLEM

(Fig. 28).
28). ItIt is
is assumed
assumed that the potential
potential energy
energy of
of the molecule
molecule depends
molecule ABA (Fig.
only
only on
on the
the distances
distances AB
AB and
and BA
BA and
and the
the angle
angle ABA.
ABA.

.A3

2

{.
L

1 )

i n

l
r

1r

L{.

28
B
-F

1t

FIG.
FIG.

I
I

A
1U

28
28

l
'1

I

(al
(of

{b)
(be

{cl
(cl

SOLUTIO!'!. The
The longitudinal
longitudinal displacements
displacements xi,
-"1, x2,
.'1::2, pa
xs of
of the
the atoms
atoms are
are related,
related, according
according
SOLUTION.
to
to (24.l),
(24.1), by
by mA(x1
mA(-"1 -I-xg)-I-mgxg
+Xs)+mn-"2 = O.
0. Using
Using this, we eliminate xg
-"2 from
from the
the Lagrangian
Lagrangian of
of the
the
longitudinal motion
motion
longitudinal

=

+

II

2-lk1[(x1-X2)
+xa2)+lmnx2
L = !mA(X1
§mA(£122-I-X32)
+§mBX°2 -Btu
[(xi "-A'2)22 +(xa-X2)2),
(xg-x2)2]1

and use
use new
new co-ordinates
co-ordinates
and

= xiX1 -I-xza,
+xa,
=

Q. -'== xi
x1 -x3.
-xa. The resit
result is
Qs

1-'mA • 2+
•
mA •
ktl-'
k1 2
alp2
Kr
= --Q
..·+ -Q.2-Q
477152
QD..22'Q-, -Q.2,
4Q'
,
4 Q.2_._ 4771 Q~
4mn3
4
4mn 2
4
}J-MA

II

»~1

L

Qa
Q,

MA

•

2

1-' =
= 2m.4
2mA -I-ma
+mn is
is the
the mass of
of the molecule.
molecule. Hence
Hence we
see that
that Q,
Q,. and
and Q,
Q. are
are normal
normal
where p
we see
co-ordinates
co-ordinates (not
(not yet
yet normalised).
normalised). The
The co-ordinate
cO-ordinate Q,
Q,. corresponds
corresponds to
to aa vibration
vibration antiantisymmetrical about
about the
the centre
centre of
of the
the molecule
molecule (xi
{x1 = is;
xa; Fig.
Fig. 28a), with frequency
symmetrical
frequency
We =
Wa
= v(k1p.]m,4m13).
y(k1fL/mAmB). The
The co-ordinate
co-ordinate Q,
Qa corresponds
corresponds to
to aa symmetrical
symmetrical vibration (xl
(XI=
-xs;;
== -x3
Fig. 28b),
28b ), with frequency Wu
wn = V (k1/"1A)(k1/mA).
Fig.
The
ya, pa of
The transverse
transverse displacements
displacements LV1.
YI,Y2,Ys
of the
the atoms
atoms are,
are, according
according to
to (24.1)
(24.1) and
and (24.2),
(24.2),
related
related by
by m,4(y1-I-y2)+mBy2
mA(y1 +Ys)+mny2 = 0, y1
YI = pa
Ys (a symmetrical bending of
of the
the molecule;
molecule; Fig.
Fig. 28c).

v

282, where
The potential
potential energy
energy of
of this
this vibration
vibration can
ca.."'l be
be written
as lk2l
where 88 is
is the
the deviation
deviation of
of the
the
The
written as
2/282;
angle
from the
angle ABA
ABA from
the value
value or,
TT, given
given in
in terms
terms of
of the
the displacements
displacements bY
by 8 =-= [(:vi
[(YI-Y2)+CYs
-y2)]/l.
*Y2)+U3 '-.v2)]l!Expressing ;vi,
y1, ya,
Ys, ya
Ys in
in terms
terms of
of s,
8, we
we obtain
obtain ~he
Lagrangian of
of the
the transverse
transverse motion
motion::
Expressing
the Lagrangian
2+.Ya2) +1}tNB}?a2-$21282
+!mn;li2 2-!k2l282
L = !mA(Y1
'§mA(J512 +J732)
=
=

mAmB •

- 1 282 - l!1 k2l282•
'"""""z2S2-y@2z2s2,
4,..,
4/1

~hence the frequency
frequency is C032
Wa2 == -v(2k2,u]m4mB).
y(2k2fLfmAmB)•
whence

v0L'1<BnsH..
Calculations of
ofthe
vibrations of
of more
more complex
complex molecules
molecules are
are given
given by
by M.
M. V
Tt Calculations
the vibrations
V.•.VoL'~Sli­
molekauqé
iya
an
eb
ol
TEIN,
M.
A.
EL'YASHEVICI:I
and
B.
I.
STEPANOV,
Molecular
Vibrations
(Kokban:;;a
nw,I ul),
(K
Molecular
Vibrations
TEIN, M.
EL'YASH1a:V1 CH and
I. STEPANOV,
I1£ft¢"'"d
ucture: .l ra-re
Moscow 1949;
1949; G.
G. HERZBERG,
HERZBERG, Molecular
Mokcular Spectra
Spectra and
and Molecular
Molecular Str
Structure:
and
Moscow
Raman Spectra
Spectra Of
of Polyatomic
Polyatomic Molecules,
Molecules, Van
Van Nostrand,
Nostrand, New
New York
York 1945.
t94S.
Raman

§24

73

oj'molecules
Vibrations ofwzoleeules

The same as Problem
PROBLEM 2. The
Problem 1, but
but for a triangular molecule ABA (Fig. 29).

JY
II

'J'
I
I

A
A

I3

A
A

20

I
}2

I

I

--~----1

-._...._.,.,--

F

_.___

B
8
L

Nl

__--'I.__--__al-_-.-I

no'
gl

If

(0)

u

bl
~b)
1

./`

\

L

(cl

FIG.
FIG. 29

By (24.1)
{24.1) and (24.2) the
the xx and
andy
components of
of the displacements u of
of the
SOLUTION. By
y components

atoms are
are related
related by
by
atoms

mA(x• -I-2¢3)+mB2¢2
+x3) +mnX'2
7?1A(rI

= 0,

°--=

mA(y1 +.V3)4-maya
+Y3) +mny2 .-:
0,
7?1A(Y1
-= O,

{y1-Y3) sin
sin a-(x1
ex -(x1 +:KI3)
+X3) cos aex = 0.
(y1-yg)

The changes
changes Eh
t:1 and
and 812
812 in
in the
the distances
distances AB
AB and
and BA
BA are
are obtained
obtained by
by taking
taking the
the components
components
The
along these
these lines
lines of
of the
the vectors
U!-U2
and U3-U22
U3-U2:
along
vectors u1
-U2 and
-3'2) cos a,
811
-pa) sin
8[} = (xi
(xl-X2)
sin 0¢+(;v1
ex +b·l-J"2)
ex,

=
812 =

-(x3-x2)
sin a-}-(Y3
ex+(J•3-y2)
ex.
--(x3
-x2) sin
-;v2) cos a.

The change
change in
in the
the angle
angle ABA
ABA is
is obtained
obtained by
by taking
taking the
the components
components of
of those
those vectors
vectors perperThe
to AB
AB and BA
BA::
pendicular to
![{:"1-x2) cos 0¢-(J'1~'°;V2)
ex-(yl-Y2) sin a]
ex)+~[
-(xa-.Y2) cos
cos a-(y3-Y2)
ex-{ya-Y2) sin
sin a].
ex).
8 = i[(."1"XQ
-I-[-(x3-x2)

l

l

of the
the molecule is
is
The Lagrangian of

.

2-lki(8h 2+812 2)-lk•1282.
!mA(i:at2 +ila2) +'§MBil
+!mni:a•22-&k1(3l12+3122)-y¢2z2s2.
L == ;m(a.2+m2)

=

=

We
x1 --x3, Q52
We use the
the new co-ordinates
co-ordinates Qu
Qa = xi
Xl -I-x3,
+xa, qsl
qo1 = .Yl-Xa,
qs2 = ;vi
Yl +Ys+ys. The components
components
of the
vectors uu are
are given
given in
in terms
terms of
of these
these co-ordinates
co-ordinates by
by :q
= %(Q=»
l(Qa 4-qu),
+qal), x3
xa = &(Qa-qu).
l(Q., -qa~),
of
the vectors
xx =
X'2 =
= -"1AQoltt1B»
-mAQa/mn, y1
Yl =
= 1}(Q324-Qa
!(qs2+Qa cot
cot al),
ex), Ya
ya = §(Q62-"Qu
l(q•• -Q.. cot
ex), Y2
-mAqa2/mn. The
The
x2
or 01),
Ya = -"1AQ=2/*HBbecomes
Lagrangian becomes

LL

2mA

= tmA
= ( -nzn
+771B

=

1 ) . 2
1-'mA .. 22
P'MA
+ --qs2
+ sln2a
.
Qa "l'?7'/Aés12
+tmAqs12+
)Qa2
.
Q52 2
4m
B
sm ex
4mn

1

-}k1QG2(

2

772 A

2725

+

)(1+

1

. 2
Sin a

2Ml4
"IB

#2

-'q-912(k1 siI12a +2k2 cos2a)--1q82" H

+qs

1

Q32

f*

27723

-

.

mB2

(2162-A1) sun a cos a.

.

sln2a

>-

=

"""'

(k1 cos2a -I-2k2 singa) -I-

74
74

§25

Oscillations
Small Oscillations

Hence
Hence we
we see
see that
that the
the co-ordinate
co-ordinate Qu
Qa corresponds
corresponds to
to aa normal
normal vibration
vibration antisymmetrical
antisymmetrical
they-axis
(x1 = x3,
xa, Y1
Y1 = -y3;
-ya; Fig.
Fig. 29a) with
with frequency
about the
y-axis (xl

wa =

We

~ ~: (l+
A
[

k1

YHA

2272A

)]~

.

( 1 + :BA SIIll2d
sin2a ) ].

mn

The co-ordinates
co-ordinates qu,
qsi, qs2 together
together correspond
correspond to
to two
two vibrations
symmetrical about
about the
the
The
vibrations symmetrical

y-axis (xl
(x1 = -»-x3,
-xa, .V1
Y1 = y3;
ya; Fig. 29b,
29b, c),
c), whose frequencies (0811
ws1, ¢0s2
ws2 are given by
by the roots
roots
of the
the quadratic
quadratic (in
(in w2)
w2) characteristic
characteristic equation
equation
of

Ki 1 +
w4-w~[!!_(
1+
mA

)]

(

)+

2k2
mA
A
k k = 0.
2mA cos2a
cos2a) + 2k2 ( 11 +
+ 2
2m
sin2a )] + 22p.ktk2
M A sin2a
I-* I 2 =
0.
"IB
ma
mA2mB
+
mn
mA
mn
mA2mB
MA

\Vhen 22a
a == or,
TT, all
all three frequencies become
become equal to
to those derived
derived in
in Problem 1.

PnoDLEM 3.
3. The same as Problem 1, but for an unsymmetrical linear molecule ABC
PROBLEM
(Fig. 30).

.cc3

2

£2

.

g,

8B
L

FIG.

I

A

30

SoLuTION. The
The longitudinal
longitudinal (x)
(x) and
and transverse
transverse (y)
(y) displacements
displacements of
of the
the atoms
atoms are
are related
related
SOLUTION.
by
by
m..1x1+mB.x2+m¢;~x3 = 0,

mAy1+mBy2+mcJ'3 = 0,

= Mclgjfa.
mcl2.va.

n1Ally1
mA1tYl =

+

+

282,
of stretching and
and bending can be
be written as &k1('8l1
!k1( 8[}))22 + %k1'(8l2)2
!k1'( 812) 2 + !k2i
The potential energy of
§k21282,
21 =:
= II1t +12+12. Calculations similar to
to those in
in Problem l1 give
where 21
give
2

(m.

+

2

)

=
kk21
l 2 ( 112
122
1t
12~ .|»_ 412
41 )
w
-(Linz, 2 = - - - - + - - +
1t2122 me
mA
mn
mA
II422
me
2

for
the transverse
transverse vibrations
and the
the quadratic
quadratic (in
(in 002)
w2) equation
equation
for the
vibrations and

I(

+

)+. (

+

)]+

1
1
1
l
w4-w2[k
+ _l)] +
w'*- we Ki1( - + _l) +kt'('
mA
mn
mn
me
2718
2718
mc
mA

for
the frequencies 0011,
wn, M22
w12 of
of the
the longitudinal vibrations.
for the

. ==00

p,klk1

NTMWBMC

§25. Damped
Damped oscillations
oscillations
§25.

So far we
we have
have implied
implied that
that all
all motion
motion takes
takes place
place in
in aa vacuum,
vacuum, or
or else
else that
that
So
!he
effect of
of the
the surrounding
surrounding medium
medium on
on the
the motion
motion may
may be
be neglected.
neglected. In
In
*he effect
reality, when
when aa body
body moves
moves in
in aa medium,
medium, the
the latter
latter exerts
exerts aa resistance
resistance which
reality,
which

to retard
retard the
the motion.
motion. The
The energy
energy of
of the
the moving
moving body
body is
is finally
finally dissipated
dissipated
tends to
tends
by being
being converted
converted into
into heat.
heat.
by
Motion under
under these
these conditions
conditions is
is no
no longer
longer aa purely
purely mechanical
mechanical process,
process,
Motion
must be
be made
made for
for the
the motion
motion of
of the
the medium
medium itself
itself and
and for
for the
the
and allowance
allowance must
and
internal thermal
thermal state
state of
of both
both die
the medium
medium and
and the
the body.
body. In
In particular,
particular, we
internal
we
general assert
assert that
that the
the acceleration
acceleration of
of aa moving
moving body
body is aa function
cannot in general
cannot
function
only
only of
of its
its co-ord.inates
co-ordinates and
and velocity
velocity at
at the
the instaht
instant considered,
considered; that
that is, there
there
no equations
equations of
of motion
motion in
in the
the mechanical
mechanical sense.
sense. Thus
Thus the
the problem
problem of
of the
the
are no
are
of a body in
in a medium is not
not one of
of mechanics.
motion of

There exists,
exists, however,
however, aa class
class of
of cases
cases where
where motion
motion in
in aa medium
medium can
can be
be
There
described by
by including certain additional terms
terms in
in the
approximately described

Damped oscillations
oscillations
Damped

§2s
§25

75

mechanical
mechanical equations
equations of
of motion.
motion. Such
Such cases include
include oscillations
oscillations with
with frequencies small
those of
of the dissipative
dissipative processes
processes in the
the
small compared
compared with those
medium. W'hen
body as being
being
When this condition
condition is fulfilled
fulfilled we may
may regard
regard the
the body
acted
o r e of
friction which
which depends
of friction
depends (for a given homogeneous
homogeneous
acted on by a fforce
medium)
velocity.
medium) only
only on its velocity.
If, in addition,
addition, this velocity
velocity is sufficiently
sufficiently small,
small, then the frictional
frictional force
can
powers of
of the Velocity.
velocity. The
The zero-order
zero-order term
in the
the expancan be expanded
expanded in powers
term in
sion is zero, since no friction
friction acts on a body
body at rest, and so the First
first nonnonsion
vanishing term is proportional to the velocity.
velocity. Thus the
the generalised
generalised frictional
frictional
vanishing
/rr acting
acting on
on a system
system executing small
small oscillations
oscillations in one
one dimension
dimension
force ffr
(co-ordinate
ax', where a()( is a positive
positive coefficient
(co-ordinate x) may be written
written fn/rr =
= --()(X,
sign indicates
indicates that
that the
the force
force acts
acts in the direction opposite
opposite to
and the minus sign
and
that of
of the velocity.
velocity. Adding
right-hand side
side of
of the
the equation
that
Adding this force on the right-hand
of motion,
motion, we obtain
obtain (see (21.4))
(21.4))
of

\\'e
We divide this by m and put
put

mx == -he-ax.
-kx-()(x.
mi?

(25.
(25.1)
1)

0(/m == ZA;
2A;
k/m =
a/m
(25.2)
kfm
= w
wou2,,
(25.2)
wo is the frequency
frequency of
of free oscillations
oscillations of the system
system in the absence
absence of
of friction,
friction,
to
coefficient or damping decrement.T
decrement. t
and AA is called the damping eoejicient
Thus the equation is
:i5+2)lx'+w02x = 0.

(25.3)
(25.3)

We
for the
We again
again seek
seek aa solution
solution xx == exp(rt)
exp(rt) and
and obtain
obtain rr for
the characteristic
characteristic

r2+2Ar+wo2 == 0,
0, whence
= -)li~.V(A2-uJ0¢2)-A±y(A2-w 0 2). The general
equation r2+2)lr+w02
equation
whence r1,2
1,2 =
solution of
of equation
equation (25.3)
(25.3) is
is
solution
xx

= C1
c1 CXP(?'1Z)
exp(r1t) + C2
cz CXP(7'2l).
exp(rzt).

Two
we, we
Two cases
cases must
must be
be distinguished.
distinguished. IfIf AA << wo,
we have
have two
two complex
complex conconof r.r. The
The general
general solution
solution of
of the
the equation
equation of
of motion
motion can
can then
then
jugate
values of
jugate values
be written
written as
as
be
x
re{A exp[-At + iV(w02-Pl2)f]}.

arbitrary complex
complex constant, or
or as
where A is an arbitrary

exp(- At) cos(wt+
cos( wt + a),
()(),
x = a exp(-At)

(25.4)

with cu
w =
= v(
wo2- A2) and
and a and
and a()( real constants.
constants. The
The motion
motion described
described by
with
v(0002-A2)
formulae consists
consists of
of damped osezlllat'ion.s°.
oscillations. It
It may be regarded
regarded as being
these formulae
of exponentially
exponentially decreasing
decreasing amplitude. The
The rate
rate of
of
harmonic oscillations
oscillations of
harmonic
decrease of
of the amplitude
amplitude is given by the exponent
exponent Pa,
A, and the "frequency"
decrease
cu
w is
is less than that
that of
of free oscillations in the absence of friction. For AA <{ we,
wo,
the
dlf'f@rence
betw
the difference between
and to
w 0 is of
of the second
second order of
of smallness.
smallness. The
The
een w and
ase in .frequency as a result of friction is to be expected, since friction
decre
decrease
in frequency as a result of friction is to be expected, since friction
retards
retards motion,
motion.
tt The dimensionless produ ,
·ng decrement
ctt AT
,.y (where T =
= 27r/
21T/W
is the
the period) is
is called the logarithmic
w is
damP• decrement.·

<<

PI'OdL1(;

76
'76

Oscillations
Small Oseillatiom'

§25

If
<{ to,
wo, the amplitude
amplitude of the damped
damped oscillation
oscillation is almost
almost unchanged
unchanged
If A~ <<
2TTfw. It
then meaningful
meaningful to consider
consider the mean
mean values
during the period
period 271/w.
during
It is then
of the squared
squared co-ordinates
co-ordinates and velocities, neglecting
neglecting the
(over the period) of
change in exp(
exp(- As)
~t) when
when taking the mean. These
These mean
mean squares
squares are evidently
evidently
change
proportional to exp(-Zh).
exp(- Ut). Hence
Hence the
the mean
mean energy
energy of
of the system
system decreases
decreases
proportional
as
E = E0
Eo exp( -- 2Az),
~t),
(25.5)
II

-

where Eo
of the
the energy.
energy.
where
Fo is the initial value of
Next, let >.
wo. Then
Then the values of r are both real and negative. The
The
A > wogeneral form of the solution
solution is
genera]
2 )]t}.
xX=
= 61
- 1}11/(A22 Cl exp{
exp{[~-y(~
- °v02)]*}
wo2)]t}+c2
exp{[~+
y(~2 -wo
(25.6)
+ Ca exp{
- [A
+ 1/(A2
w02)]t}.
(25-6)
We see that
that in this
this case, which
occurs when
when the friction
friction is sufficiently
sufficiently strong,
We
which occurs
the motion
motion consists
consists of
of a decrease
decrease in \xl,
lxl, i.e. an asymptotic approach
approach (as t ->
--+ oo)
oo)
the
the equilibrium
equilibrium position.
position. This type
type of motion
motion is called
called aperiodic damping.
to the
Finally,
~::'°. to,
Finally, in the special case where A~ =
wo, the characteristic
characteristic equation
equation has
the double
double root
root r =,
-; - A.
~- The
The general
general solution
solution of
of the differential
differential equation
equation is
then

x = (61+-21)
(c1 + czt) exp(-Ar).
exp(- ~t).

(25.7)

of aperiodic
aperiodic damping.
This is a special case of
For a system
system with more
more than
than one degree of
of freedom,
freedom, the generalised
generalised
For
frictional forces corresponding
corresponding to the
co-ordinates .'t',
linear functions
functions of
of
frictional
the co-ordinates
kg are linear
the velocities,
velocities, of
of the
the form
form
the

2
in.
L QCXfkXk·

(25.8)

I

II

frr,i = frr.¢

1:
I.:

mechanical arguments
arguments we can draw
draw no conclusions
conclusions concerning
concerning
From purely
purely mechanical
the
the. symmetry
symmetry properties
properties of the coefficients am
rxtk as regards
regards the suffixes 1.i and
k, but
but the methods
methods of
of statistical
statistical physics
physicst make it possible
possible to demonstrate
demonstrate
that in all cases
(25.9)
r1.ik
r1.ki·
lu.: == oilfibe written
as the
the derivatives
derivatives
Hence the
the expressions
expressions (25.8)
(25.8) can
can be
Hence
written as
fir,,
flr,i

of
of the
the quadratic form

F=
F

= -- 3F/35:i
oFfO.i:,

l% Z
2: 0¢i1¢
CXik·i:jXk,
i81'@k,
me
i,k

(25.10)
11)
(25.11)
(25.

which is called the dissipative function.
function.
The forces (25.10) must
must be added to
to the right-hand
right-hand side of Lagrange's
Lagrange's
The
equations::
equations
3
aL
d
d (oL)
oL OF
oF
(25,
(25.12)
12)

ox, =ox,-ox,·
ax,
(L)
II

Tt

dt
dt

336i

33i£ *

80.
Statistical
Physics, part
part 1,
1, § 121, Pergamon Press,
Press, Oxford 19
1980.
See Sta
tzlitical Physics,

§26

Forced' oscillations
oscillations under
under friction
friction
Forced'

77

The
physical significance:
The dissipative
dtssipative function
function itself
itself has an important physical
significance: it
gives the rate
rate of dissipation
dissipation of
of energy in the
the system.
system. This is easily
easily seen
seen by
gives
calculating
mechanical energy
calculating the time derivative
derivative of
of the mechanical
energy of
of the sy
system.
stem., We
have
'
have
d
3

:~ :t (L:x, ~~_ -L)
1';,;(;
=

f

Xi

3xi
L

=2(dI[L]
Xi

=

d

3

_ E . ax;OF. .

3355

a
ax;
L

)

i

locities, Euler
's theorem
Since F
F is a quadratic
quadratic function
function of
of the ve
velocities,
Euler's
theorem on
on homohomoSince
sum on the right-hand
right-hand side is equal
equal to 2F.
geneous functions
functions shows
shows that the sum
geneous
Thus
dE/dz
ZF,
dE/dt = -2F,
(25.13)
i.e. the rate
rate of
of change
change of
of the energy of
of the
the system is twice
twice the
the dissipative
dissipative
i.e.
of energy,
energy, it follows that
that
function. Since
Since dissipative
dissipative processes
function.
processes lead to loss of
F > 0, i.e. the quadratic
quadratic form (25.11) is positive definite.
of small
small oscillations
oscillations under
under friction are obtained by
by adding
adding
The equations of
the
the forces (25.8) to the right-hand
right-hand sides of
of equations
equations (23.5)
(23.5)::

2L

(25.14)
'Lmucxk+
= CXtkXk·
E"1z/¢J?1¢+ LkikXk
Zkucxk =
Gwinkk
kk
kk
equations xzf
Xk =
= Ilk
Ak exp(rt),
exp(rt), we obtain,
obtain, on cancelling
cancelling exp(rt),
exp(rt),
Putting in these equations
set of
of linear
linear algebraic
algebraic equations for the
the constants
constants Ilk
Ak::
a set

= 0.

2(Mikf2+
di)67'+ k,k)Ak =
L(mikr2+cxtkr+ktk)Ak
k
IC

(25.15)

!mtkr2 +cxikr+ktk!
= 0.
l"1zH2+
0¢u;t+kucl =

16)
(25.
(25.16)

to zero their determinant, we find the characteristic equation, which
which
Equating to
determines the possible
of r:
determines
possible values of

equation in r of
of degree
degree 2s. Since
Since all the coefficients are real,
This is an equation
its roots
roots are
are either
either real, or
or complex
complex conjugate
conjugate pairs. The
The real
real roots must be
its
the complex
complex.roots
negative rea]
real parts, since
since othernegative, and the
negative,
roots must have negative
wise the
the co-ordinates,
co-ordinates, velocities
velocities and
and energy of
of the system
system would
would increase
increase
exponentially with time,
time, whereas
whereas dissipative
dissipative forces
forces must lead to
to a decrease.
decrease.
exponentially
of
of the
the ene
energy.
rgy.
§26.
§26. Forced osc
oscillations
illations under friction
of
rced oscillations under
theo.ry
of fo
forced
under friction is entire
entirely analogous
analogous to
to
Cn i
aThe.
rhh The
t gl.Vtheory
N §22 fol' s
r
i
t
H
n
e
t
n
W
I
w
o n' Here
C11
" s without
~iNadt
~~et~ Inca S§22
oscillations
friction.
e wee oshall Cconsider
O s de 1°
rho
e 1 d
e O ffor
lc eexternal
r
i
t
P 1no
10 det
e ail thee case
of aape
. 0d".1c
f
h"
h
.
f
"d
bl
.
a
n
~
i
a
o
f
o
s
1
d
o
h
b
l
re stf
orce,
w
1C
1s
o
cons1
era
e
1nterest.
n
xt
be'
s
w

..

78

§26

Small Oscillations
Oscillations
Small

right-hand side of
of equation (25.1) an external forcefcos
yt
Adding
to the right-hand
Adding to
force f us -yr
and dividing
dividing by
by m,
m, we
we obtain
obtain the
the equation
equation of
of motion
motion::
and

(26.1)
(26.1)

.ii-1-2)L3E+ oJ02x = (ffm) COS it.

The
The solution
solution of
of this
this equation
equation is
is more
more conveniently
conveniently found
found in
in complex
complex form,
form,
and so
so we
replace cos
cos ityt on
on the
the right
right by
by exp(z•-yt)
exp(iyt)::
and
we replace

56+
x+ 2Ax'+
2..\x+ w02X
wo2x

= (ffm)
(f/m) exp(z•-yt).
exp(iyt).
=

We seek
seek aa particular
particular integral
integral in
in the
the form
B exp(i»yt),
exp(iyt), obtaining
obtaining for
for BB
We
form x = B
the value
value
the

B

y + 217%
2M.y).

- 1/2
= f/"1(<»02
f/m( wo2 -

Writing B =
= b exp(i8),
exp(iS), we have
Writing

(26.2)

bb =
= f/"1V[(<»02-v2)2+
ffmv[( woz- y2)2 + 4?*2v2]»
4A2y2], tan
tanS8 =
= 2
2Ayf(y2wo2).
(26.3)
/ ( v L 0102).
(26.3)
Finally,
Finally, taking
taking the real
real part
part of
of the expression
expression B exp(i~yt)
exp( iyt) =
= b exp[i(-yt-l-8)],
exp[i(yt + S)],
we Find
find the particular integral
integral of
of equation
equation (26.1); adding
adding to this the general
general
on the right-hand
right-hand side
side (and
(and faking
taking for
of that equation with zero on
solution of
definiteness
definiteness the case to
wo > A), we have
II

x = a exp( -At)
a) + b us(-yt
-At) cos(wt
cos( wt +
+ex)+
cos(yt + 8).
S).

(26.4)

first term decreases exponentially with
with time, SO
so drat,
that, after
after a sufFlcient
sufficient
The first
remains::
time, only
only the second term remains
time,
x = b OOS('y!
cos(yt + 8),
S).

(26.5)

The expression
expression (26.3)
(26.3) for
the amplitude
amplitude bb of
of the
the forced
forced oscillation
oscillation increases
increases
The
for the
y approaches we,
wo, but
but does not
not become infinite as itit does in
in resonance
as 'y
without friction.
friction. For
For aa given
given amplitude
amplitude f of
of the
the force,
force, the
the amplitude
amplitude of
of the
the
without
2
2
oscillations
from
v( wo - ZA2)
2A );, for
for AA <{ we,
wo, this
this differs
differs from
oscillations is
is greatest
greatest when
when 'yy = V(0)02wo
only
by
a
quantity
of
the
second
order
of
smallness.
to only by a quantity of the second order of smallness.
Let us
us consider
consider the
the range
range near
near resonance,
resonance, putting
putting 'yy == w0-1-€
wo+E withE
small,
Let
with e small,
and suppose
suppose also
also that
that AA <{ wowo. Then
Then we
we can
can approximately
approximately put,
put, in
in (26.2),
(26.2),
and
'y2-y2- oJ02
wo2 == ('y+
(y+ 0J0)('ywo)(y- wo) z~ Zw0€,
2woE, Ziky
2iAy z~ 2z7lw0,
2iAwo, so that

f

<<

or

B =
B

<<

-f/2m(E-iA)w
_f/2772(e -i)l)t.u0
0

In = f/2"1°J0V(€2 + 9l'2)!

tan
8 = A/€.
tanS=
NE.

(26.6)

(26.7)
(26.7)

A property of
of the phase
difference 8S between
between the oscillation
oscillation and the extern
externall
phase difference

the oscillation
oscillation "lags behind" the fforce.
that it is always negative,
force is that
negative, i.e. the
ore.
wo, 8S -->~0
0;, on the side 'yy > we,
wo, S -->~ -or.
-7T.
Far from resonance on the side 'yy < we,
The change
change of
of 8S from zero
zero to -or
-7r takes place in a frequency
frequency range
range near
wo
The
near to
which is narrow
narrow (of
(of the order
order of
of AA in width);
width); 8S passes through
through --}-rr
when
r when
ges
y = w0»
wo. In
In the
the absence
absence of
of friction, the
the phase of
of the forced oscillation
oscillation chan
ch~nges
'y
);
gn
si
s
ge
an
discontinuously by
by 'or
7r at
at 'yy =
= we
wo (the
(the second term in (22.
(22.4)
changes stgn);
discontinuously
4) ch
this discontinuity
discontinuity is
is smoothed
smoothed out.
out.
when friction
friction is
is allowed
allowed for,
for, this
when

§26

79

Forced oscillations
oscillations under
under friction
friction
Forced

steady motion,
motion, when
the forced oscillations
oscillations given
given
IIn
n steady
when the system executes the
by (26.5),
(26.5), its energy
energy remains
remains unchanged. Energy is continually absorbed
absorbed by
the system from
from the source of
of the
the external force
force and
and dissipated by friction.
friction.
Let J(y)
amount of
of energy
energy absorbed
absorbed per
time, which
Let
I('y) be the mean
mean amount
per unit
unit time,
which depends
on
on the frequency
frequency of
of the external
external force.
force. By (25.13)
(25.13) we have 1(v)
J(y) = ZF,
2F, where
F is the average
average value
value (over
(over the period of
of oscillation)
oscillation) of
of the dissipative
dissipative funcF
motion in one
one dimension, the expression (25.11)
(25.11) for the dissipative
dissipative
tion. For
For motion
= '§ax°2
!01:.X2 =
= >.:mx2.
Substituting (26.5), we have
function becomes F =
Ama?2. Substituting

F = Amb2y2 sin2('yt
sin2(yt +
S).
+8).
II

is
is !,, so
so that
that

The
The time
time average
average of
of the
the squared
squared sine
sine

J(y) = )tmb2y2.
J..mb2y2.
I('y)

(26.8)

I(6) = saf2)l/4m(e2+ )l2).

(26.9)

Near resonance
resonance we
have, on
on substituting
substituting the
the amplitude
amplitude of
of the
the oscillation
oscillation
Near
we have,

(26.7),
from
from (26.'7),

called a dispersion-type frequency
frequency dependence
dependence of
of the
absorption.
This is called
the absorption.
The half-width
half-width of
of the resonance
resonance curve
curve (Fig. 31) is the value of 1e1
IE I for which
The
J(E) is half its maximum
maximum value (e
(E =
= 0). It
that in the
I(6)
It is evident from (26.9) that
present case the half-width
damping coefficient
coefficient A. The
The height
height of
present
half-width is just
just the damping
2/4'm)1, and is inversely
proportional to A. Thus,
J(O) --:
= ff2/4mA,
inversely proportional
the maximum
maximum is 1(0)
///(Ol
'I/7fO)
I

.

IL_....

I

I

.,.,-.

!/2

_ - . . _

I
I
I

-2.

l
l
l

I

I

l

A

1.

FIG.
F1G.

31

6'

when the damping
damping coefficient
coefficient decreases,
decreases, the resonance
resonance curve
curve becomes
when
becomes more
The area under
under the curve, however,
however, remains
remains unchanged. This area
area
peaked. The
given by the integral
integral
is given

I 1('r)
l(y) do
dy == I 1(6)
I( E) de.
dE.
oo
ro

ro
oo

0

-w0
"We

Since I(6)
J(E) diminishes
diminishes rapidly with increasing l€l,
lEI, the region
region where l€l
lEI is
Since
of no importance,
importance, and
and the lower limit
limit may be replaced
replaced by -- oo,
oo, and
and
large is of
form given
J(E) taken
taken to
to have
have the
the form
given by
by (26.9).
(26.9). Then
Then we have
have
/(e)

ro

f2A
j2"A fro

J(E) de
dE == -- I(€)
4m
f

-ro

-oo

Q

-ro
-oo

de
dE

E2+A2
€2+ A2

of?
7Tj2
= -.

4m •

(26.10)

80

§27

,Small
~mall Oscillations
Oscillations
PROBLEM

Determine the
the forced
oscillations due
due to
to an
an external
external force
Determine
forced oscillations
force
presence
presence of
of friction.
friction.

/o exp(at)
exp(o:t)
f == to

cos ityt II]
m the
the
cos

SOLuTloN. We
We solve the
the complex equation of
of motion
motion
SOLUTION.

x+2>.~+wo~X
{fo/m) exp(at
exp(o:t+iyt)
55-1-2
M-I-w0l3x = (folm)
-I-iyt)

and then
then take
take the
the real
real part.
part. The
The result
result is
is aa forced
forced Oscillation
oscillation of
of the
the form
form
and
where
where

=

xx = bb exp(at)
exp( at) cos('yt-|cos(yt+ 8),
8),

=
tan 8 =
=
tan
b

=f0.-'rW[(w02+a2-y2+2(»)1)2+4¢»==(a+P\)2],
/o!mv[(wo2 +o: 2 -r+2o:>.)2+4r(o:+>.)2 ],

-2'Y(o:+.\)/(w02-y2+o:2+2o:.\).
-2'y(a-l-1\)](w02-»--y2+
m2-I-2a.\).

§27. Parametric resonance
There exist
exist oscillatory
oscillatory systems
systems which
which are not
not closed,
closed, but
but in which
which the
to a time variation
variation of
of the
the parameters
parameters. t
only to
external action amounts only
The parameters
of a one-dimensional
one-din1ensional system
system are
are the coefficients
coefficients m and
and k
The
parameters of
in the
If these
the Lagrangian
Lagrangian (21.3). If
these are functions
functions of
of time, the equation
equation of
of
motion is
motion
d
-(mx)+kx
dt

=

0.

(27.1)
(27.
1)

introduce instead
instead of
of rt aa new
new independent
independent variable
variable
introduce
dT = dr/m(z);
dtfm(t); this
this reduces
reduces the
the equation
equation to
to

\\re
W'e

or

T
'r

such
such

that
that

2 +mkx =
d2xjdT
= 0.
xld12+mkx

therefore no loss of
of generality
generality in considering
considering an equation
equation of
of motion
motion
There is therefore
of the form
of
d2x,~'df2 + w2(t)x = 0

(27.2)

obtained from (27.1) if m =
= constant.
constant.
obtained
The form
form of
of the function w(t) is given
given by the conditions of
of the problem.
The
Let us assume
assume that this function
function is periodic
some frequency
frequency 'yy and period
period
Let
periodic with some
T
=
hwy.
This
means
that
w(Z+
T)
T 2TTfy.
that w(t + T) = w(t),
w(t), and
and so
so the
the equation
equation (27.2)
(27.2) is
is
invariant under the
. T.
the transformation It ->
--+ l+
t+
Hence, ifif xx(t)
is aa solution
solution of
of
T. Hence,
(Z) is
the equation,
equation, so is x(t+
x(t + T). That
That is, if JC1(Z)
x1(t) and x2(t) are two independent
independent
integrals of
of equation
equation (2'7.2),
(27.2), they
they must
must be transformed
transformed into
into linear
linear combinacombinaintegrals
of themselves when
when rt is replaced by lt ++ T. It is possibles
possiblet to choose Xl
x1
tions of
and x2
+ T, they
xz in such
such a way that, when
when t ->
--+ t +
they are simply
simply multiplied
multiplied by

.

1't

A
simple example
example is that
that of
of aa pendulum
pendulum whose
whose point
point of
of support
support executes
executes aa given
given periodic
periodic
A simple

motion in
in aa vertical
vertical direction
direction (see
(see Problem
Problem 3).
3).
motion

This choice
choice is
is equivalent
equivalent to
to reducing
reducing to
to diagonal
diagonal form
form the
the matrix
matrix of
of the
the linear
linear trans
tran~­It This
formation of
of x:t(t)
and x2(t),
x2(t), which
involves the
the solution
solution of
of the
the corresponding
correspondin_g quadratlc
'luadratJc
formation
x1(t) and
which involves
secular equation,
equation, We
shall suppose
suppose here
here that
that the
the roots
roots of
of this
this equation
equation do
do not
not cQ1II1Clde°
come, de.
secular
We shall

§27

81

Parametric resonance

constants:
#1x1(I)1 X2(Z
T) = fL2X2(t).
fJ.21'2(t). The
functions
constants: x1(t
x1(t + T)
7) = flllXl(t),
x2(t + T)
The most
most general
general functions
having this
this property
property are
are
having
JC1(I) =

#1t/TI71(Il'

.7C2(Z)

#2:

1

'I

11ll2(I),

(27.3)

where ll1(t),
ll1(t), tI2(t)
ll2(t) are purely
purely periodic
periodic functions of
of time with period
period T.
The
functions must
way,
The constants
constants p.;
/Ll and
and 112
fL2 in
in these
these functions
must be
be related
related in
in aa certain
certain way.
2
2
Multiplying
.X1 + w (t)x1 == 0,
0, 562
X2 -I-w2(I)X2
+w (t)x2 =
= 00 by
by x2
x2 and xXl1
Multiplying the equations
equations 561+-2(t)x1
and subtracting,
subtracting, we
we have
have éélx2-5&2x1
.X1x2- x2x1 == d(x'1x2-x1:»22)dt
d(.X1x2- x1.X2)dt == 0,
0, or
or
respectively and
respectively
951X2-X1x2 =

constant.

(27.4)

1.

(27.5)

For any
any functions
functions x1(t), x2(t) of
of the form
form (2'7.3),
(27.3), the expression
expression on
on the leftleftFor
hand side of (27.4) is multiplied
by fLlfL2
p1p2 when tr is replaced
multiplied by
replaced by tr ++ T. Hence
Hence
equation (27.4) is to hold, we must
must have
it is clear that, if equation
=

II

fLlfL2
p-1/Ll,2

Further information
fJ-2 can
information about the
the constants fJ~1,
fLl, /L2
can be obtained from the
any integral
integral of
of
fact that the coefficients in equation (27.2) are real. If
If x(t) is any
then the complex conjugate
conjugate function x*(t) must
must also
also be
such an equation, then
an integral. Hence
up must
p.1*, !u,2*,
Hence it follows that
that al,
fLl, fL2
must be the same
same as fLl*,
fL2*, i.e.
either
/Ll =
= 1J~2*
fL2* or
or H1
/Ll and
and /L2
both real. In
In the former
former case, (27.5) gives
either #1
IJ~2 are both
2
2
.e. 1111
pa are of modulus
fLl :=: 11/fLl*,
lfL11\2 = IM
ltL2I12 = 1: the constant
constantss PA
fLl and
and fL2
modulus
#1
I1J~1*» ii.e.
unity.
In
In the other
other case, two
two independent integrals
integrals of
of equation (27.2) are
.7C1(I)

= p!/T[[1(I),

JC2(Z)

= p,"/TI72(l),

(27.6)

p (ImI
with a positive
positive or negative
negative real value of
of fL
( lfL I 5é
# 1).
1). One
One of
of these
these functions
functions
(Xl
ly with time.
(x1 or xg
x2 according
according as [al
lfL I > 11 or lp]
lfL I < 1) increases
increases CXPOI1Cl]t1211
exponentially
means that the system
system at rest
equilibrium (x =
= 0) is
is unstable:
unstable: any
any
This means
rest in equilibrium
is' sufficient
sufficient to lead to a rapidly
deviation from this state,
state, however
however small,
small, is
deviation
rapidly
called·parametric resonance.
resonance.
increasing displacement
displacement x. This is called'paramem.c
increasing
It should
should be noticed
noticed that,
that, when
when the
the initial
initial values of
of x and
and x'
x are exactly
exactly
It
happens in ordinary
ordinary resonance (§22),
zero, they remain
remain zero, unlike what happens
in which
which the
displacement
increases
with
time
(proportionally
t) even from
the displacement increases
(proportionally to 1)
initial values
values of
of zero.
initial
Let us determine the
resonance to occur
occur in the
Let
the conditions for parametric resonance
the function
function oJ(t)
w(t) differs only
only slightly
slightLy from
from a constant
important case where the
wo and
and is a simple
simple periodic
function::
value to
periodic function
H..

w2(t)
w2(z)

= W02(1
w 02(1 + h

cos ii),
yt),

(27.7)

where the
nstant hh <<
where
the Co
constant
~ 11,; we
we shall suppose h positive, as may always be
e
don
done. by
suitably choosing
choosing the
the origin of
of time. As we shall
shall see
see below,
below, paraparaby sustablY
ic res
metr
onance i
m~tnc
resHonance
iss strongest
strongest if
if the
the frequency
frequency of
of the
the function
function w(t)
is
nearly
w(t) is nearly
twice
we
twice wowo. Hence
ence we
put 'Vy =
= 20:0-}-e,
2w0 + £, where e£ <<
~ wowo.
Put

82

Small Oscillatzbns
Oscillations
Small

§27

The solution
solution of
of equation
equation of
of mot1oni°
motioni"
The

(27.8)

x =
b(t) sin(w0+%e)t,
= a(t) cos(w0+;l§~s)t+
cos(w0 +~£)t+b(t)
sin(wo+~£)t,

(27.9)

II

2 [l+hcos(2wo-t;-£)t]x
+ w02[1
+ k C08(2.cu0 +. €)¢]x = 0
x+wo

may be sought
sought in the form
may

a( t) and b(r')
b( t) are functions of
of time Whlcll
which vary slowly
slowly in comparison
comparison
where a(r)
with the trigonometrical
not
This form of
of solution
solution is, of
of course,
course, not
trigonometrical factors.
factors. Thls
In reality,
reality, the function x(t) also involves
involves terms
terms with frequencies which
which
exact. In
differ from w0-1-i1€
wo+{E by integral
integral multiples
multiples of
of 2w0+6,
2wo+E; these
these terms are, however, of
/z, and
of a higher
higher order of
of smallness
smallness with respect
respect to h,
and may
may be neglected
neglected
approximation (see Problem
Problem 1).
in a first approximation
We substitute
substitute (27.9) in (27.8) and
and retain only
only terms of
of the first order
order in
WIVe
E, assuming
assuming that et
a ~ ea,
Ea, b
Eb; the
the correctness of
of this
this assunlption
assumption under
e,
b ~ et,
resonance conditions
conditions is confirmed
confirmed by the result. The
The products of
of trigonotrigonoresonance
replaced by sums
sums::
metrical functions
functions may be replaced
metrical
cos(w0+i-€)t
cos( wo+}£)t.. CoS(2.w0+
cos(2wo+ te)t
£)t =
=

cos
~·cos

3(w0
i6):-I-§
3( wo + !£)t
+ ~ cos(w0+-.3,€)!,
cos( wo + ~£)1,

accordance with what was said above we omit
omit terms with freetc., and in accordance
quency 3(w0+
3(w 0 + ~E).
The result is
quency
e), The

(221 + b +%1wv0b)<,v0 sin(<,J0+%f)z
+ (2b- £16 +£.g1wJ0a)w0 c0s(w0+;f)r
-(2a+b£+!hwob)wo
sin(wo+!£)t+(2b-a£+Yzwoa)wo
cos(wo+{£)! =
= 0.

If
equation is to be justified,
coefficients of
of the sine
sine and
and cosine
cosine must
must
If this equation
justified, the coefficients
both be zero.
zero. This gives two linear
linear differential
differential equations for the functions
functions
both
a(t) and
and b(t). As usual, we seek solutions
solutions proportional
proportional to exp(st).
exp(st). Then
so
sa+ ;;(e+-21/zw0)b
!(E+ ~hwo)b =
= 0, =§(e--%hw0)a-sb
~( E- !hw 0 )a- sb =
= 0, and the compatibility
compatibility condition
algebraic equations
equations gives
for these two algebraic

= ;ll(%/1<»0)'
HWu.vo)2-£ 2].
].

(27.10)
(27.10)

--!hwo
d r u g < eE < éhw0
!hwo

(27.11)

82
s2 =

O - ( H9

The condition
condition for parametric
resonance is that ssis
s2 > 0.1
O.t Thus
Thus
The
parametric resonance
is real, i.e. $2
parametric
resonance occurs in the range
range
parametric resonance

on either
either side of
of the frequency
frequency 2w0.l1
2wo.l1 The
The width
width of
of this range is proportional
proportional
and the values of
of the amplification
amplification coefficient
coefficient s of
of the oscillations
oscillations in the
to h, and
of the order of
of h also.
range are of
resonance also
also occurs when
when the frequency 'yy with
with which
which the
Parametric resonance
parameter
width of
parameter varies
varies is close to any value 2010/n
2w0Jn with
with n integral. The
The width
of the
t

An
equation of
of this
this form
form (with
(with arbitrary
arbitrary y and
and iz)
h) is
is called
called in
in mathematical
mathematical physics
physics
An equation

!IIathieu's equation.
equation.
Mathieu's
The constant
constant p.
p. in
in (27.6)
(27.6) is
is related
related to
to ss by
by p.
p. =
1t The

- exp(slr/wo),
exp(srrfwo); when
when tt is
is replaced
replaced by
by
--

t+2rr/2wo, the sine and cosine in
in (27.9)
(27.9) change sign.
¢+21rf2W0,
at
are interested
interested only
only in
in the
the range
range of
of resonance,
resonance, and
and not
not in
in the
the values
values of
of s in
in th
t~a.t
H!I IIf
f we
we are
range, the
the calculations
calculations may
may be
be simplified
simplified by
by noting
noting that
that ss -=
= 0
0 at
at the ends of
of the r8N8€
range,v i.e.
•..e.
range,
in
the coefficients
coefficients a and
and bbin
(27.9) are
are constants.
constants. This
This gives
gives immediately
immediately.:=
±~hwo as
as tn
the
in (27.9)
e = _+A/m0

(27.11).
(27.11).

§27

83

Parametric
Parametric resonance
resonance

resonance
rapidly with increasing
resonance range (region
(region of
of instability)
instability) decreases
decreases rapidly
increasing n,
however,
namely as he
however, namely
hn (see Problem
Problem 2, footnote).
footnote). The
The amplification
amplification coefficient
of the oscillations
oscillations also decreases.
decreases.
efficient of
The
phenomenon of
presence
The phenomenon
of parametric resonance
resonance is maintained
maintained in the presence
of slight
slight friction, but
but the region
region of
of instability
instability becomes
becomes somewhat
somewhat narrower,
narrower.
of
of the
the amplitude of
of
have seen
seen in §25, friction
friction results
As we have
results in a damping of
oscillations
oscillations as exp(- Az).
At). Hence
Hence the amplification
amplification of
of the
the oscillations
oscillations in parametric
positive s given by the solution
exp[(s- )l)t]
A)t] with the positive
solution
metric resonance
resonance is as exp[(s~
frictionless case, and
and the limit of
of the region
region of
of instability
instability is given by
for the frictionless
the equation ss-A
-A=
= 0. Thus, with s given by (2'7.10),
(27 .1 0), we have for the resonance
range,
(2'7.11),
range, instead
instead of
of (27 .11 ),

-v[sh<»0)2-4)e] < e

<`i

q/l(hct>0)2-4)l2].

(27.12)

It should
should be noticed
noticed that
that resonance
arbitrarily
It
resonance is now possible
possible not for arbitrarily
only when
when h exceeds a "threshold" value hk.
When
small amplitudes
amplitudes h, but only
small
kg. When
= 4A/w0.
4Afw 0 • It can be shown that, for resonance near the fre(27.12) holds, hk
Lu; =
2w 0 Jn, the threshold
threshold it
hk is proportional
to A1
A11/'*,
n, i.e. it increases with n.
quency 2w0/n,
quency
proportional to
n.
PROBLEMS
PR
O B LE M s

PROBLEM 1.
1. Obtain
Obtain an
an expression
expression correct
correct as
as far
as the
the term
term in
in 112
lz2 for
for the
the limits
limits of
of the
the region
region
PROBLEM
far as
of instability
instability for
for resonance
resonance near
near y == 2w0.
2wo.
of
We seek the
the solution of
of equation (27.8) in
in the
the form
SOLUTION. We
form

=

x = to
ao cos(w0+§e)t-l>-bo
cos(wo+!.:)t+bo sin(w0-I-ie)t+a1
sin(wo+!.:)t+m cos
cos 3(w0-I-i€):-}-b1
3(wo+!.:)t+bl sin
sin 3(wo+§e)r,
3(wo+!.:)t,

which includes
includes terms
terms of
of one
one higher
higher order
order in
in hh than
than (27.9).
(27.9). Since
Since only
only the
the limits
limits of
of the
the region
region
which
of instability
instability are
are required,
required, we
we treat
treat the
the coefficients
coefficients ao,
ao, Bo,
bo, at,
a1, /21
b1 as
as constants
constants in
in accordance
accordance
of
with the
the last
last footnote.
Substituting in
in (27.8),
(27.8), we
we convert
convert the
the products
products of
of trigonometrical
trigonometrical
with
footnote. Substituting
functions
into sums
sums and
and omit
omit the
the .te
terms
of frequency 5(Wo+¥)
5( wo+!.:) in
in this approximation.
approximation. The
functions into
rms of
result is
is
·
result

+

[--a0(w0€
[ -ao( wo.: +%€2)
+!.: 2) +kw043ao+%kw02a1]
cos( wo +.1t)t
+!£)t+
+!lzwo2ao+!lzwo2ai) cos(w0

2
+
{fhw02b1]
+
+[ -bo(w0£+!.:2 )-!Izwo2 bo+!lzwo
bi] sin(wo+l.:)t+
[-b0(W0€ + € 2 ) - h W 0 2 f J 0 - | » -

SiI1(cU0+€)£

+[!lzwo2 ao-8wo2ad cos
cos 3(W0+e)¢+
3(wo+!.:)t+

+[§hW02a0-80102011

+[%hw02b0-8w02b1]
+Wrwo2bo-8wo2bd sin
sin 3(w0+§e)t
3(wo+!.:)t =
= 0.

In the
the terms
terms of
of frequency
frequency w0-I*-£6
wo+!.: we
we retain
retain terms
terms of
of the
the second
second order
order of
of smallness,
smallness, but
but in
in
In
those of
of frequency
3( w0 + !.:) only
only the
the first-order
terms. Each
Each of
of the
the expressions
expressions in
in brackets
brackets
those
frequency 3(W0+%€)
first-order terms.
must separately
separately vanish.
vanish. The
The last
last two
two give
give at
a1 == ha0/16,
lzao/16, 61
b1 =
= hbo]l6,
hbo/16, and then the first two
two
must
give w0ei§hw02+1€2-.Zwgg/32
wo£± !lzwo2 +!.:2 -lz2 wo2 /32 =
= 0.
Solving this
this as
as far
far as
as terms
terms of
of order
order 112,
lz2 , we
we obtain
obtain the
the required
required limits
limits of
of.::
Solving
6:
PROBLEM 2.
2. Determine
Determine the
the
PROBLEM

£
c

=
= i~hw0-h2w0/32.
± !lzwo-h2 wof32.

wo.
limits of the region of instability in resonance near 'yy = to.

tting y = w0+6,
SOLUTION, Pu
Putting
wo+.:, we
we obtain
obtain the
the equation
equation of
of motion
motion

SOLUTION,

Since the required I'

· ·

:ii-I-¢u02[1
x+wo2 [1 +I:
+lz cos(w0+€)t]x
cos(wo+.:)t]x == 0.
0.

.

tmtttng values of £ ~ I22,
h2, we
seek aa solution
solution in
In the
the form
we seek
fonn
Since the r;quired limiting values of e
x = ao cos(wo+ )
• .
,

x

-

`

to

"t+bo sin(w'0+.E)¢+a1
sm(wo+.:)t+al cos
cos 2(w¢,+.E)£+b1
2(wo+.:)t+bl sin
sm 2(Wo+€)£+61,
2(wo+.:)t+ct,
cos(w¢,-l-£)£-I-60

84
84

§28

Small
Small Oscillations
Oscillations

which includes
includes terms
terms of
of the
the first
t\vo orders.
orders. To
To determine
determine the
the limits
limits of
of instability,
instability, we
we again
again
which
first two

treat the
the coeEF1cients
coefficients as
as constants,
constants, obtaining
obtaining
treat

+.2

+ +

2al -Hz
2cl] cos(
[[ 2 woeag
w0231
W02C1]
we e)t
-2womo+lhwo
+hwo
cos(wo+£)t+

+
2 w0eb0 +&kwo2b1]
-I-€)t+
+[[ --2wo£bo+lhwo
bl] sin(w0
sin(wo+£)t+
+ [--3
[ -3 wn2a1
wo 2at +§hw02a0]
+!hwo2ao] cos 2(
2(wo+£)t+
+
W0 +€ )t+
+ [[ --3wo
bl +Hlwo2bo]
+lhwo bo] sin
sin 2(¢U0
2(wo+£)t+
[clwo
+lhwo ao] =
+
3 wo2b1
+€)¢ + [¢`1
wo2 +§72w02C20]
2

2

2

2

Hence
Hence at
a1 = fza0,'6,
hao,'6, b1
b1 = hb0[6,
hbo/6, c1
c1 == -Qhag,
-l'wo, and the
the limits areT
aret e£ =

2

:=:

0.
0.

-5h2w0]2-1-,
-Sh 2 "'o/24, 6£

= h2w0.~I24.
h 2 wo/24.

PROBLEM 3.
3. Find
Find the
the conditions
conditions for
for parametric
parametric resonance
resonance in
in small
small oscillations
oscillations of
of aa simple
simple
PROBLEM
pendulum whose
whose point
point of
of support
support oscillates
oscillates vertically.
vertically.
pendulum
we -}-e)¢
]q6 = 0,
w02 == g
I.
< 1) the equation ofof motion 6:{> +wo2[l
+ wo [1 -1-(4a/Z)
+(4a/l) cos(2
cos(2w
+£)t)r/>
0, where
where wo
gfl.
the parameter
parameter ft
h is
is here represented by
by 4a/l.
(27.11), for
for
Hence we see that the
4a[l. The condition (27.l1),

(go
(r/>

SOLUTIO!'!.
SOLUTION.

The Lagrangian
Lagrangian derived
derived in
in §5,
§5, Problem
Problem 3(c),
3(c), gives
gives for
for small
small oscillations
oscillations
The

161 <

example, becomes
becomes 1£1
example,

2

0

2

=

2ay(g/13).
2aV(g/13).

Anharmonicc oscillation
oscillationss
§28. Anharmoni
The whole of
of the
the theory
theory of small
small oscillations
oscillations discussed
discussed above is based
based on
The
and kinetic
k~netic energies
energies of
of the
the system
system in terms
terms of
of
the expansion
expansion of the potential
potential and
the co-ordinates
co-ordinates and
and velocities,
velocities, retaining
retaining only
only the second-order terms. The
The
of motion
motion are then
then linear,
linear, and
and in this
this approximation
approximation we speak
speak of
equations of
oscillations. Although
Although such
such an expansion
expansion is entirely
entirely legitimate
legitimate when
linear oscillations.
amplitude of
of the oscillations
oscillations is sufficiently
sufficiently small,
small, in higher approximaapproximathe amplitude
anharmonic or non-linear
non-linear oscillations) some
some minor
minor but
but qualitatively
qualitatively
tions (called
(called enharmonic
tions
different
different properties of the
the motion
motion appear.
Let us consider
consider the expansion
expansion of
of the Lagrangian
Lagrangian as far as the third-order
third-order
Let
In the potential
energy there appear
appear terms of
of degree
degree three in the coterms. In
potential energy
ordinates xi,
Xi, and
and in the kinetic
kinetic energy
energy terms containing
containing products
ordinates
products of velocities
co-ordinates, of
of the
the form
difference from the previous
and co-ordinates,
f o r XiXkXz.
XiXkxi. This difference
previous
expansion
expression (23.3) is due to the retention
retention of
of terms
terms linear
linear in xx in the expansion
expression
of the functions
functions @5k(Q)~
aik(q). Thus the Lagrangian
Lagrangian is of
of the form
·
of

L

=

rz:Z (ffumxzf -

I
-2

.

i,k
i,k

.

e.. •
•
)+
(mtkXiXkkikXiXk)
+
kzkxzxk

+! L flzkziaikxz-5%
ntklXiXkXz-l 2
L lakzxixkxz,
liklXiXkX[,
Le.:
i,k.l

as.:
i,k.l

L

(28.1)
(28,
1)

where nikl•
Niki, likl
gm are
are further
further constant coefficients.
coefficients.
If
arbitrary co-ordinates
co-ordinates xi
Xi to the normal
co-ordinates Qa
Qa.
If we change from arbitrary
normal co-ordinates
of the
the linear
linear approximation, then,
then, because
because this transformation is linear,
linear, the
of

third and
and fourth sums
sums in (28.1) become
become similar
similar sums
sums with Qa
Qa. and Q
Qa. in place
third

Tt

.Q

the width
\vidth As
f.£ of
of the
the region of
of instability in
in resonance near the
the frequencY
frequency
Generally, the

2 a>oln
wo/n is given by
by

f.£=
n2n-3J.nwof23!n-1J[(n-1)1)•,
Ac
= n2""3lz"'w0/23l"*1}[(n-1)l]Z,
57),
•
• • n 3 13
132,
1957).
2, 19
AssoeiaIi'o1"* 3.'
11 result
result due
due to
toM.
BELL (Proceedings
(Proceedings of
of the
the Glasgow
Glasgow Matlzematieal
Mathemattcal Assoctatto
M. BELL
a

§28

85

oscillations
Anharmonic oscillations

Xt and
and the velocities
velocities ix1• Denoting
Denoting the
the coefficients
coefficients in these
these
of the co-ordinates No
air and
and I*Gr6r'
fLa.PJ'• we have the Lagrangian
Lagrangian in the
the form
form
new sums by "-a.p-y

L

::=

f2:(Qa.2-wa.2Qa.2)+!
éZ(Q.,2-wa?Q
2)+é
Cl

EL Aa.p-yQa.QpQy-l 2L fLa.p-yQa.QpQy·

a,j'!y

Aa,6'yQaQ;6Q'y-

QQBJY

P'a1'5"yQaQ;8Qy'

(28.2)

\'Ve
write out
\Ve shall not
not pause to write
out in their
their entirety the equations of motion
derived from this Lagrangian.
Lagrangian. The
The important feature of
of these
these equations is
derived
that they are of
of the form
that

Qa.+ We
w~2Qa.
Qa+
Q

(28.3)
fa.(Q, Q,
Q, Q),
Q),
where
where fa
fa. are
are homogeneous
homogeneous functions,
functions, of
of degree two, of
of the co-ordinates Q
and their
their time
time derivatives.
and
Using
successive approximations, we seek
seek a solution
solution of
of
Using the method of successive
these
these equations
equations in the
the form
Q
Qal1>_|_ Qalzi3
(28.4)
Q"' =
= Q,P>+Qa.<2>,
12

::=

far»

r

·where
Qa. !2) <<
~ Qam,
Qa. (1), and
and the
the Qa.
U> satisfy
satisfy the "unperturbed" equations
equations
where Qatzn
Qa(1)
Q)l>+wa.2Qa.<l) =
= 0, i.e. they
they are ordinary harmonic
harmonic oscillations
oscillations::
Q,0)+wa2Qal1)

Q
Q fa.<D
l)

= aa
aa. COS(OJaZ+
cos( wa.t +()(a.).
=
aa).

(28.5)

Retaining only
only the
the second-order
second-order terms
terms on the right-hand
right-hand side
side of
of (28.3)
(28.3) in
Retaining
Qa.!2) the equations
the next approximation, we have for the Qaw)

Qav2) + w2Qa(2) = /(Q<1)3 Qu) gal,

(28.6)

where (28.5)
(28.5) is to be substituted on the right. This gives a set of inhomoinhomowhere
geneous linear
linear differential
differential equations,
equations, in which
sides can
can be
geneous
which the right-hand
right-hand sides
simple periodic
periodic functions.
functions. For
For example,
example,
represented as sums of simple

+ aa) cos(w6t
+ as)
Q}l>Qp> .=
= aa
aa.ap
cos(wa.t+()(a.)
cos(wpt+()(p)
a cos(oJat

Qac(1)QJ6(D

=
l2aaa5{cos[(wa
uJ3)1
COS[(wa
- OJ5)I
aa - °'5]}~
= !aa.a
+w
+ aa
Cl.a. +
+ Ag]
()(p] +
+cos[(
w"'w0 )t -l*+()(a.CJ. 0 ]}.
0 {cos[( wa. +
0 )t +

the right-hand sides
sides of equations (28.6) contain
contain terms
terms corresponding
corresponding
Thus the
oscillations whose
whose frequencies
frequencies are the
the sums
sums and
and differences
differences of
~f the eigeneigento oscillations
frequencies of the
the system,
system. The
The solution
solution of these
these equations
equations must be sought
sought
frequencies
in a form involving
involving similar
similar periodic
factors, and
and so we conclude
conclude that, in the
periodic factors,
the
seco
second
oscillations with frequencies
nd approximation, additional oscillations

wa _+ OJ 5-,

(28.7)

including the daub
double
frequencies Zna
2wa. and
and the frequency
frequency zero (corresponding
(corresponding
including
Ie frequencies
to aa constant displacement), are superposed on
on the normal
normal oscillations
oscillations of
of the
the
to
ese are called
These
called combination j1·equencies.
The corresponding
corresponding ampliamplisystem.
frequencies. The
'st€1n. Th
sy
tudes are
ar~ Pfoportional
proportional to the products azza,
aa.afl (or
(or the
aa.2) of
of the cortudes
the squares as)
responding
respon?mg normal amplitudes.
amplitudes.
high
•·
In higher
· ns, whe
f urth er terms are
· 1u d ed 111
· the
Pproxlm
ff hher Laappr
atlo
. In
o:x:.~ations,
when
are included
me
m
the expann further
t
o
e
Lagrangian,
glo n
Combination frequencies OCCUr
sio~
~ff!r:nce~~:~~~~·
occur which are the sums
d differences
of
IC t
an
han two
two Wa
wa;, and
and aa further
further phenomenon
also appears.
appears.
an
MO
than
phenomenon also

_

-

-

-

86
86

§28

Small Oscillations
Oscillations

In the
the third
third approximation,
approximation, the
the combination frequencies
frequencies include
include some
some which
which
In
coincide
coincide with
with the original
original frequencies
frequencies Wa
wa ( =
= oJa+
wa + w5-05).
wp- wp)· When
When the
the method
method
above is used,
used, the right-hand
described above
right-hand sides
sides of the
the equations of motion
motion theretherefore include resonance
resonance terms, which lead
lead to terms in the
the solution
solution whose
whose
increases with
with time. It
It is physically
physically evident, however,
however, that
that the
amplitude increases
of
the
oscillationscannot
increase
of
itself
in
a
closed
system
magnitude
magnitude of
oscillations cannot increase of itself
closed system
external source
source of energy.
energy.
with no external
In
In reality,
reality, the
the fundamental
fundamental frequencies
frequencies Wa
wa in higher
higher approximations
approximations are
wa<o> which appear
appear in the quadratic
quadratic
not equal
equal to their "unperturbed" values warn)
not
The increasing
increasing terms
terms in the
the solution
solution
expression for the potential
energy. The
expression
potential energy.
arise
arise from an expansion
expansion of the type
type
cos(w a <m +
+ !1w
)t ~
;;:; cos
cos wa(mI
waW>twa <o>t '
cos(wa(0)
Away
- lt!1w
A o aa sin
sin walMt,
*~'
a

which 'is
is obviously
obviously not legitimate
legitimate when
when zt is sufficiently
sufficiently large.
which
In going to higher
higher approximations,
approximations, therefore,
therefore, the method
method of successive
successive
In
approximations must
must be modified
modified so that the periodic
periodic factors in the
the solution
solution
approximations
exact and
and not
not approximate
approximate values
values of
of the
the frequencies,
frequencies. The
The
shall contain the exact
shall
necessary
necessary changes
changes in the frequencies
frequencies are found by solving
solving the equations
equations and
requiring that
that resonance
should not in fact appear,
appear.
requiring
resonance terms should
taking the example
example of
of enharmonic
anharmonic oscilillustrate this method
method by taking
We may illustrate
lations
Lagrangian in the form
lations in one dimension,
dimension, and
and writing
writing the Lagrangian

L
!m.X2-!mwo2x2!mO'.x3-im,Bx4.
- mw02x2 - -§mwc3
- =41ml5'x4.
L = .gm-2

The corresponding
corresponding equation of
of motion
motion is
The
.X+ wo2x == -0cx2-}8x3.
- cxx2 - ,8x3.
5é+w02x

(28.8)

(28.9)

solution as a series
series of successive
successtve approximations
approximations::
We shall seek the solution
where
.go
(1)+x(2l+x(3)$
x = xxU>+x<2>+x<3>,
xu) = a
x<ll
a cos
cos wt,
wt
(28.10)
:I

'
w=
= "Jo
w 0 + tw<l)
w<2> + .....
with the exact value of w, which in turn we express as w
o ) + we)

(The initial
initial phase in xo)
x<l) can
can always be made zero by a suitable
suitable choice
choice of the
(The
of time.) The
The form (28.9) of
of the equation of
of motion
motion is not
not the
the most
most
origin of
substituted in (28.9), the left-hand
left-hand side is
convenient, since, when
when (28.10) is substituted
convenient,
not exactly
exactly zero. We
\Ve therefore
therefore rewrite
rewrite it as
not
0)022

:~

U02

(1- )°).x.

2
x+
x =
x+ we
wo2x
= --0cx2-Bx3-cxx2-,8x3- ( 1- : ; x.
O2
2
1,002

DO

Putting
w =
Putting xx == xU)-I-x(2),
xU>+ x<2>, w
=

11)
(28,
(28.11)

w
and omitting
omitting terms of
of above
above the
w0+w(1)
0 + w<~> and
second
second order
order of
of smallness,
smallness, we obtain
obtain for x(2)
xC2) the
the equation
,X<2> +
+ cu02X(2)
wo2:xf2> = -- ota2
CJ.a2 COS2OJI
cos2wt +
+ Zcu0wMa
2wow<Da cos
cos w
wtt
56(2)
::=

+

cos Zit
2wt + 2w0wMa
2wowma cos
cos iwt.
--lza2
o c a 2 --lza2
c x a 2 cos
i.

The condition
condition for
for the resonance
resonance term
term to
to be
be absent
absent from
from the
the fight-hand
ri~ht-hd~nd Side
side
The
ussed
sc
di
n
io
•is
•
J
(1)
Q
•
.
h
h
d
-rnat1on
lSCUSsed
at
im
ox
pr
IS simply
stmp y mu)
w
In agreement
agreement '.nt
t e second
secon ap
approx•
= 0,, in
with the

§29

Resonance in
nmz-linear osczllations
oscillations
Resouanee
in 11012-linear

87

beginning of this
this section.
section. Solving
Solving the inhomogeneous
inhomogeneous linear
linear equation
equation
at the beginning
the usual
usual way,
way, we have
in the

x<2>
= - - -2+
cos2wt.
Zwl.
x(2) +- - --- cos
2wo 60102
6w02
2w02
u.a2
oca2

Putting in (28.11)
Putting
stion
on for 36(3)
xC3>

.t' =

u.a2
(xa2

x< 1 >+x<2 >+x<3>, w
w

__

12)
(28,
(28.12)

we obtain
obtain the
the equaequa= w
w0
+ w ('22 1,)9 we
0 +w<

56(3)
x(3) := - 2oM1)m42)
x<3) +
+ 0)0'2
wo2:xf3l
2a.x<ll.?..(2)- »8X(1}:a
{3:\:(1)3 +
+ 2w0w(2)x(1)
2wow<2>.?.,.(1}
. _

or, substituting on the right-hand side (28.10) and (28.12) and
and effecting aa
simple transformation,
transformation,
simple

+· w02x<3>

= a3

II

.XC3l

[!f3- 6wo2].
QUO2

cx.

+a

]

2w0w(2)

_,_
cos S3wt
wt +

+

|

I

5 a2012

60)02

i.
ja'!f3] COS
cos iwt.

3
"ECU

Equating
Equating to zero the coefficient
coefficient of
of the resonance
resonance term
term cos wt, we find the
the fundamental
fundamental frequency, which
which is proportional
correction to the
proportional to the squared
oscillations::
amplitude of the oscillations

w<2)

w(2) ::

=

" 9
DOL-'
' 35
3{3
5a..2 )
8wo- 120J03
12wo3 a2.
( 80V0
I

_

ac).

(28.13)

combination oscillation
oscillation of the third order is
The combination
33(3)

as

16w02

1°

2

3W02

-B)
!/3) cos 3wt.

COS 3OJl*.

(28.14)

§29. Resonance in
in nonnon-linear
§29..
inear oscillations
V\'hen
\Yhen the enharmonic
anharmonic terms
terms in forced oscillations
oscillations of
9f a system are taken
taken
account, the phenomena of resonance
resonance acquire
acquire new properties.
properties.
into account,
Adding to the right-hand
right-hand side of
of equation
equation (28.9) an external
external periodic
Adding
periodic force
of frequency y, we have

+ 2)l.wE:+ w02x = (am) cos it

- aux"

(29.1)

here the frictional
frictional force, with damping
damping coefficient
coefficient AA (assumed
(assumed small)
small) has also
here
Strictly speaking, when
when non-linear
non-linear terms
terms are included
included in the
been included. Strictly
been
equation of
of free oscillations,
oscillations, the terms of higher
higher order
order in the amplitude
amplitude of
equation
the
n 1 force (such as occur
the exter
external
occur if
if it depends
depends on the displacement x) should
be
also
inc
lud
ed. \'\'
also be included.
\\'ee shall
shall omit these terms merely
merely to simpli
simplify
formulae;,
fy the formulae
they
they do
do no
nott aff
affect
thee qualitative
qualitative results.
results.
ect th
Let
"w ith e., small,
Let 'V
Y =". °U0~l-e
wo +.,with
small, i.e.
i.e. y be
be near
near the
the resonance value.
value. To
To ascertain
ascertain
1
the
resulting
type
of
tmg
tyPe
of
·
·
·
·
·
·
(29. 1)
the
resu
.
mot•on, ittt is
ts not
not necessary
necessarv to
to consider
constder equation
equation (29,
motion,
gue as
llo ws , I
iff we ar
argue
as fo
follo·ws
h
.
·
".
·
·
· 1)
· I nn the
t e linear
hncar approximation,
approxtmatton, the
the amplitude
amphtude b is
ls given
gtven

88

§29

Oscillations
Small Oscillations

near
resonance, as a function
function of
of the amplitude f and
and frequency y of
of the
the
near resonance,
which we write
write as
external force, by formula (26.7), which
external
b2(E2+A.2) = fZ/4m2w02.
f2/4m2wo2.
b2(e2+)l2)

(29.2)

"Jo + !<b2,

(29.3)
(29.3)

The
The non-linearity
non-linearity of
of the oscillations
oscillations results in the appearance
appearance of
of an ampliamplitude
dependence
of
the
eigenfrequency,
which
we
write
as
tude dependence of the eigenfrequency, which
write

definite function
function of
of the enharmonic
anharmonic coefficients
coefficients (see
the constant KK being
being a deflnite
w 0 by 0J0)l'
w 0 + Kb?
Kb2 in formula (29.2) (or, more
(28.13)). Accordingly, we replace We
y-w 0 ). With 'y-w
= e,
E, the
the resulting
precisely, in the small
small difference
difference y-w0).
precisely,
y - w 00 =
equation
is
equation
or

f)2[(e - Kb'2.)2 + A2]

::

f2/4M2oJ02

(29.4)

= Kb?
Kb~ i± »\
v[(f/2mwob)2
-A_2].
eE =
/ [(f/2f~wol))2 -?l2].

Equation (29.4)
(29.4) is a cubic
cubic equation
equation in 62,
b2, and
and its real roots
roots give the ampliampliEquation
tude
tude of the
the forced
forced oscillations.
oscillations. Let
Let us consider
consider how this
this amplitude depends
depends
frequency of
of the external
external force
force for a given
given amplitude
amplitude f of
of that force.
on the frequency
Vi/'hen
powers
When f is sufficiently
sufficiently small,
small, the amplitude Inb is also small, so that
that powers
of
of b above
above the
the second
second may
may be neglected
neglected in (29.4), and
and we return
return to the
the form
b(E) given by (29.2), represented
represented by a symmetrical
symmetrical curve
curve with a maximum
maximum
of b(6)
the point
point eE =
= 0 (Fig.
(Fig. 32a). As f increases,
increases, the curve
curve changes
changes its shape,
shape,
at the
retains its
its single
single maximum, which
which moves
moves to positive
positive eE if
if
though at first it retains
though
K > 0 (Fig. 32b). At this stage only one of the three
three roots of equation
equation (29.4)
is real.
When f reaches
reaches a certain
certain value
value /k
determined below),
however, the
When
je (to be determined
below), however,
nature
nature of
of the
the curve
curve changes.
changes. For
For all f > fx
/k there
there is a range
range of
of frequencies
frequencies in
which
portion
which equation
equation (29.4) has three
three real
real roots,
roots, corresponding
corresponding to the portion
BCDE in Fig. 32c.
BCDE
The limits
limits of
of this range are determined
determined by
by the condition
condition dbl
db/dE
= oo
oo W
which
The
de =
his
the points
and C. Differentiating
Differentiating equation
equation (29.4) with
with respect
holds at the
holds
points D and
respect to
E, we have
e,
db/dE
Eb + KIJ3)/(62
Kb3)f( E2 +212
+ A_24KEb2 + 3/<2b4),
3K2b4).
dbl
de = ((- et
-4K6b2

Hence the points
points D and C are determined
determined by the simultaneous
simultaneous solution
solution of
of
Hence
the equations
equations
the
£2-4r<b2e+3r<2b4~l-A2

=

0

(29.5)

and
and (29.4).
(29.4). The
The corresponding
corresponding values
values of
of eE are
are both
both positive.
positive. The
The greatest
greatest
=
0.
This
gives
E
=
Kb2,
and
from
(29.4)
amplitude
is
reached
where
dbjdE
amplitude is reached where dbl de
0. This gives e
/<b2, and from (29.4)
we have
max :°-bmax
=

Zmwgk ;
JJ2mwo>..;

this is
is the
the same
same as
as the
the maximum
maximum value
value given
given by
by (29.2).
(29.2).
this

(29.6
(29.6))

§29

Resonance in
in non-linear
non-linear oscillations
oscillations
Resonance

89

It may be shown
pause to do so hheret)
e r e ) that, of the
shown (though
(though we shall
shall not pause
three real roots of
of equation (29.4),
(29.4), the
the middle one
one (represented by
by the
the dotted
three
part
CD of
of the
the curve
curve in Fig. 32c) corresponds
corresponds to unstable
unstable oscillations
oscillations of the
part CD
system: any action, no matter how
how slight,
slight, on
on a system in such a state causes
causes
system:
oscillate in a manner
manner corresponding
corresponding to
to the
the largest
largest or smallest
smallest root (BC
(BC
it to oscillate
DE). Thus only the
the branches
ABC and
and DEF correspond
correspond to actual
actual oscilor DE).
branches ABC
the system. A remarkable
remarkable feature
feature here
the existence
existence of
of a range
of
lations of
of the
lations
here is the
range of
frequencies
je. For
For
frequencies in which
which two different
different amplitudes of oscillation
oscillation are possib
possible.
example,
example, as the
the frequency of
of the
the external force gradually
gradually increases,
increases, the
the amplitude of
of the forced oscillations increases along ABC.
At
C
there
is a disABC.
amplitude, which
which falls abruptly
abruptly to the value corresponding
corresponding
continuity of
of the amplitude,
continuity
afterwards decreasing
decreasing along the curve
curve EF
the frequency
frequency increases
increases
to E, afterwards
EF as the
the frequency
frequency is now
now diminished,
dimini9hed, the
the amplitude
amplitude of the
the forced
forced
further. If
further.
If the
oscillations varies
varies along
along FD,
FD, afterwards
afterwards increasing
increasing discontinuously
discontinuously from
from D
oscillations
then decreasing
decreasing along
along BA.
BA.
to B and then
b

f*O

(o)

r

b

(bl

fd

6

b

(,cl
Q31

5r
I

A

D

I

FIG~
FIG;

I

I
I

I

f>&

'

.1C
5

I
I
I

F
or

3-2

To
f for which
To calculate
calculate the
the value of fx,
/k, we notice
notice that
that it is the
the value of
off
which
2
the
the two
two roots of
of the
the quadratic
quadratic equation
equation in b2
b (29.5) coincide,
coincide; for f = fy,
/k, the
the
5€ctlon
CD
section CD redu
reduces
to aa point
point of
of inflection.
inflection. Equating
Equating to zero the
the discriminant
discriminant
ces to

1't The
The Pr
proof·
oof 18 gi·ven by
Y.A
. Methods i is
llven by,, fo
forr example,
example, N.
N. N.
N. BOGOLIUBOV
BOGOLIUBOv and
andY.
A. MITROPOLSKY,
MITROPOLSKY, AsympAsymp.
.
.
.
'
. .
in Method '
h
.
~~hi 1
61.
.,
t
e
Theory
of
Non-Linear
Oscillations,
Hindustan
Publishing
Corporation,
Non-Lmear
Osczllanons.
Hindustan
Publnshlnz
Corooratlon.
of
ry
Theo
e
¢
.
In
s
1
19
6
1821h 9

'

90

Small Oscillations
Oscillations
Small

§29

32m'2w02A3/3V31r<1.

(29.7)

of (29.5), we find £2
I<b2 =
E2 =
= 3212,
3/.2, and the corresponding
corresponding double root
root is Kb2
= Z6/3
2E/3..
Substitution
of
these
values
of
b
and
e
in
(29.4)
gives
Substitution
these values
and E

-

Besides
resonance at freBesides the
the change
change in the
the nature of the
the phenomena
phenomena of
of resonance
wo, the non-linearity of the oscillations
oscillations leads also to new
new
quencies y ~ we,
of frequency
frequency close to We
w 0 are excited
excited by
by an
resonances in which
which oscillations
oscillations of
resonances
external
w 0•
external force of frequency
frequency considerably
considerably different
different from woLet
~~ !w
Let the frequency
frequency of
of the
the external
external force
force yy '*~'
y =
= !wo+E.
the
l2w0,
}w0+€. In the
0 , i.e. 'y
(linear) approximation, it causes
causes oscillations
oscillations of the
the system with
same
first (linear)
with the same
frequency
proportional to that
and with
with amplitude
amplitude proportiomil
that of
of the force
force::
frequency and

2) cos(w0+
xo)
3mw02)
x<D = (4f
( 4ff3mwo
cos(!wo + e)t
E)t

~

(see (22.4)).
(22.4)). When
When the
the non-linear terms are included
included (second
(second approximation),
approximation),
of frequency
frequency Zy
2y "~'
~ We
w 0 on the right-hand
right-hand
these oscillations
oscillations give rise to terms of
these
of the equation
equation of motion
motion (29.1). Substituting
Substituting xo)
xU> in the equation
equation
side of
side
xcz)a +g
2)3 =
x(1)2
. gt 35(2)
x<2J +2M-(2)
+ 2A.X<2> +
+ w02x(2)+
w02x(2l + ()(x<2>2
+ ,Bx<2>3
= _ ()(x<l>2,

_

using the
the Cosine
cosine of
of the double
double angle
angle and
and retaining
retaining only
only the resonance
resonance term
term
on the
the right-hand
rigl}.t-hand side,
side, we have
have
12) +2)l,;"(2)
_|_ (IJ02x(2)
_|_ Mm2)2
_|. »8x(2):a
jf-2)
+ 2.\X(2l +
w02x<2> +
()(x<2>2 +
,BJd-2>3

=

-

(8()(J2/9m2w0 4) cos(w0
cos( wo +2€)t.
+ 2E )t.
(8af2/9m2w04)

(29.8)

equation differs from (29.1)
(29.1) only
only in that
that the amplitude
amplitude f of the
the force is
This equation
replaced
proportional to f2.
replaced by an expression
expression proportional
J2. This means
means that
that the
rhe resulting
resulting
of the same type
type as that
that considered
considered above
above for frequencies
resonance is of
resonance
y z~ we,
but is less strong. The
b(6) is obtained
w 0 , but
The function
function b(E)
obtained by replacing
replacing f by
4 , and e
8 af2/9moJ04,
-8()(f2f9mwo
E by Ze,
2E, in (29.4)
(29.4)::
b2[(2E-Kb2)2+/.2] = 16oc2f4/81m4w010.
16x2f4f81m4wolO.
b2[(2e-:<blf3)2+A2]

(29.9)

Next, let the
the frequency of
of the
the external
external force be y =
= 2010-I-€.
2wo +E. In
Jn the
the first
first
x<ll =
= -(f/3mw02)
-(Jf3mwo2) cos(2w0+e)t.
cos(2wo+E)t. On'
On· substituting
approximation, we have x(1)
approximation,
2
= x(1)-I-x(2)
x<1>+x< >in equation
equation (29.1),
(29.1), we do not
not obtain terms representing
x =
representing an
occurred in the
the previous
previous case. There is,
external force
force in resonance
resonance such as occurred
external
however, a parametric resonance resulting from
from the
the third-order term prohowever,
portional
x<1>x<2>.
this is retained
retained out
out of the non-linear
non-linear
portional to the product
product xi
)x(2). If
If only this
the equation
equation for al?)
x<2> is
terms, the
xu)x(2)
.x<z> +2AX<2)
+ 2,\X(2> +
+ w02x(2)
wo2x<2> = - 2
2()(x<llx<2>
55(2)
or
or

+

2
[1 - 3mwo
C!f

lx-(2) + (U0
_x(2)) +2y
+ 2.\X<Z>
w022 1__
552

Zoxf

3??109044

Qt])t] x<2> = 0,

+E

cos(2w0 +
cos(2w0

36(2)

0,

10)
(29.
(29.10)

CadSa aS
We
t.e. an equation
equation of
of the
the type (27.8) (including
(including friction),
friction), which Ileads,
as _we
i.e.
have seen,
seen, to
to an
an instability
instability of
of the
the oscillations
oscillations in
in aa certain
certain range
range of
of frcquiinc1es.
frequencies.
have

§29

Resonance
Resonance in
in non-linear
non-linear oscillatiofzs
oscillations

91
91

however, does
does not
the resulting
ampliThis equation, however,
not suffice to determine the
resulting amplitude of
of the
the oscillations.
oscillations. The attainment of a finite amplitude involves
involves nontude
linear effects, and
and to include
include these in the equation
equation of
of motion
motion we must retain
the terms non-linear
x<2>:
also the
non-linear in elm;

5él2)+2A;i%(2*+ We2x(2) + ax'2)2-I-}3xl2)3 = (2af/3m w02)x(2) cos(2w0-l~ e)t.

(29.11)
The problem
considerably simplified
simplified by virtue of
of the
the following
following fact.
The
problem can be considerably
Putting
(2) =
Putting on the right-hand
right-hand side of (29.11) xx<2>
= b cos[(0J0+€)t+8],
cos[(wo+!E)t+S], where
resonance oscillations
b is the required
required amplitude of
of the
the resonance
oscillations and
and 8S a constant
constant
of no importance
importance in what
what follows, and
and writing
writing the
phase
which is of
phase difference which
2
product of cosines
,product
cosines as a sum, we obtain
obtain a term
term (afb/3mw02)
( cxjb f3mwo ) cos[(w0-|-l2e)tcos[( w 0 + }E)t- 8]
S]
of
of the
the ordinary resonance
resonance type
type (with
(with respect
respect to the
the eigenfrequency
eigenfrequency to
w 0 of the
the
system). The problem
reduces· to that
tha~ considered
considered at the
the beginning
system).
problem thus reduces-Ito
beginning of
only
namely ordinary
ordinary resonance
resonance in a non-linear
non-linear system, the only
this section, namely
differences
differences being that
that the
the amplitude
amplitude of
of the
the external
external force is here
here represented
represented
by afbl3w02,
cxfbf3wo2, and
and eE is replaced
replaced by
by !E.
Making this change
change in equation
equation (29.4),
(29.4),
by
e, Making
we have

b2[(}E- Kh2)2 +/.2] =
= a2_f'3b2/36m2w06.
cx2f2b2f36m 2wo6.
b2[(e-z<b2)2+Pl2]

Solving for
forb,
find the
the possible
values of the amplitude
amplitude::
Solving
b, we Find
possible values

bb=O
0,,

112

- l f + V{(°'f/6"2tJJ03)2

)\2}]»

62 = -[%-v{(f/6~tw03)2-»2}].

(29.12)

(29.13)

(29.1
4)
(29.14)

·Figure 33 shows the resulting
resulting dependence
dependence of b on eE for K > 0,
0; for KK < 0
Figure
curves are the
thereflections
of those
those shown.
shown. The
The points
the curves
reflections (in the b-axis) of
points B
3 )2-4/.2}. To
and
_+ x
and C correspond to the
the values
values eE =
= ±
v{(cxff3mwo
To the
the left of
/ {(af/3mw03)2-4A2}.
only the value b =
= 0 is possible,
there is no resonance,
and oscillations
oscillations
B, only
possible, i.e. there
resonance, and
of frequency
frequency near to
w 0 are not
not excited. Between
Between B
Band
two roots,
roots,
of
and C there are two
b=
= 0 (BC) and
and (29.13) (BE). Finally,
Finally, to
t9 the right
right of C there
there are three roots
roots
these, however,
however, correspond
correspond to stable
stable oscillations.
o!>cillations.
(29.12)-(29.14). Not
(29.12)-(29.14).
Not all these,
The value b =
= 0 is unstable
unstable on BC,]"
BC, t and
and it can also be shown
shown that
that the middle
middle
The
The unstable
unstable values of
of b are shown
shown in
instability. The
root (29.14) always gives instability.
Fig. 33 by dashed lines.
Let us examine,
examine, for example,
example, the
the behaviour
of a system
system initially
initially "at
"at rest"i
rest"t
Let
behavior of
as the
the frequency
frequency of
of the
the external
external force
force is gradually
gradually diminished. Until
Until the
the point

c

T"1: This
This segment
segment corresponds to the
the region of
of parametric resonance (27.12),
(27.12}, and
and a compMShH h0f (2
9.
3l >
4 • The
parh~shonthof <h29
·10)
and (27.8)
(27.8} gives lhl
= 2rt.ff3mwo
The condition
condition l2
l2aj/3m,;,o
> 4.\
for
=
Zafl3mwo"*.
af]3mw03l
4A for
10) and
hl
\-V IC
e Phenol
p eno
t e
whlc
ca
n
ex
+
It
should
b
rnenon
can
exist
>
hk.
corresponds
to
h
>
he.
ist
; It should be
h• nomena are
are ab
a;,se
recalled
phenomena are under consideration. If
If these
phenomena
nt9 the that only resonance
~f ~requency y. sent, the Systlim
system is
is not
not literally at
at rest, but
but executes small fc reed oscillations
Of frequency

r2gi11ed

92

§29

Small Oscillations

reached, b =
= 0, but
the state of the
the system
system passes
passes discontinuously
discontinuously
C is reached,
but at C the
to the
branch EB.
the branch
EB. As eE decreases
decreases further,
further, the amplitude of
of the
the oscillations
oscillations
decreases to zero at B. When
When the frequency increases
increases again,
again, the
the amplitude
decreases
increases
increases along BE. Tt
b

Ill

I

.-

II

lu

"\

.

E

I

cu,

I

II I

A

FIG.
FIG.

•

I

-

_

33

c

H

tI

.",.-->**F

.0

r

The
The cases
cases of
of resonance
resonance discussed
discussed above are
are the
the principal ones
ones which
which may
higher approximations,
approximations, resonances
resonances
occur in a non-linear
non-linear oscillating
oscillating system.
system. In higher
appear at other frequencies also. Strictly speaking, a resonance
resonance must occur
at
ny-I-mw0 =
and m integers,
at every frequency
frequency 'yy for which
which ny+mwo
= to
wo with
with nnand
integers, i.e. for
every
pwglq with P
= pwo/q
p and q integers.
integers. As the degree of approximation
approximation
every y =
increases,
the strength of the
the resonances,
resonances, and
and the widths of
of the
the
increases, however,
however, the
ranges in which
which they
they occur,
occur, decrease
decrease so rapidly
rapidly that in practice
frequency ranges
practice
only
w POJO/Q
resonances at frequencies
frequencies yy :::::
Pwo/q with
with small P
p and q can
can be obonly the resonances
served.
PROBLEM
PROBLE
M

Determine the
the function
function 12(6)
b(£) for
for resonance
resonance at
at frequencies
')' z
~ 3
3wo.
Determine
frequencies 'y
w0.

1l == -(f/8mw02)
SOLUTION. In
In the
the first approximation,
approximation, x1ml)
-(f/8mwo 2) cos(3
cos(3wo+£)t.
For the
the second
SOLUTION.
to -l~f)t. For
2
approximation box
xC l we
we have
have from
from (29.1)
(29.1) the
the equation
equation
appromdmation

.

x"(-1
x(1)x(2}2
9 +2Ax(-»1+w02x(2>+¢¢x(2)2+Bx(2;a
=
..
_x(2J+2M(2J+wo2x(2J+ax(2)2+tJxC2J3
= -3B
-3,SxC1Jx(2J2,

where only
only the
the term
term which gives
gives the
the required
required resonance
resonance has
has been
been retained
retained on
on the
the right-hand
right-hand
2> ==
side. Putting
Putting x<
= bb cos[(w0
cos[( roo -I-it)t+'8]
+l~:)t+8J and
and taking
taking the
the resonance
resonance term
term out
out of
of the
the product
product
side.
x(2)
of three
three cosines,
cosines, we
we obtain
obtain on
on the
the right-hand
right-hand side
side the
the expression
expression
of
2) cos[(wo-l~§f)t-2
'81(3,Sb2J/32mwo
cos[(wo+l£)t-28].
(3,8b2f/32m
w02)

2, and
Hence itit is
f by 3Bb'ff/32
w02,
be, in
is evident that b(6)
b(£) is
is obtained by
by replacing
replacing/by
3,Sb2J/32wo
and£e by
by!£,
in
(29.4)::
(29.4)
b2[(!£-Kb2)2+.\2)
(9tl2J2f212m2wo6)b4 E= Abs.
Ab4.
b2[(§e-xb2)2 -l~)l2] == (9B2f2l212m2w00)g,4

The
roots of
of this
this equation
equation are
are
The roots

b

=

0,

be

Lu

c
-°-|-

3x

A +
2 x2 -

ex +

eA

A8 _ A g
4x8

of the
the function b(6)
b(o:) for ac
K >
> 0. Only the
the value b = O
0 (the f-axis)
£-axis} and
and
Fig. 34 shows aa graph of
2)/4KA,
the branch
branch AB
AB corresponds
corresponds to
to stability.
stability. The
The point
point A
A corresponds
corresponds to
to£~:
3(4K2.\2-A
the
Gr = 3(4l~c8)l2
-A2)/4*=4,

t

T It must be noticed, however, that all the formulae derived here are valid only
when
at
eer, and
amplitude bb (and
(and also
also e)
o:) is
is sufficiently
sufficiently small.
small. In
In reality,
reality, the
the curves
curves BE
BE and
and CF
CF m
n>eet,
and at
amplitude
their point
point of
of intersection
intersection the
the oscillation
oscillation ceases;
ceases; thereafter,
thereafter, bb == 0.
0.
their

It must be noticed, however, that all the formulae derived here are valid onlY when the
the

§30

Motion
Motion in
in aa rapidly
rapidly oscillating
oscillating field
field

93

6~; 2 = (4-l~c8)l2
(4K2.\2'l"/12)/4x8A.
+A 2)/4K2A. Oscillations
Oscillations exist only for
for.:>
£~;, and
and then b
b > Br.
b~;. Since the
the state
be?
15 >- Gr,
0 is
is always stable,
stable, an
an initial "push" is
is necessary
necessary in
in order
order to excite oscillations.
oscillations.
b = 0
The formulae
formulae given above
f. This condition
A is
above are
are valid
valid only for
for small
small£.
condition is satisfied ifif.\
is small
the amplitude of
of the
the force is
is such that »c?\2[cv0
K.\ 2{ wo ~ A
Kwo.
and also the
A ~ xw0.

< <

b

-5
AI

\" ` - * " * _ /

C

1-'

FIG.
FIG.

34

Motion in
in a re
rapidly
pidly oscillating field
§30. Motion

Let us consider
consider the
the motion
motion of
of a particle subj
subject
Let
e t both to a time-independent
field of potential U and
and to a force
Held
/_f =
=

fl
wt +f2 sin wt
h cos
coswt+f2sinwt

(30.
1)
(30.1)

frequency cu
w (/1,/2
functions of
of the
which varies in time with a high frequency
(f1,f2 being functions
frequency we mean one such that w >>
~ 1
1/T,
co-ordinates only). By a "high" frequency
co-ordmates
/ T,
order of magnitude
period of the
the motion
motion which the
where T is the order
magnitude of the period
off is not
not assumed
assumed
particle would
execute in the field U alone. The magnitude
particle
would execute
magnitude off
forces due to the field U, but
shall assume
assume
small in comparison
comparison with the forces
small
but we shall
that the
the oscillation
oscillation (denoted
(denoted below
below by 5)
g) of the particle
that
particle as a result
result of this
force is small.
To simplify
simplify the calculations,
calculations, let us First
first consider
consider motion
motion in one dimension
dimension
To
field depending
depending only on the space
space co-ordinate
co-ordinate x. Then the equation
equation of
in a Held
motion
motion of the particle
particle is'
ist
I

mi? = -dU/dx+f.
-dUfdx+f.

mx

(30.2)

xx(t)
X(t) +§(t)»
+g(t),
(i) = XG)

(30.3)

It
evident, from the nature
nature of the
the field in which
which the
the particle
that
It is evident,
particle moves, that
smooth path
and at the same
same time
time execute
execute small
small oscillations
oscillations
it will traverse a smooth
path and
about that
that path. Accordingly,
Accordingly, we represent
represent the
the function
function x(t)
of frequency w about
as a sum
sum::
II

where §(t)
g( t) corresponds to these small
small oscillations.
oscillations.
where
The mean
mean value
value of the
the function go)
g(t) over
over its
its period
period 2TT/w
zero, and
and the
Zvrlw is zero,
fu
nc
tio
n
X
(t) changes
function X(t)
changes only slightly
slightly in that
that time.
time. Denoting
Denoting this average
average by a
bar,
refore ha
bar, we
we the
therefore
have
x == X(t), i.e. X(t)
X(t) describes
describes the "smooth" motion
motion of
ve x'
1't The
-ordinate x
The co
CO-ordinat

ed not
and
th x ne
ne<:d
not be
be Cartesian,
Cartesian,
and the
the coefficient
coefficient m
m is
is therefore
therefore not
not necesneces.
the
particle, n
or nee
asSumption,
howevere
J'artrcle,
nor
need
be constant
constant as
as has
has been
been assumed
assumed in
in (30.2).
(30.2}. This
This
d ltit be
BSSI-HNPtioN' however. doe8
' oes Not
not affect
affect the
the final
result (see
{see the
the last
last footnote
footnote to
to this
this section}.
final result
section).

sarilY the
the fflass
mass of
of
warily

94

Small
Small Oscillations
Oscillations

§30

averaged over
over the
the rapid
rapid oscillations.
oscillations. vVe
derive an equation
equation
the particle averaged
\fee shall derive
which
the function
function X(t).1
X( t). t
which determines the
Substituting (30.3) in (30.2) and
and expanding
expanding in powers
Substituting
powers of fg as far as the
first-order terms, we obtain
..

mX-l~m.§
mX +mg =

5 d2
d 2U
dx2

ddU
U
dx

of
oX

---g-+f(X
t)+g-.
+ f ( x , z)+§
'

(30.4)

involves both oscillatory
oscillatory and
and "smooth" terms, which
which must
This equation involves
For the oscillating
oscillating terms we can
can put
evidently be separately equal. For
put simply

to?
f ( x , Z);
mg ==!(X,
t);

(30.5)

ifg =
= -flmwz.
-Jfmw2.

(30.6)

the other terms contain the
the small ffactor
and are therefore
therefore of
of a higher
higher order
order
the
Iactor fgand
of smallness (but
(but the
the derivative g is proportional to the
the large quantity we
w2
of
and
f given
and so is not
not small).
small). Integrating equation
equation (30.5)
(30.5) with
with the
the function
function/
given by
(regarding X as a constant),
constant), we have
(30.1) (regarding

g

Next, we average equation
equation (30.4)
(30.4) with
with respect
respect to time (in the sense discussed
discussed
Next,
the mean
mean values of
of the
the first
first powers off
off and
and fgare zero, the
the result
above). Since the
is
IS

_ dU
dU

_

3of

dU

_

my
mX = ---+g-= ---+5 f =
oX
EX

do
dX

dX

1

mw2

f of

EX

involves only the function
function X(z).
X(t). This equation
equation can be written
written
which involves

'

my
mX :=
=

- dUff;ldXJ
dUerr/dX,
-

where the
the "effective
"effective potential energy" is defined
defined ask
ast
where

\ Uerr

U +j2/2mw2
+f2,12771oJ2
U
-

2 +f22),141?2w2.
U+
U12
U
+(JI
+/2 2)/4mw2.

II

Ueff =

=

(30.7)

(30.8)

Comparing this
this expression
expression with (30.6),
(30.6), we easily
easily see that
that the
the term added
added to
Comparing
just the mean kinetic energy
U is just
energy of the oscillatory
oscillatory motion
motion::
Ueff

=

U+ m§ -2,

(30. 9)
(30.9)

motion of
of the
the particle
averaged over
over the oscillations
oscillations is the
the same
same
Thus the motion
particle averaged
if the constant
constant potential
potential U were augmented
augmented by a constant
constant quantity
quantity proas if
the squared amplitude of
of the
the variable
variable Held.
field.
portional to the
t The principle
principle of
of this derivation
derivation is due
due to
to P. L. KAP1TZA
KAPITZA (1951).
1'

(JO

7
lae (30.7)
!I By
By means of
of somewhat more lengthy calculations it is easy to
to show that formu
formulae
· )

and (30.8)
(30.8} remain valid even if
if m is a function of
of x.
x.
and

§30

95

field
Motion in a rapidly oscillating field

can easily
easily be generalised
generalised to the
the case of
of a system with
number
The result can
with any number
of degrees
degrees of freedom,
freedom, described
described by generalised
generalised co-ordinates
co-ordinates qiqi- The
The effective
not by (30.8), but by
potential
potential energy is then given not

L¢1l1
a- i1.¢ft:f1.¢
ik/i/k
2w2'E
i,k

1
U+
U
+20J2

II

Uerr =
Ueff

II

- U+

1

-

L
Z laikflk.

i,z.¢
i,k

(30.10)

dike,

the quantities a-lu,
which are in general
general functions of the
the co-ordinates,
where the
f l a k , which
the elements of
of the
the matrix inverse to the
the matrix
matrix of
of the
the coefficients
coefficients ask
aik in
are the
are
the kinetic energy
energy (5.5) of
of the
the system.
system.
PROBLEMS
P
ROBLEMS

1. Determine the positions of
of stable equilibrium of
of a pendulum whose point of
of
PROBLEM 1.
with a high
('}:> V
y(g/l)).
support oscillates vertically with
high frequency ''Yy (>
(g/l)).

the Lagrangian derived in §5, Problem 3(c),
3(c), we see
see that
SOLUTION. From the
that in this
this case the
-mlay2 cos 'yt
yt sin
sin 56
r/> (the
(the quantity
quantity x being here represented by
by the
the angle
angle
variable force is fI=
= -mlag/2
rf>). The "effective potential energy" is
is therefore Uen
Uen =
= mg[--cos
mgl[ -cos gf>+(a2'y2/4gl)
r/>+(a 2y2f4gl) sin2g.f>].
sin 2r/>]. The
go).
positions of
of stable
stable equilibrium
equilibrium correspond to the minima
minima of
of this
this function.
function. The vertically
vertically
positions
(r/> =
= 0) is
is always stable. If
the condition a2'y2
a 2y2 >
2gl holds, the
downward position (of)
If the
> 2gl
the vertically
( r/> =
= or)
w) is also stable.
upward position (96
1, but
but for
for a pendulum whose point of
of support oscillates
PROBLEM 2. The same as Problem 1,
horizontally.
horizontally.

I=

SOLUTIO~- From
From the
the Lagrangian
Lagrangian derived
derived in
in iS,
§5, Problem
Problem 3(b),
3(b), we
find .f = mlay2
mlay2 cos
cos 'it
yt
SOLUTION.
we End
2
2
2
cos QUO
r/> and
and Uetl
Uen == mg[--cos
mgl[ -cos 96-I-(a2y2]4gl)
rf>+(a y2f4gl) cos2gf>].
cos rf>]. IfIf a2'y2
a2y < Zeal,
2gl, the
the position
position 96
r/> = 00 is
is stable.
stable.
cos

If a272
a 2y2 > 2gl,
2gl, on
on the
the other hand,
hand, the
the stable equilibrium position is
is given by
by cos ¢r/>
If

=

..=

2glfa2 y 2 •
2gl/a2'y2.

CHAPTER
VII
C
HAPTER V

MOTION
OF
RIGID
BODY
M
OTION O
F A R
IGID B
ODY

§3l.
§31. Angular velocity
A rigid
rigid body may
may be defined
defined in mechanics
mechanics as a system
system of
of particles such that
that
the distances
distances between
the particles do not
not vary. This condition
condition can,
can, of
of course,
course,
the
between the
by systems
systems which
which actually
actually exist
exist in nature.
be satisfied
satisfied only
only approximately
approximately by
nature.
The majority
majority of solid
solid bodies,
bodies, however,
however, change
change so little
little in shape
shape and
and size
under
ordinary conditions that
that these
these changes
changes may be entirely
entirely neglected
neglected in
under ordinary
of motion
motion of the body
whole.
considering the
the laws of
considering
body as a whole.
In what follows, we shall
shall often
often simplify
simplify the
derivations by regarding
regarding a
In
the derivations
body as a discrete
discrete set
set of
of particles,
particles, but
but this in no way invalidates
invalidates the
rigid body
rigid
assertion
bodies may
regarded in mechanics
assertion that solid
solid bodies
may usually
usually be regarded
mechanics as continuand their internal
internal structure disregarded.
disregarded. The passage
formulae
ous, and
passage from the formulae
which involve
involve a summation over
over discrete particles to those for a continuous
which
body is effected
effected by simply
simply replacing
replacing the mass of each particle
body
particle by the mass
contained in a volume
volume element
element d V (p being
being the density)
density) and
and the
the sump d V contained
by an integration
integration over
over the
the volume
volume of
of the
the body.
mation by
To
body, we use
use two systems of
To describe
describe the
the motion
motion of
of a rigid
rigid body,
of co-ordinates
co-ordinates::
a "fixed" (i.e. inertial)
inertial) system
system XYZ,
XYZ, and
and a moving
moving system
system Xl
x1 =
= x, 302
x2 =
= y,
x 3 = 2:
z which
which is supposed
supposed to be rigidly
rigidly fixed in the
the body
x3
body and to participate
participate
The origin
origin of
of the
the moving
moving system may
may conveniently
conveniently be taken
taken
its motion.
in its
motion. The
mass of
of the body.
coincide with the centre of mass
to coincide
The
position
of
the·body
with
respect
the fixed
system of
of co-ordinates
co-ordinates
The position of the°body with respect to the
Fixed system
completely determined if
if the
the position
the mowing
moving system is specified.
specified.
is completely
position of the
Let the origin
origin O
0 of the moving
moving system
system have the
the radius
radius vector
vector R (Fig. 35).
Let
The orientation
of
the
axes
of
that
system
relative
to
the
fixed
system
orientation of
of that system relative the
system is given
given
three independent
independent angles, which
which together
together with the
the three
three components
components of
by three
the vector
vector R make six co-ordinates. Thus a rigid
mechanical system
system
the
rigid body
body is a mechanical
with six degrees of
of freedom.
with
'
Let
Let us consider
consider an arbitrary infinitesimal
infinitesimal displacement
displacement of a rigid
rigid body.
It
can be represented
represented as the sum
sum of two parts. One
One of
of these
these is an infinitesimal
infinitesimal
It can
whereby the
the centre of
of mass
mass moves to its
its final position,
position,
translation of
of the body,
translation
body, whereby
but the
the orientation
orientation of
of the
the axes of the
the moving
moving system
system of
of co-ordinates is unbut
changed. The other is an
an infinitesimal
infinitesimal rotation
about the
the centre of
of mass,
changed.
rotation about
the remainder of
of the
the body
body moves
moves to its
its Final
final position.
whereby the
Let rr be the radius
radius vector
vector of an arbitrary
arbitrary point
the
Let
point P in a rigid body
body in the
moving system,
system, and
and rt the
the radius vector
vector of
of the
the same point in the
the Fixed
fixed_ syst€rn
system
moving
lacedisp
a
of
ts
the infinitesimal
infinitesimal displacement dt
dt of P con
consists
sis of d•s~lace­
(Fig. 35). Then the
d
4> X rr
splacement <I>)(
ment dR,
dR, equal
equal to
to that
that of
of the
the centre
of mass, and a2 di
displacement
rent
ntre of mass, an d
ce

no

§31

97

Angular velocity
Angular

relative to the
the centre of mass
mass resulting
resulting from a rotation
rotation through
through an infinitesimal
infinitesimal
relative
angle deb
c1p (see (9.1)): dt
dr =
= dR+dq>xr.
dR+dcJ>xr. Dividing
Dividing this
this equation
equation by the
the time
time
dz
dt during which
which the
the displacement
displacement occurs,
occurs, and
and putting'
puttingt

dr/dr
dr/dt =
= v,

we obtain
obtain the
the relation
we

dR.,/dz
dRfdt =
= V,

v = V+.Qxr.
V-I-Qxr.

dcj>ldt
= so,
.Q,
=
ad:
do
'

Z
z

XI

(31.1)

(31.2)

.xz

yy
X

FIG.
FIG.

35

The
body, and
The vector
vector V is the
the velocity
velocity of
of the
the centre of mass
mass of
of the
the body,
and is also
the translational
translational 'L'elocity
the body. The vector
vector SZ
.Q is called
called the angular
the
velocityy of the
of the
the rotation
of the body;
body; its
its direction, like that
that of
of do,
dcj>, is along
along the
the
velocity
'velocity
rotation of
y of
of rotation.
rotation. Thus the velocity
velocity v of
of any
any point
body relative to the
axis of
point in the body
of co-ordinates
co-ordinates can
can be expressed
expressed in terms of
of the translational
translational
system of
fixed system
the
body
and
its
angular
velocity
of
rotation.
velocity
of
velocity of
body and its
velocity of
It should
should be emphasised
emphasised that,
that, in deriving
deriving formula
formula (31.2), no use has been
of the fact that
that the
the origin
origin is located
located at
at the centre
centre of
of mass. The advanadvanmade of
of this choice
choice of origin
origin will become
evident when we come to calculate
calculate
tages of
become evident
the
the energy
energy of the moving
moving body.
Let us now assume
assume that the system
system of co-ordinates
co-ordinates fixed in the
the body
body is
Let
origin is not at the centre of mass
mass O,
0, but
some point
0' at
such that its origin
but at some
point O'
distance a from O.
0. Let
Let the velocity
velocity of O'
0' be V', and the angular
angular velocity
a distance
the new system
system of
of co-ordinates
co-ordinates be SF.
.Q'. \fee
vVe again consider
consider some point
P
of the
point P
r' its radius
radius vector
vector with respect
respect to O'.
0'. Then
denote by r'
in the body, and denote
= r'-l-a,
r' +a, and substitution
substitution in (31.2)
(31.2) gives v =
= V-l-Slxa-l-SZxr'.
V + .Q x a+ .Q x r'. The
The
r =
definition of V' and
and SZ'
.Q' shows
shows that
that v =
= V' + SZ'
.Q' xr'.
x r'. Hence
Hence it follows that
definition
= V-l-$l2
V +.Q xa,
.Q' = .Q.
V' =
""
No
(31.3)
The second
second of these
these equations is very
very important. We
We see that the angular
velocity of
of rotation,
rotation, at
at any
any instant,
instant, of
of aa system
system of
of co-ordinates
co-ordinates fixed in
velocity
the
y is inde
pendent of
body
independent
of the
the particular
particular system chosen. All such
such systems
the bod
told
1
an
-|To
t city
To aavoid
any
mis
d
y misun
nding it
ad be
the angular
is somew
sornewh
t unb.dersta
erstanding,
it she
should
be noted that this way of
of expressing the
is
hat
tY
veloci
I
arbitrary
g th
ve
o
a
ar
ttrary·
~A..
•
"nfi mtestma
· ·
1 rotation,
·
¢ exists
only
and not
tions*
for all Finite
finite Ilota
rotations.
· the
e vector
vector 8"'I'
extsts
on 1y cfor
wr an
an infinitesimal
1
rotatiOn,
for

98

jJotion of
of a Rigid
*Motion
Rigid Body
Eody

§32.
§32

rotate with angular
angular velocities
velocities $2
.Q which
which are
arc equal in magnitude and parallel
parallel
rotate
enables us to call .Q
angular 'z~eIorz`r|v
ulocity of
of the body.
bolzv. The
The
in direction. This enables
$2 the angwlar
velocity of
of the translational
translational motion,
motion, however,
however, does
does not have
have this
this "absolute"
velocity
property.
seen from the first formula (31.3) that, if V and
and .Q are, at any given
given
It is seen
instant, perpendicular
perpendicular for some choice of
instant,
of the origin O,
0, then V' and SZ'
.Q' are
perpendicular
for
any
other
origin
0'. Formula
Formula (31.2) shows
shows that in this case
perpendicular
other origin O',
velocities v of all points in the body are
arc perpendicular
perpendicular to SZ.
.Q. It is then
the velocities
always possiblet
possible to choose an origin
origin O'
0' whose velocity V' is zero, so
so that the
of the body
at the
the instant
instant considered
considered is a pure rotation
rotation about
about an axis
motion of
body at
through O'.
0'. This axis
axis isis called the instantaneous
instantaneous a\'is
of 1°otation,z;;
rotation.:~
through
Avis of
In what follows we shall always suppose
suppose that
that the origin
origin of
of the moving
moving
centre of mass
mass of
of the body,
body, and so the axis of
of
system is taken to be at the centre
passes through the
rotation passes
both the magnitude
the centre of
of mass. In general
general hoth
and the
the direction
direction of
of $2
.Q vary during
during the motion.
motion.
and
§32. The inertia tensor
tensor
§32.

To calculate
calculate the kinetic energy
energy of
of a rigid
rigid body, we may consider
consider it as a
To
2
discrete
particles and put
Xiiinz~2,
put T == ~
~ml' , where the
the summation
summation is
discrete system
system of
of particles
taken over
over all the particles
the body.
body. Here,
Here, and in what follows, we simplify
simplify
taken
particles in the
by omitting
omitting the suffix which
dcnumerates the particles.
notation by
which denumerates
the notation
Substitution of
of (31.2) gives
Substitution

L
Z

L

L

- m1m(.Q
( Q. xr)2.
r 12 -Z§mV2-I-21/n
V -· Q
r + T
T
~m(V -l~SI2
+,Q xxr)2
~mV2+
mV
.Qxxr+
~
T =
§m(V
-=
The velocities
velocities V
V and
and $2
.Q are
are the
the same
same for
for every
every point
point in
in the
the body,
body. In the first
The
term, therefore,
therefore, tV2 can
can be
be taken
taken outside the
the summation
summation sign, and
and Et
~m is
term,
___ the
m we put
just the mass of
just
of the
the body,
body, which we denote
denote by
by p.
fk· In
In
the second
second ter
term
ake the origin of
~mV
~mr -· VXS2
V x .Q = VXS2
V x .Q -· Ear.
~mr. Since w
wee ttake
of the
Et
V · .Q
Q xxrr = Emimoving system
system to
to be
be at
at the
the centre
centre of
of mass,
mass, this
this term
term is
is zero,
zero, because
because Emi~mr =
= 0.
moving
Finally, in
in the
the third
third term
term we
we expand
expand the
the squared
squared vector
vector product.
product. The
The result
result
Finally,
lS
is

+ l2

Z m[Q2r2-(52 - r )2

(32.1)

Thus the kinetic
kinetic energy
energy of
of a rigid
body can be written
written as the sum
sum of
of two
Thus
rigid body
The first term
term in (32.1)
(32.1) is the
the kinetic
kinetic energy of
of the translational
translational motion,
motion,
parts. The
and
and is of
of the same form as if
if the whole mass of
of the body
body were concentrated
concentrated
at
:->.t th
thee centre
centre of
of mass. The
The second
second term
term is the kinetic energy
energy of
of the rotation
rotation
with
about an axis passing
contre of mass.
mass.
with angular
angular velocity
velocity .Q
SZ. about
passing through
through the centre
It sho
uld be emphasised
should
emphasised that
that this division
division of
of the kinetic
kinetic energy into
into two parts
origin of
of the co-ordinate
co-ordinate system
system fixed
is possible only because
because the origin
Fixed in the
body has been
been taken to be at its centre of
of mass.
body
'I't

_ _

0' may,
may, of
of course, lie outside the body.
body.
ch
O'
I
h
. I
. .
be chOS€I1
osen 30
so
Inn the
V and
maY
be
t e general case where V
and $2
Q are not
not perpendicular,
~erpend~cu
ar, the orlon
~ngm
~ay
uestion)
of a
_
,
.
•
'be
t`on)
of
a
the instan
s (at
as to
to make
make V
V and
and Q parallel,
parallel, i.e.
i.e. so
so that
that the
the motior
motionl conslst
cons1sts
(at the
mstantt 111.
tn q
q s 1
as
ns.
rotation about
about some
some axis
axis together
together with
with aa translation
translation along
along that
that aaxis.
rotation

I++

_

£32

The
The inertia
inertia tensor
tensor

99

rewrite the kinetic
kinetic energy
energy of rotation
rotation in tensor
tensor form, ii.e.
We may rewrite
.e. in terms
of the components'[
componentsf xi
X-t and
and Q;
0-t of
of the vectors
vectors rrand
$2. We have
have
of
and 52.

!1 Z
L "2(Qi2xi2
m(Oi2Xi2 QiXinkxk)
- QzJciQkxk)
- ninkXtXk)
!21 Z
L "l(QiQ1a:5ikxz2
m(QiQkSikXl2 fliflkxixk)
!12 Q
nink
L "1(xz25u¢m(xl2 Sik-XiXk).
119/: Z
Xix1.¢)-

Trot = 2
Trot
=
=

Here we have used
used the identity
identity Qi
Qi =
= 8¢;Q;¢,
Siknk, where 5i/6
Sik is the unit
unit tensor,
whose components
unity for i =
= k and
and zero for i ;é
f= k. In terms
terms of the
components are unity
tensor

ZL m(xl Sik-XiXk)
- xexk)

II

lik =
Le

2
??2(XI281I/5

(32.2)

the kinetic energy
energy of
of a rigid
following expressioN
expression for the
we have finally the following
body
body::

T = §pV2+§IikQ@Qk~

(32.3)

L = !*V2+*Iu¢Q¢Qk- U.

(32.4)
(32.4)

The Lagrangian
Lagrangian for a rigid body is obtained
obtained from (32.3) by subtracting
subtracting
The
potential energy
energy::
the potential

The
variables which
The potential
potential energy
energy is in
\n general
general a function
function of
of the six variables
which define
the position
three co-ordinates
co-ordinates X, Y, Z of the
position of the rigid body, e.g. the decree
of mass
mass and
and the three
three angles
angles which
which specify
specify the relative
relative orientation of
of
centre of
centre
moving and
and Fixed
fixed co-ordinate
co-ordinate axes.
the moving
The tensor ltk
The
Iii.: is called the inertia tensor of the body. It is symmetrical,
t.e.
1.e.
(32.5)
lm = Ilia

as is evident
For clarity,
clarity, we may
may give
gtve its comevident from the definition
definition (32.2).
(32.2). For
ponents explicitly
explicitly::
ponents

lm

-Emu
m(y2+z2)
2 mxz ]
-'.Lmxy
2 +22 )
ZL m(y
- :Lmxz
Z mys
Z my
ZL m(x2
+22)
m(x2+z2)
-'.Lmyz
-'.Lmyx
-Zmw
L '"(x2+y2)
m(x2+y2)
2 max
-'Lmzy
-'.Lmzx
E

(32.6)

The components
components Ilxx,
lyy, I ZZ
zz are called the moments of
of inertia about the
The
163271 Icy,
corresponding
corresponding axes.
The inertia
inertia tensor is evidently
evidently additive:
additive: the moments
moments of
of inertia
inertia of a body
body
are the
the sums of
of those of
of its parts.
are
t In
In tthis
~his Chapter,
chapter, the
the letters
letters i,i, k,
k, Il are
are tensor
tensor suffixes
suffixes and
and take
take the
the values 1, 2, 3. The
The
T
summ~uo';
be used
used, ii.e.
summation signs
signs are
are omitted,
omitted, but
but summation
sununation over
over
summa
on rule will aalways
l a s be
.e. summation
2 3
theedva
m' 2,
isi ximplieYd
implied
whenever
a so
suffix
occurs
in aany
expression.
Such a suffix
suffix IS
is
• 3S iS
'
'n
n yexp
re ssto
' n . Such
'
the
1,
h never
1
Hi x OCC
Ur s ttwice
wlce
1
dummy
a
d aueds
Fo w
alleVa|LIeS
c
11
call
a suffixe
my sriffix F r
2 etc. It is
· obvious
•
h at
2 = As/lr
=
A2,
that
=
A
B,
A12
b.
Or
example,
AtE•
=_A
•
B,
At
AzAz
=
A
,
etc.
It
IS
obvwus
t
example,
A:Bz
_ s can
bee repla d
dummY
- suffixes, except
.
replaced
already
here in
in the
the can
expres
ce by
by any other like
hke
except ones which
wh•ch
a I rea d y appear
exp
elsewhere
sion concerned.
elseW
resston
cconcerned.

-

of a Rigid Body
Motion of

100

§32

If the body
If
body is regarded
regarded as continuous,
continuous, the sum
sum in the definition
definition (32.2)
integral over the volume
volume of
of the body
body::
becomes an integral
Ick
fix =

Jf p(xf-Stk-XtXk)
dV.
- xzxk) dv.
P(~W26ik

(32.7)

Like
reduced
Like any symmetrical
symmetrical tensor of
of rank two, the inertia tensor can
can be reduced
appropriate choice of the directions
directions of the axes
diagonaL form by an appropriate
to diagonal.
Xl,
princzpal axes of
X1o kg,
X2, x3.
Xs. These directions
directions are called the principal
of inertia, and the
corresponding values
values of
of the
the diagonal
diagonal components of
of the tensor are
are called
called the
the
pn"ncipal
of inertia; we shall denote them by II,
I2, 13.
Is. When the
prince])al moments of
/1, 12,
x1, xg,
x2, x3
xs are so chosen,
chosen, the kinetic
kinetic energy
energy of
of rotation takes the
the very
very
axes xi,
simple form
Trot

=

(I1Q12+I2Q22-l-I3Q32).

(32.8)

None of
principal moments of inertia
None
of the three
three principal
inertia can
can exceed
exceed the sum of
of the
For instance,
instance,
other
er two. For
oth

11-1-12 =

2 m(x12+x22+2x3-2) 2

Em(x12+x22) = I3.

(32.9)
{32.9)

whose three
three principal moments
moments of
of inertia
inertia are all different
different is called
A body
body whose
an asymmetrical top. If
If two are equal
equal (al
{II=
I2 as
f= I3),
Is), we have a symmetrical
== In
top. In this
this case the direction
direction of
of one
one of the principal
x1x2-plane
top.
principal axes in the xlxg-plane
may be chosen
If all three
principal moments
chosen arbitrarily.
arbitrarily. If
three principal
moments of
of inertia
inertia are equal,
of inertia
inertia may be chosen
chosen
the body
body is called a sphe-,:ical
spherical top, and the three axes of
any three
three mutually perpendicular axes.
arbitrarily as any
The
principal axes of inertia
The determination
determination of the principal
inertia is much
much simplified
simplified if
the body
symmetrical, for it is clear
clear that the position
of mass
the
body is symmetrical,
position of the centre of
the principal axes must have the same symmetry as
and the directions of
of the
and
the body.
For example,
example, if the body
body has a plane
of symmetry, the centre of
of
the
body. For
plane of
which also contains
contains two of
of the principal
principal axes of
of
mass must lie in that plane,
plane, which
plane. An obvious
obvious case of this
while the third
third is perpendicular
inertia, while
perpendicular to the plane.
kind is a coplanar
coplanar system
system of
of particles.
particles. Here
Here there
there is a simple relation
relation between
kind
between
three principal
moments of
of inertia.
inertia. If
If the plane of the system
system is taken as
the three
principal moments
2,
xs =
= 0 for every
every particle,
soh
= Zmx22,
J:..mx2 2, 12
I2 =
= J:..mx1
X1x2-plane, then x3
the xlxg-plane,
particle, and so
11 =
Zmx12,
13
Is =
= 2m(x12+x22),
J:..m(x12+x2 2), whence

Is
= I1-1-I2.
Is=
h+h.

(32.10)

If
of symmetry
symmetry of
of any
any order,
order, the centre
centre of
of mass
mass must
must lie
If a body
body has an axis of
that axis, which
which is also one
one of
of the principal
of inertia, while
while the other
on that
principalaxes of
perpendicular to it. If
If the axis is of order
order higher
higher than
than the second,
second,
two are perpendicular
symmetrical top.
top. For
For any
any prin~ipal
the body
body is a symmetrical
principal axis perpendicular
perpendicular to the
axis of
of symmetry
symmetry can be turned
turned through
through an angle different from
from 180O
180° about
about the
choice of
of the perpendicular axes is not unique, and
and this
this can
can
latter, i.e. the choice
happen only
only if
if the body
body is a symmetrical
symmetrical top.
happen
e
ne o0f~ th
the li
particular case here is a collinear
collinear system
system of
of particles. If
line
the
A particular
If the
o
S
article, an
system is taken
taken as the
the x3-axis,
x3-axis, then al
x1 =
= x2
x2 =
= 00 for
every
and so
ery pparticle,
system
for ev

§32

The inertia
inertia tensor
tensor
The

two of the principal moments
moments of
of inertia
inertia are equal
equal and
and the third
third is zero
zero::
II

Is = 0.
13

(32.11)

II

2:ZMBQ,
mxs2,

h = I2=
I2 =
11

101

Such
property which dis
Such a system
system is called a rotator. The
The characteristic
characteristic property
distintinguishes
bodies is that
guishes a rotator
rotator from other
other bodies
that it has only
only two, not three,
three, rotational
rotational
degrees
X! and
and xg
x 2 axes: it
degrees of
of freedom,
freedom, corresponding
corresponding to rotations about the xi
is clearly
clearly meaningless
meaningless to speak
speak of
of the
the rotation
rotation of a straight line about
about itself.
itself.
further result
result concerning
concerning the calculation
calculation of
of the
Finally, we may
may note one'
one further
Finally,
inertia tensor. Although
Although this tensor has
has been
been defined
defined with respect to a system
system
of
if"the
of co-ordinates
co-ordinates whose origin
origin is at the centre
centre of mass (as is necessary
necessary if'the
fundamental formula
formula (32.3)
(32.3) is to be valid), it may
may sometimes
sometimes be more confundamental
veniently
Fm
calculating a.similar
a.similar tensor
tensor I'
ik =
= 2??2(38'I28i],;
~m(x' z2Sikveniently found
found by first calculating
- x'
x'¢x';,),
1x' A;),
defined
If the distance
OO' is reprerepredefined with respect
respect to some
some other
other origin
origin O'.
0'. If
distance 00'
sented
sented by a vector
vector a,
a, then r =
= r"-l-a,
r' +a, Xi
x, =
= x',;+ai,
x' i +a,; since,
since, by the definition
definition
0, Emf
~mr =
= 0, we have
of O,

I m+p@%m-%@)
I' ik = ltk
+ p{a 2Stk- aiak ).
Fu.:
I in if l'ik
Using
this formula,
formula, we
we can
can easily
easily calculate
calculate Iik
Using this
[ k is known.

(32.12)

PROBLEMS
PROBLEMS

PROBLEM
l . Determine the
PROBLEM 1.
the principal
principal moments of
of inertia for the
the following types of
of molecule,
cule, regarded
regarded as
as systems of
of particles
particles at Fixed
fixed distances apart: (a) a molecule of
of collinear
collinear
atoms, (b) a triatomic
triatomic molecule which is an
an isosceles triangle
triangle (Fig. 36),
36), (c) a tetratomic
molecule which is an equilateral-based tetrahedron ((Fig.
F i g . 37).

'Et

I x2
I
I

I

mI

(a)

FIG.
FIG.

am. .

Ha
in

i} o

I4

SOLUTION.

m

36

11
It = /2
l2
1

=; Z.L

=

1

"'

P

agsb
a ..b

O

/al;
I

2
7N¢l.7Nblg,b2)
mamblab ,

1lf271

O

FIG.
FIG.

O

37
37

I 3=
= 00,,
la

whe
re Ma
where
ma is
is the
mass of
of the
the ath
ath atom,
atom, lab the
the distance
distance between
between the
the ath
ath and
and bth
bth atoms,
atoms, and
and
the mass
the
the summation includes
of atoms
atoms in
in the
the molecule.
molecule.
one term for every pair of
.
is only one term
term in
in the sum,
sum, and the result is
is obvious'
obvious: itit is
is
For a diatomic molecule there is
the Product
product of
of the
the reduced
reduced mass
mass of
of the
the two
two atoms
atoms and
and the
the square
square ooff the
the distance
distance between
between
the
them:
them: 11
h =:'
= 12
I2 ==
= m1m2l2;"(m1+m2)mtmJ2j(mt +m2)X2 == mahllLL
(b) The
The centre
centre of
of mass
mass is
is on
on the
the axis
axis of
of symmetry
symmetry of
of the
the triangle,
triangle, at
at a distance X2
mw/,_,
(b)
from its
its b32861(h
base (h beln_i-Ihe
being the height
height of
of the
the triangle),
triangle). The
The moments
moments ooff inertia
inertia are
are 11
It=
2mtm2h2ffL,
= 2m1m2}22/p,
h2 == Qui
lmla2, Ia == I•+I
•.
2.
1
(c) The
The c;;ntre
_of mass
mass is
is on
on the
the axis
axis of
of symmetry
symmetry of
of the
the tetrahedron,
tetrahedron, at
at aa distance
distance
(c)
frltre .of
X
m2h/P.
rorn its
Its bese
base (iz
(h being
being the
the height
height of
of the
the tetrahedron).
tetrahedron). The
The moments
moments oof
inertia
f inertia
Xa3 = m
zhl/u. ron

so

.

Motion of
of a Rigid Body

102

are 1,
= 31r11m2h2/p+&m1a2,
3mlm2h 2/p.+lm·a2,
are
/1 = 12 =

1a == Graz.
mw2. If
If
la

1nl =
=
Ml

regular
/3 = m1a2,
regular tetrahedron
tetrahedron and
and 11
1, = 12
h == la
mw2.

§32

1n2, h
h = Vv(2f3)a,
the molecule is aa
my,
( 2 / 3 ) G . the

PROBLEM 2.
2. Determine
Determine the
the principal
principal moments
moments of
of inertia
inertia for
the following
homogeneous
PROBLEM
for the
following homogeneous
bodies: (a)
(a) aa thin
thin rod
rod of
of length
length I,l, (b)
(b) aa sphere
sphere of
of radius
radius R,
R, (c)
(c) aa circular
circular cylinder
cylinder of
of radius
radius R
R
bodies:

and height h, (d)
(d) a rectangular parallelepiped of
of sides a,
a, b, and
and e,
c, (e)
(e) a circular cone
cone of
of height
and
hh and
and base radius R,
R, (f)
(f) an ellipsoid of
of seriates
semiru.es a,
a, b,
b, e.
c.

2 1a = 00 (we
SOLUTION. (a)
(a) 11
1, == 12 = 1*.w/2,
(we neglect theI
the' thickness of
of the rod).
rod).
SOLUTION.
1\p.l , 13
(b) 11
-I~I2-l~I3 = 2pIr2
h =
= 12 = la
la == §,uR2
fp.R 2 (found
(found by
by calculating
calculating the sum 11
1,+12+la
2pfr2 dV).
dV).
(c)
(c) 11
h == 12
h =.}MR2+§-hZ),
=. !JL(R2 +lh 2), 13
1a = 1,122
lp.R 2 (where the
the x3-axis
xa-axis is
is along the axis of
of the
the cylinder).
cylinder).
2+c2), 12
(d) 11
'g,u(b2~l-eZ),
11 =
= np.(b
h == 112l*(a2'l'C2)J
].1~p.(a2+c2), 13
1a = 1123.11-(02-I~b2)
].\p.(a 2+b2) (where the axes
aXes xi,
XI, kg,
X2, Ra
X3 are
are
along
the sides
sides a,
a, bi'
b, ec respectively).
respectively).
g the
alon
(e) We
We first
first calculate
calculate the
the tensor
tensor IIik
I'tk with
with respect
respect to
to axes whose origin
origin is
is at
at the vertex of
of
(e)
the
the cone (Fig. 38).
38). The calculation is
is simple ifif cylindrical co-ordinates are used,
used, and the
the result
is I'1
1'1 = I'2
1'2 = %#(%.R2+h2),
fp.(!R2+h 2), I'3
I'a = ,'bp.R2.
of mass is
is easily shown to
to be
be on
on the
the
is
TSUPR2 The centre of
of the
the cone and
and 'at
·at a distance aa =
= -ih from
from the
the vertex. Formula
Formula (32.12) therefore
therefore·gives
gives
axis of
1, = 12 = I'1-p02
1'1-p.a2 == é%#(R2+:l>h2),
i\rp.(R 2+!h2), la
la = I'3
I' a = i'%f»R2~
itp.R2.
11

oh

XI, XI

I

If

f

/-

._./

' - u r q n i

l-\

2/

'Xa

x

j,
xI

X2

J

FIG.
FIG.

38
38

(f)
(f) The
The centre
centre of
of mass
mass is
is at
at the
the centre
centre of
of the
the ellipsoid,
eilipsoid, and
and the
the principal
principal axes
axes of
of inertia
inertia are
are
along the
the axes
axes of
of the
the ellipsoid.
eilipsoid. The
The integration
integration over
over the
the volume
volume of
of the
the ellipsoid
ellipsoid can
can be
be reduced
reduced
along
to one
one over
over aa sphere
sphere by
by the
the transformation
transformation x == of,
af, yy = b-a7,
b7J, z == .c{,
which converts
converts the
the equaequato
et, which
tion of
of the
the surface of
of the ellipsoid x2/a2-l~y2/[22
x 2fa 2+y2fb2+22/e2
+z2fc2 =
= 1 ihto
into that of
of the
the unit sphere
f2+7J2+{2
9
+ + ¥ ==1 1..

For example,
example, the
the moment
moment of
of inertia
inertia about
about the
the x-axis
is
For
x-axis is
1,
/1

= pp III
Iff (Y2+z2]
(y 2+z2 ) dx
dx do
dy do
dz

--°=

pabcfff (b2n2-I-e2§2)
(b27)2+c2 { 2) ifdf df;
d7J dt!
d{
= pabejjf
2 ),
= §abcI'(b2
}abcl'(b2+c
=
-l~c2),

where I'I' is
is the
the moment
moment of
of inertia
inertia of
of aa sphere
sphere of
of unit
unit radius.
radius. Since
Since the
the volume
volume of
of the
the ellipsoid
ellipsoid
where
is 41rabC/3,
£14522+e2),
4TTabcf3, we End
find the
the moments
moments of
of inertia 11
1, =
= !p.(b
+c2), 12 =
= iIL»(a2+¢2)1
!p.(a2+c2), la
1a :=°.
= %p(a2-I-192).
!p.(a2+b2).
Detennine the
the frequency of
of small oscillations of
of a compound pendulum (a
PROBLEM 3. Determine
rigid body
body swinging
swinging about
about aa Fixed
fixed horizontal
horizontal axis
axis in
in aa gravitational
gravitational Held).
field).
rigid
SOLUTIO~. Let
Let Il be
be the
the distance
distance between
between the
the centre
centre of
of mass
mass of
of the
the pendulum
pendulum and
and
SOLUTION.

the aaxis
to;
is

t e
ertia an
about which
which itit rotates,
rotates, and
and or,
a:, B.
{3, y the
the angles
angles between
between the
the principal
principal axes
axes of
of in
inertia
and ~he
about
i
axis of
of rotation
rotation. We
We take
take as
as the
variable co-ordinate
co-ordinate the
the angle
angle ¢,
<P betwcf't*
betweenhethe
th~~~:ttyicaJ
of
axis
velocity
her t cal
the variable
· of
·
The
. through
and
T
the
and aa line
Ime
through the
the centre
centre of
o f mass
mass perpendicular
perpendicu 1ar to
to the
t h e axis
axts
o f rotatioHrotauon.
the-.eprincipOf
r

.'

.'

= MB,
lt/>, and
and the
the components
components of
of the
the angular
angular velocity
velocity alo
along
g

the centre
centre of
of mass
mass is
is V
the
V =

principa1
l

a

§32

The inertia
inertia tensor
tensor
The

103

small, we End
axes of
of inertia
inertia are
are ¢
</> cos
cos a,
ex, </>g cos
cos B,
{3, <5
</> cos
cosy.
Assuming the
the angle
angle ¢r/> to
to be
be small,
find the
the
axes
Y- Assuming
potential energy
energy U
U = ,u.gl(1-cos
p.gl(l-cos ¢>)
r/>) A.:
~ 59gl$b2.
}pglrf>2 • The Lagrangian IS
is therefore
therefore
potential
2 +HI.
%~1%2+%(I1
L ::= !p.l2</>

r

CoS2a-l-I2
cos 2 cx+J2 Cos2B-l-I3
cos 2f3+h 00s2?)¢°-%/1g/¢2~
cos2y).f, 2 -!,<glrf>2.

of the
the oscillations is consequently
The frequency of

-I-I3 C0S2'y).
+12 cos2§
w 2 = #8/.f'(:J2+I1
pgl/(p.l2 +It cos2a
cos 2cx+h
cos2{3+h
cos 2y).
we

are thin
PROBLEM 4.
4. Find the kinetic energy
energy of
of the system
system shown in
in Fig,
Fig. 39:
39: OA
OA and AB
ABare
uniform rods
rods oflengthl
oflengthl hinged together
together at
at A.
A. The rod
rod OA
OA rotates
rotates (in
(in the
the plane of
of the
the diagram)
about O,
0, while
while the
the end
end B
B of
of the
the rod
rod AB
AB slides along Ox.
Ox.
about
I yy
I

.a
A

I

I

I

L

0 ' __
Q

'1____________
----x
_____5____x
B

FtG. 39
39
FIG.

of the
the centre of
of mass of
of the
the rod
rod OA
OA (which is at
at the
the middle of
of
SOLUTION. The velocity of
the rod) is }l.f,, where go
rf> is the
the angle AOB. The kinetic energy of
of the
the rod
rod OA
OA is therefore
the
+i:I€52I2, where Mp. is the
= %fJ2€52
f;p.l2.f>2+lf</>
the mass of
of each rod,
rod.
T1 =
of the
of mass of
of the
the rod
rod AB
AB are
are X =-= it
~~ cos
cos go,
rf>, Y
Y
the centre of
The Cartesian co-ordinates of
= i}l
sin 96,
rf>. Since the
the angular velocity of
of rotation of
of this rod
rod is also
also ii,
.f,, its
its kinetic energy is
t sin
=
kinetic energy of
T 2 =°= %p(X2_l.Y2)
!p.(X2+Y2)+lf</>
= %n/2(1
f;p.i2(1 +8
+8 sin2¢)<£2+%I¢52.
sin 2 r/>)</> 2+lf¢ 2. The total kinetic
of this
-I-iIq§22 =
T2
2
the~efore T =
= §pl2(l
!p.l2(1 +3
+3 sin2sf>)(52.
sin r/>)</>2, since I =
= 11.2.l2
system is therefore
1\p.l2 (see Problem 2(a)).

g

the kinetic
kinetic energy of
of a cylinder
cylinder of
of radius
radius R
R rolling on
on a plane,
plane, if
if the
the mass
PROBLEM 5. Find the
of the
the cylinder is so
so distributed that one of
of the
the principal
principal axes of
of inertia
inertia is parallel
parallel to the
the axis
of
of the
the cylinder
cylinder and at a distance
distance a from
from it, and
and the
the moment of
of inertia
inertia about that principal
of
axis is I.
I.
axis
r/>t, be the
the angle
angle between the
the vertical and
and a line from
from the
the centre of
of mass
the axis of
of the
the cylinder (Fig. 40).
40). The motion of
of the
the cylinder at any
any instant
perpendicular to the
SOLUTIO~.
SOLUTION.

Let
Let

Lu.

l

R

/'

//

JZ.

/r

FIG.
FIG.

I

.

.,r

IJ'

J

40

be regarded
regarded as
as aa pure
pure rotation
rotation about
about an
an instantaneous
instantaneous axis
axis which
which coincides
coincides with
the
with the
l l y be
Imay
line Where
the cylinder
cylinder touches
touches t_he
t_he plane.
plane. The
The angular
angular velocity
velocity of
of this
this rotation
rotation is
is <f,,
sinee
45. since
Where the
line
the arugular
angular velocity
velocity of
of rotation
rotation about
about all
all parallel
parallel axes
axes is
is the
the same.
The centre
centre of
of mass
mass is
is at
at aa
same. The
the
2 +R22. -2 R
(¢ 12
therefore
is
distance
v!LI
+_,R
-2aR
cos</>)
from
the
instantaneous
axis,
and
its
velocity
is
therefore
velocity
its
and
axis,
instantaneous
the
from
¢)
cos
stance
dl
R'
/( a 2 + R 2 -2 a cos
V
= $
<f,·v(a·+R--2aR
cos</>).
The total
total kinetic
kinetic energy
energy is
is
V=
¢>)- The

T ==-= §*f1(a2+R2-2¢'1R
h•(a 2 +R2 -2nR cos
cos ¢>)ql>2+{,]<f>'2•
</>)r/>2 +!fc/> 2 •
T

104

Motion of
of a Rigid Body

§32

energy of
of a homogeneous cylinder
cylinder of
of radius a rolling inside
PROBLEM 6. Find the kinetic energy
a cylindrical surfagze
surfas:e of
of radius R
R (Fig. 41).
41).
al/
V /

2'

I

/

/I
/

/'

..

/

/

-

..I
- I
" " - / ~ ¢ , .I / y / / / 4 '

/ '

41

FIG.
FIG.

SoLUTION.
We use
use the
the angle
angle 56
r/> between
between the
the vertical
and the
the line
line joining
the centres
centres of
of the
the
SOLUTION. We
vertical and
joining the
cylinders. The
The centre
centre of
of mass
mass of
of the
the rolling
rolling cylinder
cylinder is
is on
on the
the axis,
axis, and
and its
its velocity
is V
V ==
cylinders.
velocity is
.f,(R -a). We
We can
can calculate
calculate the
the angular
angular velocity
velocity as
as that
that of
of aa pure
pure rotation
rotation about
about an
an instantaneous
instantaneous
'MR-a).
V/a = q6(R-a)/a.
= Vfa
axis
axis which
which coincides
CGincides with the
the line
line of
of contact
contact of
of the
the cylinders;
cylinders; it is
is Q
f.!=
.f,(R-a)fa.
If la
I a is the
the moment of
of inertia about
about the
the axis of
of the
the cylinder, then
If
-

=

2fa 2 =
T :=.= mR-a)2<£2+iI==(R
lp.(R-a)2t/>2+lfa(R-a)2rf>
= if»(R-0)2s52»
-ip.(R-a)2.f,2,
-a)2¢2/Q2

la being
being given
given by
by Problem
Problem 2(c).
2(c).
13

PRoBLEM 7.
7. Find
Find the
the kinetic
kinetic energy
energy of
of aa homogeneous
homogeneous cone
cone rolling
rolling on
on aa plane.
plane.
PROBLEM

SOLUTION.
denote .by
the angle
angle between
between the
the line
line OA
OA 'in
-in which
cone touches
touches the
the
SOLUTION. We denote
which the
the cone
by 08 the

plane
plane and some
some fixed direction
direction in
in the
the plane
plane (Fig. 42).
42). The centre
centre of
of mass is on
on the axis of
of the
the
cone, and
and its
its velocity
velocity V
aO us
cos a,
a:, where Za
2a: is the vertical
vertical angle of
of the
the o
cone
the
gone,
V = of?
n e and aa the

zz

-n

X

O in'

6'

..

4I

0

-in

-==;;-=__1111

-=°:

yy

1

I

A

FIG.
FIG. 42

distance of
of the
the centre
centre of
of mass
mass from
the vertex.
vertex. The
The angular
angular velocity
velocity can
can be
be calculated
calculated as
as
distance
from the
that of
V/a sin a
al cot
of a pure rotation about the
the instantaneous axis
axis OA:
OA: Q
f.! = V{a
a: = 0
cot a.
a:. One of
of
the principal axes of
we take another (xg)
of inertia (xg)
(xs) is
is along the
the axis of
of the
the cone, and we
(x2) perpendicular to
to the
the axis
axis of
of the
the cone
cone and
and to
to the
the line
line OA.
OA. Then
Then the
the components
components of
of the
the vector
vector $2
dicular
is parallel to
to OA) along the
the principal axes of
of inertia are Q
f.! sin
sin a,
a:, 0, Q
f.! cos a.
a:. The kinetic
(which is
energy is thus
energy

n

T =

&f1¢1262 CL 2 a + / 1 9 2

.

cos4a
oos2a -I-&I362-

= 3ph292(1
Jp./1202(1 +5
+S cOs2a)/40,
cos 2a:)/40,
=

S1n2d

where hhis
the height of
of the
the cone, and
and In,
It, 13
la and
and a have been
been given in
in Problem 2(e).
where
is the

PROBLEM 8.
8. Find
Find the
the kinetic
kinetic energy
energy of
of aa homogeneous
homogeneous cone
cone whose
whose base
base rolls
rolls on
on aa plane
plane
PROBLEM
and whose
whose vertex
vertex is
is fixed
fixed at
at aa height
height above
a~ve the
the plane
plane equal
equal to
to the
the radius
radius of
of the
the base,
base, so
so that
that
and
the axis
axis of
ofthe
cone is
is parallel
parallel to
to the
the plane.
plane.
the
the cone

_

are and
SOLUTION. We
We use
use the
the angle
angle 08 between
between aa fixed
fixed direction
direction in
in the
the pl
plane
and the
the .Pr;j~t:o
SOLUTION.
V =' QQ
re of
of the
axis of
of the
the cone
cone on
on the
the plane
plane (Fig.
(Fig. 43).
43). Then
Then the
the velocity
velocity of
of the
the cent
centre
of mass
mass is,s
•
of
the axis

projection

I

Angular
;nomentum of
of aa rigid
rigid body
body
/Ingulaf ,momentum

§33

105

the
OA
the notation
notation being
being as
as in
in Problem
Problem 7.
7. The
The instantaneous
instan.taneous axis
axis of
of rotation
rotation is
is the
the generator
generator OA
which
where the
lane. The
which passes
passes through
through the
the point
point where
the cone
cone touches
touches the
the plane.
The centre
centre of
of mass
mass is
is at
at aa
slance a
di
in a
V/a sin
distance
a Ssin
a from
from thi
thiss axis,
axis, and so Q
f.! == Vfa
sin aa == Bfsin a. The components
components of
of the
the
vector
along the
the principal
principal axes
axes of
of inertia
inertia are,
are, if
if the
the x2-axis
x2-axis is
is taken
taken perpendicular
perpendicular to
to the
the
vector $2 along
of the
the cone and to the
the line OA,
OA, Q
f.! sin
sin aa = 9,
0, 0,
0, Q
f.! us
cos aa = 0 cot
cot on.
a. The kinetic energy
axis of
is
is therefore
therefore

n

i$...1

=

T

= 5u0292+%II02+§I392
i1La202+lft02 +lfa02 cot2¢
cot 2a

=

=

..-=

J,.h202(sec2a+5)/40.
= 3ph202(sec2a
-I-5)/40.

z

y
y

-~.

FIGFm.

43

PROBLEM 9. Find
Find the kinetic energy of a homogeneous ellipsoid which
rotates about
about one
one
which rotates
of its axes (AB
(AB in
in Fig.
Fig. 44)
44) while
that axis itself rotates about a line CD
CD perpendicular to it
it
of
whi je that
and passing
passing through
through the
the centre
centre of
of the
the ellipsoid.
ellipsoid.
and

SOLUTION. Let
AB (i.e. the
SoLuTION.
Let the angle of
of rotation about
about CD
CD be 0,
8, and
and that about
about AB
the angle

between CD
CD and
and the
the xi-axis
xt-axis of
of inertia,
inertia, which
is perpendicular
perpendicular to
to AB)
AB) be ¢tf>. Then
Then the combetween
which is
the axes of
L, ifif the
ponents of
of $2 along the
of inertia are
are 6
0 cos ¢,
</>, 6
0 sin is,
4>, .f,,
the x3-axis
xa-axis is
is AB.
AB. Si-nce
Si-nce the
ponents
centre
centre of
of mass,
mass, at
at the
the centre
centre of
of the
the ellipsoid,
ellipsoid, is
is at
at rest,
rest, the
the kinetic
kinetic energy is

n

f
6,---"*

ID

I

I

,*

____

,
I I-»....
I!'_..---F
I
"\
" ~- ~
I
I

r

,g

W
YN

\J l B
2:"""*"'",.--»
u
I
r .
'Pdf
I

-

-*`._

*

;

luv

I

I

.

|I

A

I

,J

I

FIG.
FIG.

A

`

|
,D

I

'v* XI I v
/

//(fH./

A

an

T =
=!(It <=<>s2¢
cos2tf>+h
sin2tf>)02+!Ja.f,2.
+12 sinai)02+§I=£2.

I

a
.__L__

I

2'

1I

.1

Y

:.c
I

FIG.
FIG.

4-4
44

¢5

45

PROBLEM 10.
10. The same as Problem 9, but
but for the case where the axis ABis not
not perpendicular to
to CD
CD and
and is an axis
axis of
of symmetry of the
the ellipsoid
ellipsoid (Fig.
(Fig. 45).
45).
lar

n

SOLUTION. The components of
of $2 along the
th~ axis AB
AB and
and the other two
two principal
principal axes of
of
inertia, which
are perpendicular
perpendicular to
to AB
AB but
but otherwise
otherwise arbitrary,
arbitrary, are 6006
0 cos ona cos tf>,
0 cos aax
inertia,
which are
QS, 0
x
xsin
52 oos2a-1-§I3($-I-6
x sin 95,
q,, $4.0
.f>+O sin a,
a. The
The kinetic
kinetic energy is T =
= iII
lft82
cos2a+ila(./>+O sin a)2.

body
§33.
Angular momentum of
of a rigid
rigid body
3. Angular
§3

The value
value of
of the
the angular
angular momentum
momentum of
of a system
system depends,
depends, as we know, on
the
with respect
which it is defined.
the point with
respect to which
defined. In
In the
the mechanics
mechanics of
of a rigid
rigid body,

Motion of
of a Rigid
Rigid Body
Motion

106

§33

the most
most appropriate point to choose
choose for this purpose is the origin
origin of the
system of co-ordinates,
co-ordinates, i.e. the
the centre
centre of mass of
of the body,
body, and in
moving system
what follows we shall
shall denote
denote by M the
the angular
angular momentum
momentum so defined.
defined.
what
According to formula
formula (9.6),
{9.6), when
when the origin
origin is taken at the
According
the centre of mass
of
angular momentum
momentum M is equal
equal to the "intrinsic" angular
angular
of the body, the angular
momentum resulting from
from the motion
motion relative
relative to the centre of
of mass. In the
definition M = Zmrxv
~mrxv we therefore replace v by .Qxr:
definition
Q x r:
mr x (Qxr)
2mrx
(.Qxr)

notation,
or, in tensor notation,

Z

=

§m[,-2
'L;m[r2.Q-r(r·.Q)],
SO-r(r-SIZ)],

II

=

II

M

Mi
Mi =:
= 2 m(x¢-Q¢;-x»,;x;,,Q;,¢)
m(xl92Qi-xixkQk) =
:--- .Qk
Qk 211z(x¢'28m-xixk).
2m(xl2 Sik-XiXk)·

Finally, using
using the definition
definition (32.2)
(32.2) of the inertia
inertia tensor, we have
have
Finally,

.Mi
Mi = like.
JikQk·

(33.1)

If
XI, .x2,
X3 are the same as the principal axes of
of inertia, formula
If the axes xl,
JC2, kg
(33.
(33.1)
1) gives

IlNlt

II

M1

M2

L092,
_

Ma

13523.

(33.2)

II

In particular, for a spherical top,
top, where
where all three principal moments of inertia
inertia
are equal,
equal, we have
have simply
are
M = J.Q
IQ,,
(33.3)
M

the angular
angular momentum vector
vector is proportional to, and
and in the
the same
same direci.e. the
angular velocity
velocity vector. For
an, arbitrary
arbitrary body, however,
however, the
the
tion as, the angular
For an.
vector M is not in general in
in the same
same direction
direction as SZ,
.Q; this happens
happens only
vector
when the body
rotating about
about one of its principal
principal axes of inertia,
inertia.
when
body is rotating
Let us consider
consider a rigid body
moving freely, i.e. not
not subject to any
any external
external
Let
body moving
forces. \Ve suppose
suppose that
that any uniform
translational motion,
motion, which
which is of no
forces.
uniform translational
of the body.
removed, leaving
leaving a free rotation of
interest, is removed,
As in'any
in·any closed
closed system,
system, the angular
angular momentum
freely rotating
momentum of the freely
rotating body
body
is constant.
constant. For
For a spherical
spherical top
top the
the condition
condition M = constant
constant gives
gives $2
.Q = conIS
stant; that is, the
the most
most general
general free rotation
rotation of a spherical
spherical top
top is a uniform
uniform
stant,
rotation about
about an axis fixed in space.
rotation
The case of a rotator
rotator is equally
equally simple.
simple. Here also M =
= ISO,
J.Q, and
and the
the vector
vector
The
,Q is perpendicular to the axis of
of the rotator. Hence a free rotation of a rotator
$2
is a uniform
uniform rotation in one plane about an axis perpendicular to that plane.
plane.
The law of
of conservation
conservation of angular
angular momentum also
also suffices
suffices to determine
The
of a symmetrical top.
top. Using the fact
that the
the more
more complex
complex free rotation of
the
fact that
,
principal
axes
of
inertia
x
x2
(perpendicular
to
the
axis
of
symmetry
{x3)
principal
of inertia xl,1 xg (perpendicular
symmetry (kg)
the top) may be chosen
chosen arbitrarily,
arbitrarily, we take the
the xi-axis
x 2-axis perpendicular to
of the
containing the
the constant
constant vector
and the
the instantaneous
instantaneous position
position
the plane
plane containing
vector M and
=: 0.
of the x3-axis.
X3-axis. Then M2
M2 =
= 0, and
and formulae
formulae (33.2)
{33.2) show
show that
that !22
0. This
This
QUO =
means that the
the directions
directions of M,
M, $2
,Q and
and the axis of the
the top
are at every instant
means
top are
SZ x r
in one plane
plane (Fig.
(Fig. 46). Hence,
Hence, in turn,
turn, it follows that
that the
the velocity
= Qhxr
velocity v -..:=
tO
· on
· of
· at
·mstant perpendiclllar
to that
t at
pmnt
on the
t h e axis
ax1s
of the
the top
top is
IS
at every
every instant
perpend"cular
1
of every
every point
of

The equations of
of motion of
of a rigid
rigid body

§34

107

plane. That is, the axis of
of the top
top rotates uniformly
uniformly (see
{see below)
below) about
about the
the
direction
M, describing
direction of M,
describing a circular
circular cone. This is called regular precession
of the
the top.
top. At
At the same time the top
top rotates uniformly about
about its
its own
own axis.
of
MM

`

I

I.

/

\

1'

/

.».-»'°'

».»'°"""'***

*

I

.f2 pr

"*~.°\.

'-....__.__

1

'

I

I

I

/'

I

x

,4I

/

:
I

I

Ins.

"*l---...
I

|

_n

. \

|

I

|

i-

\

IIX3
/1

X' \\

Fm.46
FIG.
46

The
The angular
angular velocities
velocities of
of these
these two
two rotations'
rotations' can easily be expressed
expressed in
terms
terms of
of the
the given angular
angular momentum
momentum M and
and the
the angle 6(} between
between the axis
top and
and the direction
direction of
of M.
angular velocity
velocity of
of the
the top
top about
about its
of the top
M. The angular
the
component
!23
of
the
vector
.Q
along
the
axis
:
own axis is just
just the component QUO of
vector $2 along the

QUO = M3lI3 = (M//3) cos 6.

(33.4)

To determine
determine the rate of
of precession
vector $2
.Q must be resolved into
To
precession Qpr, the vector
M. The
The first of these gives no displacement
displacement
components along x3
x3 and along M.
components
of the
the top,
top, and the
the second
second component
component is therefore the
the required
required
of the axis of
angular velocity
velocity of
of precession.
shows that
that QD,
Dpr sin
sin 60 =
= 91,
!21, and, since
angular
precession. Fig. 46 shows
Q1
I1 ::= {M/h)
I1) sin 6,
!21 =
= Ml/
M1/h
0, we
we have
have
(33.5)
Qpr : : MlI1.

§34
The equations of
of motion of
of a rigid body
body
§34.. The

Since a rigid
rigid body
body has, in general,
general, six degrees
degrees of freedom,
freedom, the general
general
Since
of motion
motion must be six
six in number. They can be put
put in a form which
equations of
which
thee tim
timee derivatives of
of two
two vectors,
vectors, the
the momentum and
and the
the angular
angular
gives th
gives
momentum of
of the
the bod
body.
momentum
y,
The iirst
first equation
~qu~tion is
is obtained
obtained by
summing the
the equations
equations P
p= f
The
by ssimply
imply summing
h
for eac
each particle
the body,
body, P
p being
being the
the momentum
momentum of
of the
the particle
particle and
and ff the
the
partIcle IIn
n the
for

Motion
Motion of
of aa Rigid
Rigid Body
Body

108

§34

II

force acting
body P = Ep
acting on it. In
In rems
terms of the total momentum
momentum of the body
~p = [LV
fi-V
and total force acting
acting on it F :::
= 211
~f, we have
and

P/do =
ddP/dt
= F
F..

(34.1)

viaR.
F =
= -oUfoR.

(34.2)
(34.2)

Although
been defined
defined as the
the sum
sum of all the forces f acting on the
Although F has been
various
particles, F actually
various particles,
particles, including
including the forces
forces due to other
other particles,
actually
the forces
forces of
of interaction between the
the particles
includes only
only external forces:
forces: the
includes
particles
composing the body
cancel out, since
since if there are no external
external forces
composing
body must cancel
the momentum
momentum of the
the body, like that
that of
of any
any closed system,
system, must be conserved,
conserved,
i.e. we must have F = 0.
If U is the potential
potential energy
If
energy of a rigid body
body in an external
external field, the
the force
F is obtained
obtained by differentiating
differentiating U with
with respect
respect to the co-ordinates
co-ordinates of the
of mass
mass of
of the
the body:
centre of
body :

For, when
when the body
body undergoes
undergoes a translation
translation throu
through
SR, the
the radius
gh a distance 8R,
every point
point in the body
changes by 8R,
SR, and so the change
change in the
vector rt of every
body changes
potential
energy is
potential energy

2 ( viar ) ·~8Srr == 8SRR -· 2_oUfor
§aul3r == -8
8SU
U == 2_(oUfor)
R .· 2
8 R.
-SR
2_ ff == --FF ·SR.

It may
may be noted
noted that
that equation
equation (34.1)
{34.1) can
can also be obtained
obtained as Lagrange's
It
equation
L/ 3 R,
equation for the co-ordinates
co-ordinates of the
the centre
centre of mass, (dldr)3Ll3V
(dfdt)oLfoV =
= 3oLfoR,
with the Lagrangian
Lagrangian (32.4), for which
which

aoLfoV
Li r v = p.v
fi-V =

3oLJoR
R == FF..
P
L/ 3 R ..= -3U/3
- oUJoR
P,1
Let us now derive
derive the
the second
second equation of motion, which
which gives the
the time
time
Let
angular momentum
momentum M.
To simplify
simplify the derivation,
derivation, it is
derivative of the angular
derivative
M. To
convenient to
to choose
choose the
the "fixed" {inertial)
frame of
of reference in such
such a way
convenient
(inertial) frame
that the centre of
frame at the
the instant considered.
of mass
mass is at rest
rest in that
that frame
We have M
= (d/dt)2rxp
(d/dt)~r x p = '2i°xp+Zrxp.
·~r x p + ~r xp. Ou
Ourr choice
choice of the
the frame
frame of
of
eVe
NI =
means
that
the
value
of
i'
{with V = 0)
that
of rat
the instant considered
considered is
at the
reference (with
same as vv ==- i tt.. Since
Since the
the vectors
vectors v and p =
= my
mv are parallel, i'xp
rxp ::= 0.
the same
Replacing 13
p by the force f, we have finally
Replacing

where
where

°:

dM/dr
dM/dt = K,

(34.3)

K
r xf.
K = 2_rxf.

(34.4)
(34.4)

Since
Since M has
has been
been defined
defined as the
the angular
angular momentum
momentum about
about the
the centre of
mass (see
{see the beginning
beginning of §33), it is unchanged
unchanged when we go from one inertial
inertial
another. This is seen from formula (9.5) with R =
frame to another.
== 0. We can therethe equation of motion
motion (34.3), though derived
derived for a particular
particular
fore deduce that the
ivity
of reference,
reference, is valid in any other
other inertial
inertial frame, by Galileo's
Galileo~s relat
relativity
frame of
principle.
total
· called the
· the
the L"k
total
K is
Th e vector rxf
IS called
t h e moment of
of the
the force
force ...~f,, and
and so K
IS d
The
r x f is
body~ Like
torque, i.e. the
the sum
sum of
of the
the moments
moments of
of all
all the
the forces acting on
on the
the bo Y· I e
torque,

§34

The
The equations of
of motion of
of a rigid
rigid body

109

total force F, the sum
sum (34.4) need include
include only the external
external forces:
forces: by
the total
the
law
of
conservation
of
angular
momentum,
the
sum
of
the
moments
of
the
of conservation
angular momentum,
sum
forces in a closed system
system must be zero.
the internal forces
The
The moment
moment of a force, like the angular
angular momentum,
momentum, in general
general depends on
the choice
choice of the
the origin
origin about
about which it is defined. In (34.3) and (34.4) the
the
are defined
defined with respect
respect to the
the centre of
of mass
mass of
of the
the body.
moments are
When
When the origin
origin is moved
moved a distance
distance a,
a, the
the new radius
radius vector
vector r' of each
each
point in the body
r - a . Hence K = 2rxf
= 2r"xf+
Zaxf or
point
body is equal to
tor-a.
~rxf=
~r'xf+~axfor
K'+axF.
K = K'
+ a x F.

(34.5)
(34.5)

Hence we see,
see, in particular,
that the value
value of the
the torque
torque is independent of
Hence
particular, that
the
force F =
body is said
the choice of
of origin
origin if the total force
= 0. In this case the
the body
said to
he
acted on by a couple.
couple.
be acted
Equation
EILl352
Equation (34.3) may be regarded
regarded as Lagrange's
Lagrange's equation
equation (d/dt) oLfo!J.
=
= 81,/3q>
oLfo.p for the "rotational co-ordinates". Differentiating
Differentiating the Lagrangian
Lagrangian
(32.4) with respect
respect to the components
components of the vector
vector SZ,
!J., we obtain
obtain ala
oLfoO.i
(32.4)
L Q
=
Ii°'¢Q/lc5 =
potential energy
= lu,O.k
= Mi. The
The change in the potential
energy resulting
resulting from an
infinitesimal
S.p of the body
body iiss 8U
SU = -- ~f.Sr
= -215
-~f.S.pxr
infinitesimal rotation
rotation 8<l>
Ef- 8r =-'
5q>xr
<l>-2rxf=
= - 8S.p-~rxf
= -K-8q>,
-K·S.P, whence
K =
K
q >== K
aula
oU/o.P
K..

-

oUfo.p,
via4>,

(34.6)

that a
Li r 4> = oLJo.p
Let us assume
assume that
that the vectors
vectors F and K are perpendicular.
vector a
Let
perpendicular. Then a vector
that K' given by formula (34.5) is zero and
can always be found such that
SO
so

K
x F.
(34.7)
K =
= a
ax
(34.7)
The
The choice of
of a is not unique, since the addition to a of any vector parallel

to F does not affect equation
equation (34.7).
(34. 7). The
The condition
condition K'
K' =
= 0 thus gives a straight
straight
not a point,
the moving
moving system
system of
of co-ordinates.
co-ordinates. When
line, not
point, in the
\Vhen K is perpendiperpendicular to F, the
the effect
effect of
of all the applied forces can
can therefore
therefore be reduced to that
of a single
single force F acting
acting along this line.
of
that of
of a uniform
field of force,
forre, in which
which the force
force on a particle
Such a case iiss that
uniform Field
particle
= eE, with E a constant vector
vector characteriSing
characterising the
the field and
and e characterising
is f =
particle with respect
respect to the
the {ield.T
field. t Then F =
= EBB,
E~e,
the properties of a particle
~er x E. Assuming
Assuming that
that EE
~e go
t= 0, we define a radius
radius vector to
ro such
such that
K = 2erxE.
r

2

to
or/E e,
ro =
= 2erf2
e.

simply
Then the total torque is simply

X F.
K=K = to
rox

(34.8)

(34.9)

the effect of th
.e field
Thu
Thus,
when
rigid body
uniform field, the
th_e
s, whe
body moves in a uniform
n a rigid
reduces to
to the
the action
action of
of a single
single force F applied
applied at the point whose
whose radius
radius
reduces
.8). Th
(34.8).
The
of this point is entirely determined by
vector
or is (34
by the
vect
position of
e position
t For
For example,
exa~pl~, in
in aa uniform
uniform electric
electric field
E is
is the
the field
field strength
strength and
and ee the
the charge;
charge; in
in aa
field E
avitational field
uniform
gravttatlOnal
field E
E is
is the
the acceleration
acceleration g
g due
due to
to gravity
gravity and
and ee is
is the
the mass
mass m.
m.
form gr
uni

Motion of
of a Rigid
Rigid Body
Body
Rlotion

110

§35

properties
properties of the body
body itself.
it-self. In a gravitational
gravitational Held,
field, for example,
example, it is the
centre of
of mass.

Eulerian angles
§35. Eulerian
As has already
already been
mentioned, the motion
motion of
of a rigid
rigid body
body can
can be described
described
been mentioned,
by means
means of
of the three
three co-ordinates
co-ordinates of
of its centre of
of mass
mass and
and any
any three
three angles
angles
by
which determine
orientation of the axes xi,
x1, JC2,
x2, x3
X3 in die
the moving system
system of
determine the orientation
co-ordinates relative
relative to the Fixed
fixed system
system X, Y, Z. These angles
angles may often
often be
co-ordinates
conveniently taken
taken as what
what are
are called
called Eulerian angles.
conveniently

z

n Z

(>

x

v

..

.f

\

\\

r

\\

\

/

/

/

I

I

\.
/

1'\
\

x

, -

/'

.r

\

''
\.

1

1

I/

O
-»

¢

I

I

I

/

/
I

/I

I

/'\

X2

\\

\

\

=»1.

\

\
\

I

'I

1'

~-

g

y

~<

X

\ ..
/'
\\
\_' _......._____
/
_
,,,
_
_
_
. _*

FIG.
FIG.

,f

/~'

47

Since we are here
here interested
interested only
only in the
the angles
angles between
between the
the co-ordinate
co-ordinate
Since
axes, we may
may take the
the origins
origins of the
the two
two systems
systems to coincide
coincide {Fig.
(Fig. 47). The
x1x2-plane intersects
intersects the fixed X
XY-plane
some line ON, called the
moving xlxg-plane
Y-plane iin
n some
the
Zine of
line
of nodes.
nodes. This
This line is evidently
evidently perpendicular to both the
the Z-axis
Z-axis and
and the
the
x3-axis; we take
take its positive
positive direction
direction as that of
of the
the vector
vector product ZXX3
z x X3
xi-axis,
{where
and X3
X3 are unit
x3 axes).
(where z and
unit vectors along the Z and x3
We take, as the
the quantities defining
defining the position
the axes JCi,
x1, x2,
x2, x3
X3
posit i on of the
the angle 0(}between
z~and xg
X3 axes, the angle ¢>
c/>
relative to the axes X, Y, Z the
between the Z"2nd
the X-axis and
and ON,
ON, and
and the
the angle go
f between
the x1-axis and ON.
between the
between the
The angles (I,
and
¢,
are
measured
round
the
Z
and
cf> and if;
measured round the and xg
x3 axes respectively
respectively in the
corkscrew rule. The angle 6(} takes
takes values from 0 to 1-r,
TT,
direction given
given by the corkscrew
direction
and ¢c/> and .1,
21-r.T
if; from 0 to 2TT.t

z

t The
The angles
angles 08 and
and 95-ln
t/>-t>r are
are respectively
respectively the
the polar
polar angle
angle and
and azimuth
azimuth of
of the
the direction
direction
The angles
angles 68 and
and lTT-1/l
are respectively
respectively the
polar angle
angle
the polar
The
r-5l' are

xa with
respect to
to the
the axes
axes X,
X, Y,
Z.
with respect
X3
Y, Z.

and azimuth of
of the
the direction Z
Z with respect
respect to
to the
the axes xi,
Xl, x2,
x2, xs.
and

§35

111

Eulerian angles

Let
Let us now express
express the components
components of the angular
angular velocity
velocity vector
vector $2
.Q alo
along
ng
the moving
l ) xg,
x2, kg
X3 in terms of
of the Eulerian
Eulerian angles and
and their
their derivatives.
derivatives.
moving axes Xx1.
To
To do this, we must Find
find the components along those
those axes of
of the
the angular
angular
velocities
0, of),
~. go.
~- The
The angular velocity
velocity 6(j is along the line of
of nodes ON,
ON, and
and
velocities 6,
its components
z/1, 62
H sin .1,,
components are 61
81 = 68 cos 1/J,
82 = --8
1/J, 63
03 = 0. The
The angular
angular velocity~
along the Z-axis;
component along the X3-3.XiS
x3-axis is 953
¢3 = is
¢ cos 6, and
<:iw
43 is along
Z-axis, its component
X1X2-plane ¢.
¢sin
latter along the
the. Xl
X2 axes,
axe~. we
in the xlxg-plane
sin 6. Resolving the latter
Xl and xi
have Q51
¢1 = gb
cf> sin
sin 6esin
sin .1,,
1/J, 952
c/>2 = al,
cf> sin
sin 6ecos ./,.
1/J. Finally,
Finally, the angular
angular velocity ,~kg
along the
the x3-axis.
is along
components along each axis, we have
Collecting
Collecting the components

e,

e.

QUO =
¢5 sin 6 sin»,!'+6 cos lf,
!21
=~sinOsinifi+Ocosifi,
QUO
Q2

= d~ sin
6 cos'f/-6
sine
cos 1/1- esin'j',
sin 1/J,

QUO
cos 6 + ,,;,
!23 =
=~cosO+~.

)

(35.1)

If
x1, xg,
x2, x3
X3 are taken to be the principal
inertia of the body,
If the axes xl,
principal axes of inertia
the Eulerian
Eulerian angles is obtained
obtained by
rotational kinetic
kinetic energy
energy in terms of the
the rotational
substituting (35.1) in (32.8).
_
substituting
For a symmetrical
symmetrical top (I1
{h = 12
h go
t= I3),
h), a simple
simple reduction
reduction gives
For
Trot

_

= §11(¢i2 sin26 + 62) + %I3(¢3 cos 6 + .1;)2.

(35.2))
(35.2

expression can also be more simply
simply obtained
obtained by using
using the
the fact that
that the
This expression
of directions
directions of
of the principal
x1, kg
x2 is arbitrary
arbitrary for a symmetrical
symmetrical
choice of
principal axes xl,
ON, i.e. ¢,
ifi = 0, the compotop. If
If the x1
Xl axis is taken along the line of nodes ON,
of the angular velocity
velocity are simply
nents of

9,

QUO

QUO

=

sin 6,

QUO =

cos9+',5.

(35.3)

simple example
example of
o{ the
the use of the Eulerian
Eulerian angles, we shall
shall use them
them
As a simple
determine the free motion
motion of a symmetrical
symmetrical top, already
already found in §33.
to determine
We take the
the ZZ-axis
of the fixed system
system of
of co-ordinates
co-ordinates in the
the direction
direction of the
axis of
constant angular
angular momentum
momentum M of the top.
top. The
The x3-axis
x3-axis of the moving
moving system
system
constant
the axis of
of the
the top,
top; let the A11-aXiS
x1-axis coincide
coincide with
with the line of
of nodes
nodes at
along the
is along
the instan
instantt considered.
considered. Then the components
components of
of the vector
vector M are, by
{35.3), m1
M 1 = 1191
hrl 1 =
= 116,
hO, MY
M2 =
= 1192
hrl 2 = I1¢,3
h¢ sin 6,
e, m3
M3 =
= 1303
hrl3
formulae (35.3),
=I
3(¢ cos 6e+a,b).
+ ~ ). Sinoe
Since the A31-aXlS
XI-axis is perpendicular
perpendicular to the Z-axis, we have
=I3(¢>
.All
M1 =
= 0, M2
M2 = M
M sin
sine,
M3 = M
M cos 6.
e. Comparison gives
6, Ma
II

-Q;

0,

1143
m,
h~ = M,

h(~ ¢os@+¢•)
cosO+~)=
M cos e.
I3(¢3
= Mcos9.

(35.4)

The Flrst
first of these equations
equations gives 6e = constant, i.e. the angle between
between the
The
the top
top and
and the
the direction of
of M
M is constant. The second equation gives
of the
axis of
angu
city of
lar vol
the angular
velocity
of precession
if> = M/I1,
Mfh, in agreement
agreement with {33.5).
(33.5).
the
precession 95
d
eq
thir
ua
Finally,
~ird
equation
gives
the
angular
velocity
with
which
the
top
the
tio
»
gives
the
angular
velocity
with
which
the
top
11Y
n
Fin3
about
its
rotates
about
Its
own
mcis:
O;J
=
(Mfla)
cos
e.
rotate'-3
own axis; QUO
(M/I3)
9.
•

of aa Rigid
Rigid Body
Motion of

112

§35

PROBLEMS
PROBLEMS

PROBLEM
PROBLEM 1.
1. Reduce
Reduce to
to quadratures
quadratures the problem of
of the
the motion of
of a heavy
heavy symmetrical
top whose lowest point is
is fixed
fixed (Fig.
(Fig. 48).
48).

SOLUTION.
SoLUTION. We
We take the
the common origin of
of the
the moving and
and fixed systems
systems of
of co-ordinates
co-ordinates
at the
a gravitathe fixed
fixed point O
0 of
of the
the top,
top, and
and the
the Z-axis vertical. The
The Lagrangian of
of the top
top in
in:;~
tional Held
field is L = MIN
!(11 +,12)
+ ,..1 2) (92
(0 2 +,$2
+t/> 2 sin26)+y3(.L+,5
sin 2 0)+ lfa(.f,+.f> cos 9)2-#gl
0)2- p.gl cos 6,
0, where P
p. is the mass
of
from its
fixed point
of the
the top
top and
and Il the
the distance
distance from
its fixed
point to
to the
the centre
centre of
of mass.
mass.

\

\

z

Z

x5

*2

\

\+

¢5

x

y
Y

wx,

FIG.
FIG.

N

48

co-ordinates .1,
1/J and
and </>are
two integrals of
of the
the motion
motion::
The co-ordinates
£6 are cyclic. Hence we have two

P¢ = EL/EM
oLjO.f, =
1%

la(.f,+rf> cos 0)
8) =
=constant=
Ma
I3(¢+$
constant E M3

+I#

p¢
constant E m,,
M .:-.
= aLlaql=
oLfo.f>= (I'I
(l't sin2@+13'¢os20;¢
sin 2 0+la.cos2 0)rf,+Jaif, cos 0
0 ==constant=
M,,

(1)

(2)

where I'1
J'1 == 11
h -I-plz;
+p.l2; the
the quantities p¢
P¢ and
and P¢
p<l> are
are the
the components
components of
of the
the rotational
ro~ational angular
angular
where
momentum about
about O
0 along
along the
the xg
xa and
and Z
Z axes
axes respectively.
respectively. The
The energy
energy
momentum
E = §I'1(92

+42 sin2l9) -l-§13(8lv +5 cos 0)2 -I-,ugl cos 19

is also conserved.
conserved.
equations (1)
(1) and
and (2)
(2) we find
From equations
find

rf, = (Mz-M3
(M.-Ma cos 6)/I'1
8)/1'1 sin26l,
sin2 0,

.f, _

-

M3
0 M:-M3
Ma _
M.-Ma cos 0
cos 0
1
cos
1'1 sin26
sin2 0
·
133
I'1

(3)

(4)
(4)

(5)
(5)

.f, from
from the energy
energy (3) by
by means
rp.eans of
of equations
equations (4)
(4) and
and (5),
(5), we obtain
obtain
Eliminating rf, and :or
where
where

2 + Uen(0),
E'
51'162+
E' == !1'10
Uen(B),

E'

Uen

=
=

M3
Ma22
E-ngl
- 2 3- -ugl,
2Js
" '
(M,-.Ma COS
cos 9)8
8)2
(M2-A13
.
-p.gl(l-cos 0).
ll).
2
2I'1 sin30
stn ll
2I'1
-#EIU -cos

(6)
(6)

§35

11
1133

Eulerian angles
angles
Eulerian

Thus we have

fI

III

t =

dO
dB
.
v{2[E'- Ucn(0)]/1'1};9
V{2[E'-U¢n(9)]/I"1}

(7)

this is an elliptic integral.
.kg are then expressed in
integral. The angles ¢.
1/J and tf>
in terms
terms of
of 0 by
by means of
of
integrals obtained from
from equations (4)
(4) and
and (5).
(5).
The range
range of
of variation of
of 00 during
during the
the motion
motion is
is determined
determined by
by the
the condition
conditionE'
~ Uerr(0).
Ue 11(0).
The
E' ,Z
function U]n(0)
Uen(O) tends
tends to
to infinity
infinity(if
Ma ;é
#- MY)
Mz) when 0Otends
to 00 or
or1r,
and has minimum
a minimum
The function
(if Ms
tends to
or, and
between
the
between these
these values.
values. Hence
Hence the
the equation
equation E'
E' == Uen(0)
Uen(O) has
has two
two roots,
roots, which
which determine
determine the

limiting values
values 01
01 and
and 02
02 of
of the
the inclination of
of the
the axis of
of the
the top
top to
to the
the venical.
limiting
vertical.
When 00 varies
varies from
from 61
Ot to
to 02,
02, the
the derivative
derivative <5
tf, changes
changes sign
sign ifif and
and only
only ifif the
the difference
difference
When
M,Ms cos 00 changes sign in
in that range of
of 0.
0. IfIf itit does
d0es not
not change sign,
sign, the axis of
of the
top
M,
-Ms
the top
precesses monotonically
monotonically about
about the
the vertical,
vertical, at
at the
the same
same time
time oscillating
oscillating up
up and
and down.
down. The
The
precesses
latter oscillation
oscillation is
is called
called nutation;
nutation; see
see Fig.
Fig. 49a,
49a, where
the curve
curve shows
shows the
the track
track of
of the
the axis
axis
latter
where the
on the
the surface
surface of
of a'
a· sphere
sphere whose
whose centre
centre is
is at
at the
point of
of the
the top.
top. IfIf tf,
does change
change sign,
sign,
on
the fixed
fixed point
,kg does
the direction of
of precession is
is opposite on
on the
the two
two limiting
limiting circles,
circles, and
and so the
the axis of
of the
the top
top
the
describes loops
loops as itit moves round the
the vertical
venical (Fig.
(Fig. 49b). Finally,
Finaily, if
if one of
of 01,
Ot, 02
02 is a zero of
of
describes
tf, and
and 0 vanish
vanish together
together on
on the
the corresponding
corresponding limiting
limiting circle,
circle, and
and the
the path
path
,g
of the axis is of
of the kind
kind shown in
in Fig.
Fig. 49c.
of

Mz-Ms
cos 0,
0,
M;
-Ma cos

162

(a)
(o)

(b)

FIG.
FIG.

49

(cl
(cl

PROBLEM 2.
2. Find
Find the
the condition
condition for
for the
the rotation
rotation of
of aa top
top about
about a vertical axis
axis to
to be
be stable.
stable.
PROBLEM

SOLUTION. For
For 00 = 0,
0, the
the Ra
xs and
and Z
Z axes
axes coincide,
coincide, so
so that
that M3
M 3 == My,
Mz, E'
E' == 0.
0. Rotation
Rotation
SOLUTION.
about this axis is stable if
if 0 = 00 is a minimum of
of the
the function
Uen(O). For
For small 0 we have
about
function U¢¢¢(0).
Uen z
~ (M32/8I'1-§p.gl)02,
(Ma 2 /81'1-lp.gl)02, whence
the condition
condition for
for stability
stability is
is M32
Ma 2 > 4I'1pgl
41'1p.gl or
or (232
Oa 2
Uetl
whence the

> 4I'Ip.g//I32.
4l'zp.glfla 2•
>

PROBLEM 3.
3. Determine
Determine the
the motion
motion of
of aa top
top when
when the
the kinetic
kinetic energy
energy of
of its
its rotation
rotation about
about
PROBLEM
its axis is large compared
compared with its energy
ene~y in
in the
the gravitational field
''fast" top).
top).
its
field (called a "fast"
SOLUTION. In
In aa first
approximation, neglecting
neglecting gravity,
gravity, there
there is
is aa free
precession of
of the
the
SOLUTION.
first approximation,
free precession

axis of
of the
the top
top about
about the
the di
direction
of the
the angular
angular momentum
momentum M,
M, corresponding
corresponding in
in this case
case
axis
lection of
to the
the nutation
nutation of
of the
the top;
top; according
according to
to (33.5),
(33.5), the angular velocity
velocity of this precession is
to

S 2 n u ° M lI'I.

(1)
(1)

In the
the next
next approximation,
approximation, there
there is
is aa slow
slow precession
precession of
of the
the angular
angular momentum
momentuf"!l M
M about
about
In
the vertical
vertical (Fig.
(Fig. 50),
?0). To
To determine
determine the
the rate
rate of
of this
this precession,
precession, we
we average
average the
the exact
exact equation
equation
the
of motion
motion (34-3)
(34.3) dM]dt
dM/dt = K
K over
over the
the nutation
nutation period.
period. The
The moment
moment of
of the
the force
of gravity
gravity
force of
of
on the
the top
top is
is K-'_-'p-fnaxg
K=,.Jnaxg, where
where no
113 is
is aa unit
unit vector
along the
the axis
axis of
of the
the top.
top. It
It is
is evident
evident
vector along
on
€t1'y that the
from syn1Il'1
synunetry
the res
result
of averaging
averaging K
K over
over the
the "nutation
"nutation cone"
cone" is
is to
replace Na
na by
by
from
to replace
ult of
its comp0
component
cos aex in
in the
the direction
direction of
of M,
M, wher
wheree acx is the angle
angle between
between M and the
its
M / M ) cos
"ent ((M/M)
tOP- T'hu3
axis of
of the
the top.
Thus we
we have
have dm/dr
dM/dt == -on/m)g><m
-(p.l/M)gx M cos
cos a,
cx. This
This shows
shows that
that the
the vector
M
vector M
axis

114

Motion
of a Rigid Body
Rfotiofz of

§36

precesses
precesses about
about the
the direction
direction ooff gg (i.e.
(i.e. the
the vertical)
vertical) with
with aa mean
mean angular
angular velocity
velocity

Slpr =
go,

-(p./'J.J)g COS
COS G
a:
-(/1/Lu)§

I:

S1nu·
which is small compared with Qm.

(2)
(2)

Nor

'
I
f

, ............ ------f

'/'

/'J'

In"

--_ __

n

4 '

.

**

\

'\

I

\

"

.

I

I1

\

1

I

('r2nu
I'

X

I

'\

"'--

\

\

\

0'
*i

IA' ' /

FIG.
FIG.

50

In this approximation
approximation the quantities
quantities M
M and
and cos aa: in
in formulae (1)
(1) and (2)
(2) are constants,
constants,
In
although they
they are
are not
not exact
exact integrals
integrals ooff the
the motion.
motion. To
To the
the same
accuracy they
they are
are related
related
although
same accuracy
to
to the
the strictly
strictly conserved
conserved quantities
quantities EE and
and Ma
M3 by
by Ma
M3 == M
M cos
cos a,
a:,

E 3 gif"
1

.J

o

cos~a

13
-....

-

o

sm-a

+__

Fl

O

§36. Euler's equations
The equations
equations of
of motion
motion given
given in §34 relate
relate to the fixed system
system of
of coThe
dPfdt and
and dM/dt
dMfdt in equations
equations (34.1)
(34.1) and
and (34.3)
{34.3)
ordinates: the derivatives op/dt
of change
change of
of the vectors
vectors P and
and M with respect
that system.
rates of
are the rates
respect to that
The simplest relation
relation between the components of
of the
the rotational
rotational angular
angular
The
of a rigid
rigid body
and the
the components
components of
of the
the angular
angular velocity
momentum M of
momentum
body and
hmvever, in the moving
moving system
system of
of co-ordinates
co-ordinates whose axes are the
occurs, however,
occurs,
of inertia.
inertia. In order
order to use this relation,
relation, we must
must first transform
transform
principal
principal axes of
of motion
motion to the moving
moving co-ordinates
co-ordinates xi,
x1, x2,
equations of
the equations
xg, xs.
xg.
Let
dA/dt be the rate
rate of
of change
change of
of any
any vector
with respect to the fixed
Let dA,idt
vector A with
system of co-ordinates.
co-ordinates. If
ddcs not
not change
change in the moving
moving system
system,,
system
If the vector A does
that
rate of change
change in the fixed system
system is due only
only to the rotation,
rotation, so that
its rate
dAJdt
QxA; see §9, where
where it has
has been pointed out
out that
that formulae
formulae such
such as
dA
/ do == SZxA,
1)
(9.
(9.1) and
and (9.2)
(9.2) are valid for any vector..In
vector. "In the general
general case, the right-hand
right-hand
side includes
includes also the rate
rate of
of change of
of the
the vector
vector A with
with respect
respect to the moving
moving
side
system. Denoting
Denoting this rate
rate of
of change
change by
by d'A/dt,
d' Afdt, we obtain
system.
dA
d'A
d'A
= -+QxA.
+ .Q xA.
dt
dt
dr
dt

-

(36,
(36
.1)
1)

i

§36

11
5
115

Euler's equations
equations
Euler

Using this
this general
general formula, we can
can immediately
immediately write
write equations
equations (34.1) and
(34.3) in the form

-+S'2xP
- + £2xP = F,
dt
dd'P
'p

d'M
d'M
--+S'2xM
+ $2 x M = K.
dt

{36.2)
(36.2)

Since the
the differentiation
differentiation with
with respect
respect to time is here
here performed in the
the moving
moving
system
system of
of co-ordinates,
co-ordinates, we can take the components
components of
of equations
equations (36.2)
{36.2) along
along
the
putting (d'P/dt)1
the axes of that
that system, putting
(d'P/dt)! =
= dp1/dt,
dP1/dt, ...,
... , (d'M/dr)1
{d'Mfdt)! =
= dm1/dz,
dM1fdt,
... , where the suffixes 1, 2, 3 denote
denote the
the components
components along
along the axes xl,
x1, JC2,
x 2, x3.
x3 •
...,
In
In the
the first equation we replace P by
by iv, obtaining

fLV,

1
fL(
d; +
f!2Vs- Q3v2)
f!sV2)
~( dt + Q2V3-

dv1

Q1

Q2V1

F1,

=

F2,

=

Fs.

l

II

Val
~( dr
dv3
fL(d;s
+ V2»( dt +f!1V2-n2v1)
dv22
Qs V1- Q1
fL(d; +
+f!sV1-f!1Vs)

=

J

I}

(36.3)

If the axes xl,
If
x1, xg,
x2, x3
xs are the principal
principal al-nes
axes of
of inertia, we can put
put M1
M 1 = I191»
hf!1,
second equation
equation {36.2),
obtaining
etc., in the second
(36.2), obtaining

I1
(I3-I2)Q2Q3 h dQlldt+
df!1/dt+(ls-h)f!2f!s
=
12
- I3)Q3Q1
/2 dQ2/dt
df!2/dt + (11
(hls)f!sf!l =
=
la
+(12-I1)Q1Q2 =
Is dQ3/dz
df!sjdt+(h-h)f!1f!2

Ki,
K1.

_ Kg.
Ks.
K2,

1

}

(36.4)

equations.
These are Euler's
Eule1"s equations.
rotation, K =
= 0, so that
that Euler's
Euler's equations
equations become
In free rotation,
become

dQ1/di-l-(13-I2)Q2Q3/I1
df!1/dt+{Is-h)f!2f!s/h =
= 0,
dQ2/dI+(I1-I3)Q3Q1/I2
df!2/dt+(h-ls)f!sf!1/h =
= 0,
dQ3/df-}-(12-I1)Q1Q2/I3
df!s/dt+(l2-h)f!1f!2/ls =
= 0.
0.

W
I

(36.5)

As an example,
example, let
let us apply these
these equations
equations to the free rotation of
of a symsymh =
= 12,
h, we Hnd
find from
metrical top, which has already
already been discussed. Putting
Putting 11
metrical
third equation
equation QUO
0 3 == 0, i.e. !1
= constant. We then
then write
write the first two
the third
QUO3 =
equa
tions as Q1
equations
nl =
= -wfl2,
-wf!2, QUO
Q2 =
= wQ1,
wf!l, where
w = Q3(I3-11)/I1
f!s(ls-h)/h

(U

(36.6)

is a cons
constant.
Multiplying the second
second equation
equation by i and adding,
adding, we have
tant. Multiplying
d(Q1+iQ2)/dz
d(f!1 + i0.2)jdt =
= iw(Q1+i§22),
iw(f!1 + if! 2), so that
that Q1+iQ2
!11 + if! 2 =
= A exp(iwt), where A is a
which may
may be made
made rea]
real by aa suitable
suitable choice of the origin
origin of time.
constant, which
constant,
Thus
fl1 =
=A
coswt
Q1
A COS
ii

0.2 -=A
sin iwt.
QUO
A sin
i.

(36.7)
(36.7)

I
Motion of
of a-Rigid
Rigid Body

116

J

I
I

§37

This result
perpendicular
result shows
shows that
that the component
component of the angular
angular velocity
velocity perpendicular
to the axis
axis of
of the top
top rotates with
with an angular velocity
velocity w, remaining of
of constant
2
2
magnitude
V(Q12++ Q22).
= y(01
02 ). Since
Since the component
component QUO
Os along
along the axis of the
magnitude A =
top
top is also
also constant, we conclude that
that the
the vector SZ
n rotates uniformly with
with
angular velocity
unchanged in magniw about
about the axis of
of the top,
top, remaining
rel'\laining unchanged
magnivelocity cu
tude. On
relations M1 = I1Q1,
On account
account of
of the
the relations
ftOl, M2 =
= I2Q2,
h02, 11/13
lVls =
= I3Q3
fsOs between the components
M, the angular
components of £2
n and
and M,
angular momentum
momentum vector
vector M evidently
evidently
of the top.
the axis of
executes a similar motion with
with respect to the
description is naturally
naturally only
only a different
different view of
of the motion
motion already
already
This description
discussed
discussed in §33 and §35, where it was referred
referred to the fixed system
system of
of oocoIn particular,
particular, the angular
angular velocity of
of the vector
vector M (the Z-axis
Z-axis in
ordinates. In
ordinates.
about the x3-axis
xs-axis is, in terms
terms of
of Eulerian
Eulerian angles, the same
same as
Fig. 48, §35) about
the angular velocity
velocity -.;.
- ~- Using
Using equations
equations (35.4), we have

=

M
cos 6 .
Mcose.

II

1/J
~/3

6* = M
Mcos6
cosO=
cosO
I3 .-.¢>c/> cos

Is

(1- - -1) ,

('13Is Ih')1 '

or
or-~= Q'3(I3-II)/I1,
O.s{ls-h)/h. in agreement
agreement with (36.6).

§37. The
The asymmetrical top
\Ve shall
shall now
now apply Euler's equations to the
the still
still more
We
of
of the free rotation
rotation of
of an asymmetrical
asymmetrical top,
top, for which
which all
inertia are different.
different. We
\Ve assume
assume for definiteness
definiteness that
that
inertia

complex problem
three moments
moments of
three

la
> 12
11Is>
h >
>ft.

(37.
(37.1)
1)

:.'lf12 Jh2
_V/s2
i'1'/12
22
32
- - - + - -+
+ - - = 2E,
2£,
-*--+
11
/3
13
h
h.
Is

(37.3)

Two integrals
integrals of
of Euler's
Euler's equations
equations are known already
already from the laws of
of
Two
conservation of
of energy
energy and
and angular
angular momentum
momentum::
conservation
2+
ZE,
/1Q122 +
IgQ22
/3 Q322 =
/101
+hfi2
+IsOs
= 2E,
(37.2)
/12 Q12 +/22Q22 -1-132932 == Mg,
magnitude M of
of the angular
angular momentum
momentum are given
given
where the energy E and the magnitude
equations, written
written in terms of the components of
of the
the
constants. These two equations,
vector M, are
1112
Jl1 2 + .V-22
-'l/2 2 + *W32
Jfs 2 = M/2.2•

(37.4)

From these
these equations
equations we
\Ye can
can already
already draw
draw some
some conclusions
conclusions concerning
concerning
From
of the motion.
motion. To
To do so, we notice
that
equations
(37.3)
and (3'7.4),
(37.4),
the nature of
notice that equations (37.3) and
regarded as
<lS involving
involving co-ordinates
co-ordinates 111,
J/1, 111
ills, are respectively
respectively the equation
equation
regarded
JU2,
2 , 1M3,
of an ellipsoid
ellipsoid with semiaxes
semiaxes V(2FI1)9
y{2Eh), V(2E12)»
y{2EI2), y{2E/g)
and that
that of
of a sphere
of
I(21213) and
the
ertia of
of radius JY.
Jl. \Yhen
vector M
to the
the axes of
of in
inertia
of the
of
When the vector
M moves relative to
5
two eudfaces,
top, its
its terminus
terminus moves
moves along
along the
the line
line of
of intersection
intersection of
of these
these t,vo
':rfac~s.
top,
h
oi
ips
ell
Fig. 51 shows
shows aa number
number of
of such
such lines
lines of
of intersection
intersection of
of an
an ellipsmd wit
Wlth
Fig.

I

§37

I

l
I

11
'7
117

The asymmetrical
asymmetrical top
top
The

spheres
spheres of
of various
various radii. The
The existence
existence of
of an intersection
intersection is ensured by the
obviously valid inequalities
inequalities
obviously

2Eh < M2 < ZEISS,
2Els,
ZEISS

(37.5)
(37.5)

that the
the radius
radius of
of the sphere (37.4)
(37.4) lies between
the least
least and
and
which signify
signify that
which
between the
greatest seriates
semiaxes of
of the ellipsoid
ellipsoid (3'7.3).
(37.3).
x,

'I

/t

un'

.-1\

|

X3

Fi *

'

.

,_

'I

I
FIG.
FIG.

"~`

1

*a

'-v.

"=~=:.

.m

*2

51

Let us examine
examine the way in
in which these
these "paths"'I'
"paths"·;· of
of the terminus
terminus of the
Let
vector M change as M varies (for
{for a given value of E). When
M2 is only slightly
slightly
vector
\Vhen MY
than
2Eh,
the
sphere
intersects
the
ellipsoid
in
two
small
closed
curves
greater
greater
ZEI1,
sphere intersects
ellipsoid
corresponding poles of
of the ellipsoid,
ellipsoid; as M2 ->
-+ ZEI1,
2Eh,
x 1-axis near the corresponding
round the al-axis
shrink to points
\Vhen M2 increases, the curves
these curves shrink
points at the poles. When
= ZEISS
2Eh they
they become two
tv.ro plane
plane curves (ellipses)
become larger, and for M2 =
of the ellipsoid
ellipsoid on the x2-axis.
M2 increases
increases
which intersect
intersect at the poles of
which
JC2-21XiS. When Ma
again appear, but
but now round
the poles
further, two separate closed
closed paths again
further,
round the
poles on
the x3-axis
x3-axis;, as M2 ->
-+ ZEISS
2Els they shrink
shrink to points
points at these
these poles.
the
of all, we may
may note
note that, since the
the paths are closed,
closed, the motion
motion of
of the
First of
must be periodic;
during one period the vector
vector
vector M relative
relative to the top
top must
vector
periodic, during
and returns to its original
original position.
M
conical
surface and
M describes some coni
cal surface
essential difference
difference in the-nature
the-nature of
of the paths near
the various
various
Next, an essential
near the
poles of
of the ellipsoid
ellipsoid should
should be noted.
noted. Near
x1 and
and xg
xs axes, the paths lie
Near the xl
enti
rely
neighbourhood of
of the corresponding poles,
which
entirely in the neighbourhood
poles, but the paths which
pass near
near the poles
poles on the JC2-3.XiS
x2-axis go.elsewhere
go.elsewhere to great
great distances
distances from
from those
difference corresponds to a difference
difference in the stability of
of the
the rotapoles. This difference
of the top
top abo
about
its three axes of
of inertia. Rotation about
about the
the Xl
Xl and
and x3
xs
tion of
ut its
tion
axes (corresponding to
to the
the least
least and
and greatest of
of the
the three moments of
of inertia)
corresponding
cur"es
. descri
. bed by
·
h e vecwr .:o~
,.....
tt The
POnding curves
T he cofr€s
descnbed
by the
t h e termin
termtnus
oftthe
po/hodes.
vector $2 are called polhodes.
us of

I

Motion of
of aa Rigid Body

118

§37

is stable, in the
the sense
sense that, if the top
top is made
made to deviate
deviate slightly
slightly from such
such a
state, the
the resulting
resulting motion is close to the original
original one. A rotation about
about the
the
X2-2:llXiS,
x2-axis, however,
however, is unstable!
unstable: a small
small deviation
deviation is sufficient
sufficient to give rise to a
motion
motion which
which takes the top
top to positions
positions far from its original
original one.
To determine
determine the time dependence
dependence of
of the
the components
components of
of $2
Q (or
(or of
of the
the comcomTo
ponents
proportional to those
ponents of
of M,
M, which
which are proportional
those of
of 52)
Q) we use Euler's
Euler's equations
equations
(36.5). Vile
We express
express Q1
01 and
and QUO
Os in terms of
of QUO
0 2 by means of
of equations
equations (37.2)
and (37.3)
(37.3)::
2)-h(ls-h)02
2]/h(ls-h),
Q12
- M2)
- I1),
- I2(I:3- I2)Q22]/I1(I3
012 =
= [(2EI3
[(2E/s-M
(37.6)
2],'ls(ls- Il),
Q32
ZEI1) - I2(I2 -I1)Q22],'Ia(I3Os2 = [(m2[(M 2-2Eh)-h(h-h)02
h),

and substitute
substitute in the
the second
second equation
equation (36.5), obtaining
obtaining
and

df!-3/dt
d02/dt =
= (13-I1)Q1Qa/I2
(ls-h)010s/h
2] x
2)-h(ls-h)02
=
- 1142)
- I2(I3- I2)Q222
= v{[(2EI3
y{[(2E/s-M
x [(M22EI1)-I2(I2-I1)Q22]}/I-;,(I1I3).
[(M2..:..._2Eh)-h(h-h)022]}/hv(Itls).

II

(37.7)
Integration of
t(02) as an elliptic
elliptic integral.
of this equation
equation gives the function
function t(Q2)
In reducing
reducing it to a standard form
form we shall
shall suppose for definiteness
definiteness that
that
2Eh; if
if this inequality is reversed,
reversed, the suffixes 1 and 3 are interchanged
interchanged
M2 > ZEISS;
in the following
following formulae. Using
Using instead
instead of
of zt and
and QUO
02 the new variables
variables
I2)(M22ZEI1)/I1I2l3],
T
-r ~ IV[(I3ty[(ls-h)(M
-2Eh)/h1Js],
(37.8)
ss = Q2V[I2(1s-12)/(2513-M2)]»
02y[h(ls-h)/(2E/s-M 2)],
parameter k2 < 1 by
and defining a positive parameter
2-2Eh),
2)/(13 -/2)(M
k2 =
- I1)(2EI3- M2),'(I3I2)(M2ZEI1),
(37.9)
= (I2
(h-h){2Els-M
we
we obtain
obtain

f

s

T

T =

0
0

ds
y[(1-s2)(1-k2s2)]'
V[(1-~*2)(1-k2s2)]'

the origin
origin of
of time being
being taken
taken at an instant
instant when Q2
02 =
= 0. When
When this integral
integral
inverted we have a Jacobian
elliptic function
function s =
= sn Ta
-r, and this
t~is gives QUO
02
is inverted
Jacobian elliptic
as a function
function of
of time; 91(1)
01(t) and
and Q3(t)
Os(t) are algebraic
algebraic functions
functions of
of 92(1)
0 2(t) given
given
by (37.6). Using
Using the definitions
definitions cn
en 7°-r = y{1-sn2-r),
-v'(l-k2
sn2-r),
\/ (1 -sn21~), dn 1'-r = x/(1
-k2 sn2v),
bY
we find
QUO
01 =
= V[(2EI3-M2)/I1(I3-I1)l
v[(2E/s-M 2)/h(ls-h)] CI11",
cn-r, )
QUO
(37.10)
02 =
= v[(2E13-mz)/12(13-12)1
v[(2Els-M2)/h(ls-h)] SHE,
sn-r,
2-2Eh)/ls(ls-h)]
Qs
[(r7W22EI1)/I3(I3- I1)1 dIor.
= v[(M
dn-r.
Os =
These are periodic
periodic functions,
functions, and
and their
their period
period in the variable
variable 1'-r is 4K,
4K,
complete elliptic
elliptic integral
integral of
of the first kind
kind::
where K is a complete

= oJ
0

II

M

K

1l

du
ds
Jh
du
v[(l-s2)(1-k2s2)] =
y(l-k2
sin2u)
(1 - k2 sin'2u)
v[(1¥s2)(14k252)]
0

(37.
(37.11)
11)

§3w
§37

119

;

The
period in tr is therefore
The period
therefore

The asymmetzrieal
asymmetrical top
top
The

2-2Eh)].
T=
[I1I2I3,'(I3-I2)(]W2-2EI1)].
= 4K
4Kv[hhls,'(ls-h)(M

(37.12)
(37.12)

After
After a time
time T the vector
vector $2
Q returns
returns to its original
original position
position relative
relative to the
axes of
return to its original
of the top.
top. The
The top
top itself, however,
however, does
does not
not return
original position
position
relative to the Fixed
fixed system
system of
of co-ordinates,
co-ordinates; see below.
relative
For 11
h == I2,
/ 2, of course, formulae (37.10) reduce
reduce to those
those obtained
obtained in §36
For
2
for a symmetrical
symmetrical top:
top: as 11
h ->~Ig,
--+ /2, the parameter
parameter k2
k --+ G,
0, and the elliptic
elliptic
functions degenerate
degenerate to circular
circular functions:
functions: sn 1'-r ->
--+sin
-r, cn
en 1'r ->~
--+cos
functions
sin 1-,
cos -r,
--+ 1, and
and we return
return to formulae
formulae (36.7).
{36.7).
dn 1'-r ->
2
When M2
M =
= ZEISS
2Els we have QUO
01 = Qg
02 = 0, QUO
Os = constant, i.e. the vector $2
Q
is always parallel
parallel to the kg-axis.
xs-axis. This case corresponds
corresponds to uniform rotation
rotation of
of
2
the top about the xi-axis.
xs-axis. Similarly,
Similarly, for IH?
111 = ZEISS
2Eh (when
(when -r-r E= 0) we have
uniform
uniform rotation about
about the JC1-3.XiS.
x1-axis.
Let
Let us now
now determine the
the absolute motion of
of the
the top
top in space
space (i.e. its
motion
motion relative to the fixed system
system of co-ordinates
co-ordinates X, Y, Z). To
To do so, we
use the Eulerian
Eulerian angles
angles go,
if, ¢,,
c/>, 6,
e, between
between the axes xl,
Xl, xg,
X2, x3
Xs of
of the top
top and
and the
axes X, Y, Z, taking the fixed
Fixed Z-axis in the direction
of
the
direction
constant vector
vector M.
constant
M.
Since the polar
polar angle
angle and
and azimuth
azimuth of
of the Z-axis
Z-axis with respect
respect to the axes
xl,
a - ,L (see the footnote to §35), we obtain
obtain on
x1, xg,
x2, x3
x3 are respectively 0eand J_7r-if
taking
1, xg,
taking the components
components of
of M along the axes xx1,
x2, x3
xs
L

Hence
Hence

I3Q3l"IW,
cose6* = lsOsfM,
cos

and from formulae (37.10)

tangy

=

Jl

r"

Msin6
M sine sin',b
sin if= M1
M1 = I1Q1,
h01,
M
M sin
sine6 cos»,b
cos if =
= M2 -=
= I2Q~2,
h02,
JW
M us
cose9 =
= Z1/I3
Ms =
= I3Q3.
fsOs.

I1Q1/I2Q2,

/ [13(m2-2E11)§m2(13-- I1)] dllT,
us 6* =
cose
= 1'\![ls(M2-2Eh)/1112(ls-h)]
dn-r,
~

2),,/IQ
1I3-I1)] CI11'/SHT,
tan»,b
tanif =
= v[I1(I3-I
v[h(/s-h)/h(ls-h)]
cnrfsnr,

(37.13)

(37.14)
(37.15)

if as functions
functions of
of time; like the components of
of the
the
which give the angles 6eand ,L
Q, they
they are periodic
periodic functions,
functions, with period
(37.12).
period (3'7.12).
vector so,
ae (37.13), and to calculate it we
The angle al,
cf> does not
not appear
appear in forkful
formulae
The
return toto. formulae
formulae (35,
( 35.11),
), which
which express
express the components
components of
of $2
Q in terms
must return
es. Eliminating 9
of the time
time derivatives of
of the Eulerian
Eulerian angl
angles.
efrom
from the
the equaof
cos
,L65
sin
./,,
we
obtain
gt;
sin
6
sin
+
9
cos
95,
QUO
sin
6*
t~ons 01 = ~
e if+ e if, !22 = J e if- e if,
obtain
cos
go)/sin
H,
and
then,
formulae
(3'7.13),
c/>
=
(Olsinif+02cosif)fsin
e,
using
(37.13),
it, = (91 S111',b-1'Q2
2 )M/(ft 2 012 +
(37.16)
I2Q22)M/(IFQ12
d¢~»/dz
dcf>/dt = (/1Q12+
(hil12, + h02
+ 122Q22).
/ 2 2 !122).
) is obtained
ction ¢(f
The
functio~
</.>(t)
obtained by
by integration,
integration, but
but the
the integrand
integrand involves
involves
he fun
T

tions in
functions
in aa complicated
Complicated way.
way. By
By means
means of
of some
some fairly
fairly complex
complex
lliptic func
eelliptic

M0tz.on
Motion of
of aa Rigid
Rigid Body
Body

120

§37

the integral
integral can
can be expressed
expressed in terms
terms of
of theta
theta functions
functions;,
transformations, the
shall not
not give the calculations,T
calculations, t but only
only the
the final result.
we shall
result.

The function
function ¢,(¢)
q,(t) can
can be represented
represented (apart from an arbitrary
arbitrary additive
additive
constant) as a sum
sum of
of two terms
terms::

W)

one of
of which is given by

.

exp[21qf>1(
Z)] =
exp[wp1(t)]
=

=

¢1(t)+¢2u),

(37.17)

01 --za j &01(~
T +icx
za ).
&01(~
( T -icx)
)/ 01 (--+
),

s

.

2:

s

2:

.

(37.18)

wheree 901
&01 is a theta function and
and acx aa real
real constant such
such that
that
wher

sn(2iaK)
sn(2icxK) =
= iV[I3(M2+ZEII)/I1(2EI3-Il/I2)];
iv[ls(M 2 - 2Eh)/h(2Els- M2)];

(37.19)

Tare
The function
function on
on the right-hand side
side
K and T
are given by (37.11) and (37.12). The
of
IT, so that
of (37.18) is periodic,
periodic, with period
period !T,
that ¢>1(*)
q,1(t) varies by 21:
27T during a time
T. The
The second term in (37.17) is given by

¢>2(l)
q,2(t) ===
= 2111/T',
2TTtjT',

M
M

11

il

-=----T'

`

z,

901y(Z.a)
&01'(icx)

T s&01(icx)
ea)
TTT

'

(37.20)

This function
T'. Thus
function increases by Zn
27T during
during a time T'.
Thus the motion in go
¢> is a
combination of
of two periodic
periodic motions,
motions, one of
of the periods
periods (T)
( T) being
combination
being the same
as the period
period of
of variation
variation of
of the angles ,L
if and 9,
e, while the other
other (T')
(T') is incomincommensurable
mensurable with T. This
This incommensurability
incommensurability has the result
result that
that the top does
not
not at any
any time
time return
return exactly
exactly to its original
original position.
PROBLEMS

PROBLEM
x3-axis or
PROBLEM 1.
1. Determine
Determine the
the free
free rotation
rotation of
of aa top
top about
about an
an axis
axis near
near the
the xa-axis
or the
the

x1-axis.
Jc1-axis.

SOLUTION. Let
Let the
the kg;-axis
xa-axis be
be near
near the
the direction
direction of
of M.
M. Then
Then the
the components
components M1
M1 and
and Ms
Ms
SOLUTION.
are
are small
small quantities,
quantities, and
and the
the component
component Ma
Ma = M
M (apart
(apart from
from quantities
quantities of
of the
the second
second and
and
higher
higher orders of
of smallness). To
To the same accuracy the
the first two
two Euler's equations (36.5) can
be written
written dM1/dt
dM1/dt =..'° QOM2(1
floM2(1-la/12),
dM2/dt 3 NOM1(I3lI1
f.loMl(la/JI-1),
flo=
Mila. As
-.I3!I2), dM2/dt
-1). where 00
1
MII3.
be
usual we
we seek
seek solutions
solutions for
M1 and
and M2
M2 proportional
proportional to
to exp(iwt),
exp(iwt), obtaining
obtaining for
the frequency
frequency w
usual
for M1
for the
w

=

=

e:
w

=

n0

Ma"/(-Is - 1 )

of M1
M1 and Ma
Ms are
The values of

M1 = MaJ
M1

la

=

Ju- -1)(- ~1)]~
la
11

-1) cos
cos wt,
wt,

la
I.

MARJ( 11~:I --1)

M2 ==
M2
= Maj ( a -1) sin
sin wt,
wt,

(1)
(1)

(2)
(2)

where aa iiss an
an arbitrary
arbitrary small
small constant.
constant. These
These formulae
give the
the motion
motion of
of the
the vector
vector.M
where
formulae give
M
relative
frequency w,
relative to
to the
the top.
top. In
In Fig.
Fig. 51,
51, the
the terminus
terminus of
of the
the vector
vector M
M describes,
describes, with
with frequency

small ellipse
ellipse about
about the pole
pole on the
the x3-axis.
xa-axis. I,
a small
To deterrmne
determine the
the absolute
absolute moti
motion
of the
the top
jn space,
space, we
we calculate
calculate its
its Eulerian
Eulerian 8ngl€s.
angles.
on of
To
top in
In the
the present
present case
case the
the angle
angle 08 between
between the
the xi-axis
x3-axis And
the Z-anus
Z-axis (direction
(direction of
of M)
M) is
is small,
small,
And the
In
These are
are given
given by
by E.
E. T.
T. WI-IITTAKER,
WHITTAKER, A
A
Tt These

•
•
• s of Particle~
namics of Particle;
Treatise
Treatise on
on the
the Analytical
Analyttcal Dy
DynanJ'c

and Rigid
IUgid Bodies,
Bodin, 4th
4th ed.,
ed., Chapter
Chapter VI,
VI, Dover,
Dover, New
New York 1944.
and

§37

12
1211

The asymmetrical
asymmetrical top
top
The

-cos H)
and by
byformulae(37.14)
tan W,
= M1/M2, 62
02 ..'z:~." 2(1
2(1-cos
0)2(1-MafM)
z (M12
(M1 2+Ma
+Ms2)/M2;
and
formulae (37.14) tan
of( =
=- 2(1
"M3/347 21
2)/M2 S

substituting (2),
(2), we
obtain
substituting
we obtain
92
02

tan
tan

=

[ 2~: -1) cos2wt+ e:
I' I ..1) sin wt]..

.;.1/J =

be
a2[(

I

1/[I1(I3"/2)//2(I3-I1)]
v'[II(la-12)/12(la-II)] cot
cot wt,

-1) oos2wt+

2
-1) sin?wt

To
To find
find ¢>,
4>, we note
note that,
that, by
by the
the third formula (35.1),
(35.1), we have,
have, for
for 60

Hence
Hence

(3)
(3)

< 1,1, QUO
no z Us
na zz ¢+¢.
.f,+.f>.
i v
i v

(4)
45 = Qof-'A
(4)
omitting an
an arbitrary
arbitrary constant
constant of
of integration.
integration.
omitting
A
clearer idea
idea of
of the
the nature
nature of
of the
the motion
motion of
of the
the top
top is
is obtained
obtained ifif we
consider the
the change
change
A clearer
we consider
in
in direction
direction of
of the
the three
three axes
axes of
of inertia.
inertia. Let
Let 111,
DI, 112,113
n2, ·na be
be unit
unit vectors
vectors along
along these
these axes.
axes. The
The
vectors
Y-plane with
vectors n1
n1 and
and nz
n2 rotate
rotate uniformly
uniformly in
in the
the X
XY-plane
with frequency 90,
flo, and
and at
at the
the same
same time
time
execute small
small transverse
transverse oscillations
oscillations with
with frequency to.
w. These oscillations are
are given by
by the
the
execute
Z-components of
of the
the vectors:
·
Z-components
vectors .

z M1/M == fn/(13II2-1)
av'(la/12-1) cos wt,
wt,
Z M2lM
M2/M ==
= aV(I3[I1--1)
av'(la/li-1) sin wt.
wt.
fmz
i z z -"

nlz
mz

For
For the
the vector
vector no
na we
we have,
have, to
to the
the same
same accuracy,
accuracy, 1131
naz z 60 sin
sin Qs,
tf>, 7!3
nayy zz -6
-8 cos tf>,
naz zz 1.
1.
95, 713:
(The polar
polar angle
angle and
and azimuth
azimuth of
of no
na with
with respect
respect to
to the
the axes
axes X,
X, Y,
Z are
are 68 and
and gb-151-r;
tf>-!rr; see
see
(The
Y, Z
the footnote
to §35-)
§35.) We
We also
also write,
using formulae
formulae (37.13),
(37.13),
the
footnote to
write, using
713:
naz =

68 sin(Q4)t-511)
sin( f.lot-1/l)

== 60 sin
sin Do:
not cos
cos »,£'-6
1/J-0 cos
cos Dot
not sin
sin
=
=

;

.;,1/J

¢.,¢-a/ (_ -1)
"J (I1-1)
Qa J (~: -1 )+ J (~: -1 )] cos(no+w)t+
+ +
-la[
L/ (i= 1)+/ (-E

Similarly

(M2/M)
(M2/M) sin Q0¢-(M1/M)
not-(MI/M) OOS
cos Hot
not

n

la
aJ ( _
~: -1) sin
sin (lot
not sin
sin wt-aJ ( ~: -1) cos
cos Quit
not COS
cos w
wtt

J(~: -0]
wJ(~: -1)-~/(13

+§a
+la[

-l )-

e:

cos( Q0

w)t

£20 - w)t.
-1)] cos(
cos(no-w)t.

Ll(-?-1)+l(? H

Qa
1
sin(9o+w)t+
nay= -!a[J
( ~: -1 )+ J (:: --1)]
sin(no+w)t+

7'l3 y " '

+la[J ( ~: -1 )~ J
+
4 ¢ ( 1 ) ~ / (la--1)]
) 1 sin(no-w)t.
la

sin( Ho

cu) L

From this
this we
see that
that the
the motion
motion of
of fig;
na is
is aa superposition
superposition of
of two
two rotations
rotations about
about the
the Z-axis
Z-axis
From
we see
with
frequencies Noi'
no± w.
with frequencies
PROBLEM 2. Determine
Determine the
the free
free rotation
rotation of
of aa top
top for
for which
which M2
M 2 == 2E/3.
2Els.
PROBLEM

SOLU
TION. This case
SOLUTION.
case corresponds
corresponds to
to the
the movement
movement of
of the
the terminus
terminus of
of M
M along
along aa curve
curve
thro
ugh the
through
the pole on the xi-axis
x2-axis (Fig. 51). Equation
Equation (37.7)
(37.7) becomes d.t[d1°
ds/dT == 1-s*,
1-s•,
1'
[(Is-I1)(I3-I3)]I1I3]f20, ss == Q2fN0,
== MlI3
7' = - 11/
tv'[(ls-ll)(ls-ls)/I!ls)no,
nz/no, where 00
no=
M/ls '=
= 2E]M.
2E/M. Integration
Integration of
of
this equation
equation and
and the
the use
use of
of formulae
(37.6) gives
gives
formulae (37.6)

n1 = QoV[/2(/3-13)/I1(I3-/1)]
nov'(J2(Ja-Js)/Jl(Ja-l!)) sech
sech T,
T,
Q1

n2 == QUO
no taiih
tanh 1',
.,.,
£22
!13 = !'loV[I2(I3
!lov'[h(ls-1!)/la(ls-1!)]
.,._
Qs
-/1)[/3(I3 -11)] sech 'r.

}

(1)
(1)

To
olute mo
tion of
To describe
describe the
th~ abs
abs_olut~
motion
of the
the top,
top, we
we use
use Eulerian
Eulerian angles,
angles, defining
defining 08 as
as the
the angle
angle
ection of
Z- 16s
M) and
een the
betw
between
the ~7";
') (dir
(dr~tron
ofl\{)
and the
the xg-axi8
X2-axis (not
(not the
the x3-axis
as previously).
previously). In
In formulae
formulae
x3-axis as
(37.14) and
and (37.· ).'w
ch r¢late
relate the
the components
components of
of the
the vector
vector $2 to
to the
the Eulerian
Eulerian angles,
angles, we
we
(37.14)
which

6

n

[Motion
M~tion of
of aa Rigid
Rigid Body
Body

122

§38

must cyclically
cyclically permute the
the suffixes
suffixes 1, 2, 3 to 3, 1, 2. Substitution of
of (1) in these formulae
=tanh
T, ¢
tf> = 90:
f.lot+constant,
tan 'J'
1/J =
= V'
,I[Ia(I2-1
then gives cos 90 =
tank 'r,
+constant, tan
[I3(I2-I1).!I1(I3--I2)].
1 )fl1 (Ia-12)].
-->- oo, the
the vector $2
Q asymptotically approaches the
the
It is seen from these formulae that, as t ->
x2-axis, which itself
itself asymptotically approaches the
the Z-axis.
xi-axis,

Rigid bodies
bodies in
in contact
§38. Rigid
The
The equations
equations of motion
motion (34.1) and (34.3) show that
that the conditions
conditions of
equilibrium for a rigid body can
can be written
vanishing of
of the total
total force
equilibrium
written as the vanishing
and
and total
total torque on the body
body::
II

F

f

0,

K

rxf

0.

(38.1)

Here the summation
body, and
summation is over
over all the external
external forces
forces acting on the body,
and r
vector of
of the "point of
of application",
application"; the origin with respect
respect to
is the radius vector
the torque
torque is defined
defined may be chosen
chosen arbitrarily,
arbitrarily, since if F =
= 0 the
which the
value of
of K does not
not depend on this choice
choice (see (34.5)).
(34.5)).
If
system of
of rigid bodies
bodies in contact,
contact, the conditions
conditions (38.1) for
If we have a system
each body
body separately
separately must
must hold
hold in equilibrium.
equilibrium. The
The forces
forces considered
considered must
must
include those
those exerted
exerted on each body by
by those
those with which
which it is in contact. These
include
forces at
at the
the points of
of contact
contact are called
called readimzs.
reactions. It
It is obvious that
that the mutual
mutual
forces
reactions
bodies are equal
reactions of
of any
any two bodies
equal in magnitude
magnitude and
and opposite in direction.
general, both
both the magnitudes
magnitudes and
and the directions
directions of
of the reactions
reactions are
In general,
found by
by solving simultaneously
simultaneously the equations
equations of
of equilibrium
equilibrium (38.1) for all the
bodies.
some cases, however,
howeYer, their
their directions
directions are given
given by the conditions
conditions
bodies. In some
of the problem. For
For example,
example, if two
two bodies
bodies can
can slide freely
freely on each
each other,
other, the
of
reaction
them is normal
normal to the
the surface.
surface.
reaction between
between them
If
two bodies
bodies in contact
contact are in relative motion,
motion, dissipative
dissipative forces of
of friction
If two
friction
addition to the reaction.
reaction.
arise, in addition
possible types of
of motion
motion of bodies
bodies in contact-sliding and
There are two possible
rolling. In sliding,
sliding, the reaction
reaction is perpendicular to the surfaces
surfaces in contact,
friction is tangential.
tangential. Pure rolling, on the other
other hand,
hand, is characterised
characterised
and the friction
by the fact that
that there
there is no relative
relative motion
motion of
of the bodies
of
bodies at the point
point of
contact; that
that is, a rolling
rolling body
every instant as it were
were fixed to the point
point
contact,
body is at every
of contact. The
The reaction
reaction may
may be in any
any direction,
direction, i.e. it need
not be normal
normal
of
need not
the surfaces in contact. The friction
friction in
-in rolling appears as an additional
additional
to the
torque which
which opposes
opposes rolling.
rolling.
torque
If
friction in sliding
sliding is negligibly
the surfaces
surfaces concerned
concerned are
If the friction
negligibly small, the
said to be perfectly smooth. If, on the other
other hand, only
only pure
without
said
pure rolling without
sliding is possible,
and the friction in rolling
rolling C2111
Ca!l be neglected,
surfaces
sliding
possible, and
neglected, the surfaces
said to be perfectly
are said
p e f e d l y rough.
both these
these cases the frictional
frictional forces do not
explicitly in the proIn both
not appear explicitly
blem, which is therefore
therefore purely
mechanics. If, on the other
other hand,
hand, the
purely one of mechanics.
properties of
of the friction play
play an essential
essential part
part in determining
determining the
the no¢ion,
motion,
then the latter is not
not a purely
mechanical process (cf. §25).
then
purely mechanical
0~ freedom
egrees of
Contact between
between two
two bodies
bodies reduces
reduces the
the number
number of
of their
their d
degrees
freedom
Contact
g such
.
h
f
f
.
H"
h
.
discussmg
such
ussin
disc
in
Wit
the
case
o
ree
motwn.
It
erto,
m
as
compared
case
as compared with the
of free motion. Hitherto,

§38

123

Rigid bodies in contact
contact

problems, we have taken
using co-ordinates
taken this
this reduction
reduction into account by using
which
correspond directly
directly to the actual
actual number
number of
of degrees
degrees of freedom,
freedom. In
which correspond
rolling,
however, such a choice
choice of
of co-ordinates
co-ordinates may
may be impossible.
impossible.
rollin
g, however,
The condition
condition imposed
imposed on the motion
motion of
of rolling
that the velocities
velocities
The
rolling bodies
bodies is that
of
of the points
points in contact
contact should be equal,
equal; for example,
example, when a body rolls on a
Fixed
fixed surface,
surface, the velocity
velocity of
of the point
point of
of contact must be zero. In the general
general
case, this condition
condition is expressed
expressed by the equations of
of constraint, of the form

E Can££iz
'i

0,

(38.2)

where
Cat are
are functions
functions of
of the co-ordinates only, and
and the suffix aex denumerwhere the Cai
ates the equations.
equations. If
If the left-hand sides
sides of
of these equations are not
not the total
time derivatives
derivatives of
of some
some functions°of
functions ·of the co-ordinates,
co-ordinates, the equations
equations cannot
time
between the
be integrated.
integrated. In other
other words,
words, they
they cannot
cannot be reduced
reduced to relations
relations between
co-ordinates only, which
which could
position of
bodies
could be used to express
express the position
of the bodies
in terms of
of fewer
fewer co-ordinates, corresponding
corresponding to the
actual number
number of
of degrees
degrees
the actual
of freedom.
freedom. Such
Such constraints
constraints are said to be non~lzolonomic,
non-holonomic, as opposed
opposed to
of
holonomie
relations between the co~ordinates
holonomic constraints, which
which impose
impose relations
co-ordinates only.
Let
Let us consider,
consider, for example,
example, the'rolling
the· rolling of
of a sphere
sphere on a plane.
plane. As usual,
translational velocity
velocity (the velocity
velocity of
of the centre of the
we denote by V the translational
sphere), and
and by $2
Q the angular
angular velocity
velocity of rotation.
rotation. The
The velocity
velocity of
of the
sphere),
the point
of
plane is found by
of contact
contact with
with the plane
by putting
putting r =
= -an
general formula
an in the general
v =
= V-I-S2Xr,
V + Q x r; a is the radius of the sphere
sphere and
and n a unit vector
vector along the
normal to the plane. The
The required
required condition
condition is that
that there should
should be no sliding
sliding
normal
the point
point of
of contact, i.e.
at the

V-aS'2xn
= 0.
V
- a f ! xn =

(38.3)

This cannot be integrated: although
although the velocity
velocity V is the total time derivative
derivative
This
of
of the radius
radius vector
vector of
of the centre of
of the sphere, the
the angular
angular velocity
velocity is not
not in
general the total
total time derivative of
of any
any co-ordinate.
co-ordinate. The
The constraint
constraint (38.3) is
general
therefore non-holonomic.T
non-holonomic. t
therefore
the equations
equations of
of non-holonomic
non-holonomic constraints cannot
cannot be used
Since the
used to reduce
reduce
number of
of co-ordinates, when
when such
such constraints are present it is necessary
the number
necessary
to use
use co-ordinates which
which are not
not all independent. To
To derive
derive the correspondto
L"agrange's equations,
equations, we return
of least
least action.
action.
ing L'agrange's
return to the principle of
The existence
existence of the constraints
constraints (38.2) places certain
certain restrictions
The
restrictions on the
possible values
values of
of the
the variations
variations of
of the co-ordinates: multiplying
multiplying equations
equations
possible
by 8t,
St, we find that the variations
variations 891
Sq1 are not
not independent, but are
(38.2) by
relate
related
d by
(38.4))
(38.4
0.
Caigqi
t

2i

It maY
rnay be
be not
noted
the similar constraint
constraint in
in the
the rolling of
of a cy
cylinder
is holonomic. In
In
ed that the
linder is
It
. the
tation ha
the I:'1Xis
axis of
of ro
rotation
has
fixed direction in
in space, and
and hence
hence Q =-_
= dqajdr
d<f>/dt is the
the total
s a fixed
the aogle
derivative of
of _the
angle ¢>
</> of
of rota
rotation
of the
the cylinder about
aboUt its axis.
mds. The condition (38.3) can
tion of
derivative
that
that C356
case

e . t
.
therefore
be
integrated,
and gives
a
efoI'¢ b
ther
IN
s, Curated, and 81ves a
centre of
of mas
rnass.
centre

n

. between
.
relation
between the
the angle
angle ¢,
<f> and the
the co-ordmate
co-ordinate
of the
relation
of

124

Rigid Body
Motion of
of a Rigid

§38

must be -taken into
into account in varying
varying the action. According
According to
to
This must
L
ran e's method
if conditional
Lagrange's
method of
of End
finding
conditional extrema,
extrema, we must
must add to
to the inteinteag _g
.
g .
grand in
in the variation
variation of
of the action
action
grand
I

aL d AL
ss J
~sq,
q; [oL --S
f 3oq, -~(o~)]
oq,H dt
dt (3g5¢
Lil

=
8S =

ii

1

the left-hand
left-hand sides of
of equations
equations (38.4)
(38.4) multiplied
multiplied by undetermined
undetermined coefl:icicoefficiAa (functions
(functions of
of the co-ordinates), and
and then
then equate the
the integral to
to zero.
ents Aa
In so doing
doing the variations
variations Sq;
Sq, are regarded
regarded as entirely
entirely independent,
independent, and
and the
In
result
result is

AL
_
2
oqt ) oLgooq, = 2:A.acai·
dt ( 3qg

d AL
~(oL)
_

=

(38.5)

AaCai'

aa

equations, together
together with
with the constraint equations (38.2),
(38.2), form
form a comThese equations,
plete
unknowns QUO
of equations
equations for the unknowns
q, and
and Aa.
plete set
set of
The
The reaction
reaction forces do not
not appear in this treatment,
treatment, and
and the contact
contact of
of
the bodies
bodies is fully allowed
allowed for by means
means of
of the constraint
constraint equations.
equations. There
is, however,
bodies in
however, another
another method
method of
of deriving
deriving the equations
equations of
of motion
motion for bodies
contact, in which the reactions
are
introduced
explicitly.
The
essential
reactions
introduced explicitly. The essential feature
Ienzbert'ss Principle,
of this method,
method, which is sometimes
sometimes called d'_4
d' Alembert'
principle, is to write
of the bodies
bodies in contact
contact the equations.
equations.
each of
for each

up/
Dr =
dP/dt
=

22: f,f,

dM/dz
dM/dt --=
=

22: rr xxf,f.

(38.6)
(38.6)

wherein
wherein the forces f acting
acting on
on each body
body include
include the reactions.
reactions. The
The latter
initially unknown
unknown and
and are
are determined, together with the motion
motion of
of the
are initially
by solving
solving the equations.
equations. This method is equally
equally applicable
applicable for both
body, by
both
holonomic
and non-holonomic
non-holonomic constraints.
holonomic and
PROBLEMS

1. Using
Using d'A1embert's
d'Alembert's principle,
principle, find
find the
the equations of
of motion
motion of
of a homogeneous
homogeneous
PROBLEM 1.
sphere rollin
rolling
on aa plane
plane under
under an
an external
external force
force FF and
and torque
torque K.
K.
sphere
g on

SOLUTION. The
The constraint
constraint equation
equation is
is (38.3).
(38.3). Denoting
Denoting the
the reaction
reaction force
at the
the point
point of
of
SOLUTION.
force at
contact between
between the
the sphere
sphere and
and the
the plane
plane by
by R,
R, we
we have
have equations
equations (38.6)
(38.6) in
in the
the form
contact
form
p. dV[dr
dV/dt

= F +R,

(1)
(1)

I d$°2,'d¢
cln/dt == K--an
K-an XR,
xR,

-

V =an

(2)
(2)

.

where we
we have
have used
used the
the facts
facts that
that P
P = ,UV
p.V and,
and, for
for a spherical
spherical top,
top, m
M _= IQ.
IQ. Differentiating
Differentiating
where
the
n
the constraint
constraint equation
equation (38.3)
(38.3) with
with respect
respect to
to time,
time, we
we have
have V = a $'Zxn
xn. Subs
Substituting
tituting iin
equation (1)
(1) and eliminating Q by
by means of
of (2),
(2), we
we obtain .(I/ap)(F-l-R)
-(I/ap.)(F+R) = Kxn-aR+
Kxn-aR+
+an(n+an(n • R),
R), which relates R,
R, F and
and K
K.. Writing
Writing this
this equation
equation in
in components
components and
and substitutsubstituting II = spa
fp.a 22 (§32,
( §32, Problem 2(b)),
2(b)), we
we have
ing
5
2
RE = -Ky- -F2,
7
7a

2
55
R 11 = - -Kz- ;.Fish
-Fy

Ry

.

7a
7aKI

7

. .
'

R, == --Fe,
-F,,
R:

. .

•
.
•
•
in (1),
where
xv-plane. Finally,
substxMtxng
these
where the
the plane
plane is
IS tal-:en
taken as
as the
the .xy-plane.
Frnally,
sub strtutrng
t h ese ¢xpresslo"5
expreS sions on

W
we
e

§38

125
125

Rigid bodies in contact
Rigid

-1Fx+

.

obtain the
the equations
equations of
of motion
motion involving
involving only
only the
the given
given external
external force
force and
and torque
torque::
obtain

dV..~=
5
dVz = ~(Fz+
K 11 ) .

dt
7p.
a _
dr
71"
d Vy11 _ 5
KJ.:
dV = ~(Fu- Kz )·.
dt
71"
a
dt
7p.
a

The components
components Qs,
f.!..,, fly
f.l 11 of
of the
the angular
angular velocity
velocity are
are given
given in
in terms
terms of
of Vz,
by the
the constraint
constraint
The
Vi, V
V,11 by
equation (38.3),
(38.3); for
f.!, we have
have the
the equation
equation §pa2
fp.a 2 di22fdt
df.l./dt = KG,
K,, the
the z-component
z-component of
of equaequaequation
for Q;
(2).
tion (2).

PROBLEM 2.
2. A
unifom1 rod
rod BD
BD of
of weight
P and
and length
length Il rests
rests against
against aa wall
as shown
shown in
in
PROBLEM
A uniform
weight P
wall as
Fig. 52
52 and
and its
its lower
lower end
end B
B is
is held
held by
by aa string
string AB.
AB. Find
Find the
the reaction
reaction of
of the
the wall
wall and
and the
the tentenFig.
sion in
in the
the string.
string.
sion
O

i

I .
h
J

.

II

+'*'8

I
I

T

re

_Y -_l

A

FIG.
F1G.

5
B

'//
.0

52

SoLUTION. The
The weight
weight of
of the
the rod
rod can
can be
be represented
represented by
by aa force
force P
P vertically
downwards,
SOLUTION.
vertically downwards,
applied at
at its
its midpoint.
midpoint. The
The reactions
reactions RB
RB and
and Re
Rc are
are respectively
respectively vertically
vertically upwards
upwards and
and
applied
perpendicular to
to the
the rod;
rod; the
the tension
tension T in
in the
the string
string is
is directed
directed from
B to
to A.
A. The
The solution
solution
perpendicular
from B
of the
the equations
equations of
of equilibrium
wves R
Rc
(Pl/4h) sin
sin2a,
P-Rcsin
a, T
Rccos
a.
sin a,
T = R
e cos a.
of
equilibrium gives
e = (Pl]4h)
2 a, RB = P-Rc
PRoDLEJ\1 3.
3. A
A rod
rod of
of ~eight
p'has
has one
one end
end A
A on
on aa vertical
vertical plane
plane and
and the
the other
other end
end BB on
on
PROBLEM
weight P
is held in
in position by
by two
two horizontal strings AD and
and BC,
BC,
a horizontal plane (Fig. 53), and is

41 -*,

A

__

C`LrI

.

\~6'

1I Ra

QQ

Z

I

fP'
¢

I

J7é

I
I

I

0'

FIG.
FIG.

53

_|

I

B

126

1.1!otion of
of aa Rigid
Rigid Body
Ilfotion

§39

the latter
latter being
being in
in the
the same
same vertical
vertical plane
plane as
as AB.
AB. Determine
Determine the
the reactions
reactions of
of the
the planes
planes and
and
the
the tensions
tensions in
in the
the strings.
strings.
the

SOLUTION. The
The tensions TA
T A and TB
Tn are from A to D and from B
B to C
C respectively. The
SOLUTION.
and RB
Rn are
are perpendicular to the
the corresponding planes. The solution of
of the
reactions RA and
the
equations of
P, TB
of equilibrium
equilibrium gives
gives RB
Rn =
= P,
Tn = !P
cot ex,
a:, RA = TB
Tn sin B,
{3, TA
T A = TB
Tn cos (3.
=}P cot
B-

PRODLE!II 4. Two rods
rods of
of length Zl and
and negligible weight are
are hinged
hinged together,
together, and
and their
their ends
ends
PROBLEM

-are
'lre connected
connected by
by aa string
string llB
(Fig. 54),
54). They
They stand
stand on
on aa plane,
plane, and
and a force FF is
is applied
AB (Fig.
at
at the
the midpoint
midpoint of
of one
one rod.
rod. Determine
Determine the
the reactions.
reactions.

\

\

tI Ra

1~'-..v

~==T~d
:L_

*I

A

4__
I
I

FIG.
FIG.

SOLUTION.
T acts
A from
from
SOLUTIO:-:. The
The tension
tension T
acts at
at l1

54

5B

.

A
from B
A to
to B,
B, and
and at
at B
B from
B to
to A. The
The reactions
reactions RA
RA

and
A and
and RB
Rn at
at l1
and B
B are perpendicular
perpendicular to
to the
the plane. Let
Let Re'
R c be the
the reaction on
on the
the rod
rod AC
AC at
at
the hinge
hinge;, then
then aa reaction
reaction -- RRec acts
acts on
on the
the rod
rod BC.
BC. The
The condition
condition that
that the
the sum
sum of
of the
the moments
moments
the
of the
the forces RB,
Rn, T
Tand
acting on the
the rod
rod BC
BC should be zero shows that
that R
Rc
acts along
along
of
and --Rc
R e acting
e acts
BC. The remaining conditions
conditions of
of equilibrium (for the
the two
two rods separately)
separately) give
give RA
RA = iF,
~F.
RB
-}F, Re'
Rn == !F,
Rc == ;lF
!F cosec a, T = iF
!F cot
cot a,
a:, where a
a is the
the angle
angle CAB.
CAB.

§39. Motion in a non-inertial
non-inertial frame of
of reference
point we have always used inertial frames of reference
reference in discussdiscussUp to this point
of mechanical
mechanical systems.
systems. 'For
, For example,
example, the Lagrangian
Lagrangian
ing the motion of
2 - U,
L0
"W02U,
Lo =
= !mvo

(39.1)
(39.1)

_ acLL
cr
Er

<39.2)
_2 )
(39

and the corresponding
corresponding equation
equation of
of motion
motion an
m dv0/dt
dv0 fdt =
- - 3oUlcer,
Ufor, for
for a single
particle in an external
external field are valid only
only in an inertial
inertial frame. (In this section
section
particle
suffix 0 denotes
denotes quantities
quantities pertaining
pertaining to an inertial
inertial frame.)
the suH'ix
Let us now consider
consider what
what the equations
equations of
of motion
motion will be in anon-inertial
a non-inertial
Let
frame
of reference.
The basis of
of the
the solution
solution of
of this problem is again
again the
frame of
reference. The
of least
least action,
action, whose validity
validity does
does not
not depend on
on the
the frame of
of
principle of
reference chosen. Lagrange's equations
equations
reference

d / 3L`
~(
oL)
dt \ 3ov
dr
V 1

•
•
•
n
likewise va
valid,
but the Lagrangian Is
is no
no longer of
of the
the form (39°1)ti2ndLtO
(39.l), _andLto
are likewise
lld, but
derive it
it vue
v-e must
must carry
carrv out
out the
the necessary
necessary tra
transformation
of the fun
functiOn o.
o.
derive
nsf orm atio n of

Motian in
in aa non-inertial
non-inertial frame
frame
Motz•on

§39

of reference
reference
of

127

This transformation
transformation is done
done in two steps. Let
Let us First
first consider
consider a frame of
of
ith a translational
K' .which
:which moves w
with
translational velocity
velocity V(z)
V(t) relative
relative to the
the.
reference K'
The velocities vo and v' of a particle
particle in the frames Ko
K 0 and
inertial frame Ko. The
K'' respectively
respectively are related
related by
by
·
K
v0
V(r).
vo = v'+
v' + V(t).

(39.3)

Substitution of this in (39.
(39.1)
Lagrangian in K':
Substitution
1) gives the Lagrangian
r2 U.
U.
L'
mv'2+mv'-V-I-m\
L' == !mv'2+mv'·
V +imV2-

Now V2(t) is a given
given function
function of
of time, and
and can
can be written
written as the
the total derivaderivaNow
tive with respect
L' can thererespect to tr of
of some
some other function,
fllnction; the
the third
third term
term in L'
omitfed. Next,
v' = dr'ldt,
dr' fdt, where
where r'
r' is the radius vector
vector of the parfore be omitted.
Next, v'
K'. Hence
tide in the frame K'.
1' ticle
II

dr'/dt = d(mv
- mr' .· dv/dr.
MV(t)-v'
mV(t)·v' = mvmV ·dr'fdt
d(mV .·r")/dt
r')/dt-mr'
dV/dt.

Substituting in the
the Lagrangian and
and again
again omitting
omitting the
the total time derivative,
derivative,
Substituting
we have finally

L'
L' =

~mv'2-mW(t)·r'm2)'2- mW(t)-r' -

U,
U,

where W =
= dv/dz
dVfdt is the
the translational
translational acceleration
acceleration of
of the frame K'.
K'.
where
The Lagrange's equation
equation derived
derived from
from (39.4)
(39.4) is

oU
dv'
W
NU
--z.
m- == ---mW(t).
m
do
2r' m ( )
dt

cr'

(39.4)

(39.5)

accelerated translational
translational motion
motion of
of a frame of
of reference
reference is equivalent,
equivalent,
Thus an accelerated
regards its effect
effect on the
the equations of
of motion
motion of
of a particle, to the
the application
application
as regards
of a uniform
uniform field of
of force equal
equal to the mass of
of the particle
multiplied by the
the
of
particle multiplied
acceleration W, in the direction
direction opposite
opposite to this
this acceleration.
acceleration.
acceleration
Let us now bring in a further
further frame of reference
reference K, whose origin
o~igin coincides
coincides
Let
w
that of
of K',
K', but
rotates relative
relative to K'
K' with angular
angular velocity
velocity S2(r).
Q(t).
with that
but which
hich rotates
K executes both a translational and
and a rotational motion
motion relative
the
Thus K
relative to the
Ko.
inertial frame Kg.
The
velocity
v' of
of the particle
composed of
of its velocity
velocity v
v
The velocity v'
particle relative to K'
K' is composed
K and
and the
the velocity
velocity Q
Q xx rr of
of its rotation
rotation with K:
K: v' =
= v-l-.QXr
v+QXr
relative to K
relative
r' in the
the frames
frames K and
and K'
coincide). Substitut(since the radius
radius vectors
vectors rrand
(since
and r'
K' coincide).
ing this in the Lagrangian
Lagrangian (39.4), we obtain
obtain

L

%-m1;l2 + my -$2 Xr-1-§m(S'Z

Xr)'2-

mV \7- r-

u.

(39.6)

the general
general form of the
the Lagrangian
i.agrangian of
of a particle in an arbitrary, not
not
This is the
necessarily
inertial, frame
frame of
crf reference. The rotation of
of the
the frame
frame leads to the
necessarilY inertial,
appearance in
in the
the Lagrangian
Lagranf;ian of
of aa term
term linear
linear in
in the
the velocity
velocity of
of the
the particle.
appearance
particle.
To
galcul
To calculate
the derivatives
deriv'ltiYcs appearing
appearing in
in Lagrange's
Lagrange's equation,
equation, we
atc the
we write
write

128

Rigid Body
Motion of
of a Rigid

the
the total
total differential
differential

§39

dL
mdv-S2 x r + mv-S2 xdr +
dL == my-dv+
mv·dv+mdv·S'2xr+mv·S'2xdr+

+
m(S2 xr)(S2 xdr)-mW·dr-(oUfor)·dr
X dr)- mW-dr- (3U/8r) - dr
+m(Q
xr)·(s-2

= my-dv+
mdv»S2 x
r + mdr-v X
52+
mv·dv+mdv·s-2
xr+mdr·v
xs-2+

+
m(S2 xr)
xr) xS2-drmW-dr- (3U/3r)-dr.
+m(Q
xS'2·dr-mW·dr-(oUfor)·dr.

The terms
terms in
in dv
dv and dr
dr give
The

3oLfov
L/ 3 v

+

= my
mv+ mf!
ms-2 Xxr,
r,

3L/31'--=
r.
oLfor = mvxS2-|-m(S2xr)xQ.-mWmvxS2+m(S'2 xr) xs-2-mW- aula
oUfor.

Substitution
Substitution of
of these
these expressions
expressions in
in (39.2)
(39.2) gives
gives the
the required
required equation
equation of
of

motion::
motion

= -oUfJr-mW+mrxQ+2mvxS'2+mS'2x(rxS'2).
mdvfdt =
mdvldt
aUla-fnw +.mr xQ.-I-2mvxS2+mS2 x(r XSZ). (39.7)

that the
the "inertia forces"
forces" due to the rotation
of the frame consist
consist
We see that
rotation of
of three terms. The
The force
force mrx
Q is due to
to the !!on-uniformity
of the
the rotation,
of
mr X $2
non-uniformity of
but the
the other two terms
terms appear
appear even
even if the rotation
rotation is uniform.
uniform. The force
force
but
2mvxS'2 is called
called the
the Coriolis force;
unlike any
any other
other (non-dissipative)
(non-dissipative) force
force
2mvxS2
force, unlike
considered, it depends
depends on the velocity
velocity of
of the particle.
force
hitherto considered,
particle. The force
mS'2x(rxQ)
called the centrifugal force.
plane through rand
mSg x(rxS2) is called
force. It
It lies in the plane
r and
SB,
s-2, is perpendicular
perpendicular to the axis of
of rotation
rotation (i.e. to so),
s-2), and is directed
directed away
from the axis. The
The magnitude
magnitude of
of this force is mpQ2,
mpD.2, where
where p is the distance
distance
from
of
of the particle from
from the axis of
of rotation.
Let us now consider
consider the
the particular
of a uniformly
uniformly rotating
Let
particular case of
rotating frame with
translational acceleration.
acceleration. Putting
Putting in (39.6) and (39.7) $2
Q =
= constant,
no translational
= 0, we obtain
obtain the Lagrangian
Lagrangian
W =

L =
m'v2+mv-S2
= !mv2+
mv·s-2 xr+t§-m(S2
xr+!m(s-2 xr)2- U
and
of motion
and the equation of

-

marv/dt
Znzv xS2+mS2
oUf&+2mv
xS'2+mS'2 x(r
x(r XSZ).
xs-2).
mdv/dt = - 3U/3r+

The
The energy
energy of
of the
the particle
particle in
in this
this case
case is
is obtained
obtained by
by substituting
substituting

p
P

=

in E = p-vp· v- L, which gives

3Ll3v
oLfov

+

= my
mv+ 17152
ms-2 xr
Xr

E=
= é~"w2-%"z(9
!mv 2 -l~(Q xr)2+
xr) 2 + U.

(39.8)

(39.9)

(39.10)

11 )
(39.
(39.11)

It
should be noticed
noticed that
that the energy'
energy contains
contains no term linear
linear in the velocity.
It should
The rotation
term depending
rotation of
of the frame simply
simply adds
adds to the
the energy
energy a term
depending only
only
on the
the co-ordinates
co-ordinates of
of the particle
arid proportional
proportional to the
the square of
of the
on
particle and
angular velocity. This additional
additional term - m
!m(Q
angular
( Q xxr r)2
) 2 is called the centrifugal
pot.ential
potential energy.
The velocity
velocity vv of
of the
the particle relative
frame of
of
rota ting frame
The
relative to
to the uniformly
uniformly rotating
by
o
K
e
to its velocity
velocity vo
v 0 relative
to the
the inertial frame
fram Ko by
reference is related to
relative to
(39
" xr.
(39.12)
-12)
vo = v+ ....

of

Motzon
Motion in
in aa non-inertialframe
non-inertial frame of reference
reference

§39

129

The momentum
momentum p (39.
(39.10)
of the particle
therefore the same
The
10) of
particle in the frame K is therefore
as its momentum
momentum P0
po =
= my()
mvo in the frame KoKo. The
The angular
angular momenta
momenta
Mo
x p 0 and M =
x p are likewise equal. The
particle
M0 =
= rrxpo
= rrxp
The energies of the particle
the two frames are not
not the same,
same, however.
(39.12) in
in the
however. Substituting v from (39.12)
2
=$W
l'V02+
11), we obtain
= !mvo2-mvo·S'2xr+U=
~m'v02-mv0 • S2xr-l- U = fmvo
E=
+U-mrxv
(39.11),
obtain E
(39,
U-mrxv00 •.Q.
SZ.
The First
first two
two terms are the
the energy
energy ET
Eo in the frame KoKo. Using
Using the angular
angular
momentum
momentum M,
M, we have

E ==
= E0-M-SZ.
Eo-M·s-2.

(39.13)
(39.13)

This formula
formula gives the law of
of transformation
transformation of
of energy
energy when
when we change
change to a
This
uniformly
been derived
particle,
uniformly rotating
rotating frame. Although
Although it has been
derived for a single
single particle,
g~neralised immediately
immediately to
to any system of
the derivation can
can evidently
evidently be generalised
the
particles, and
and the
the same
same formula
formula (39.13)
(39.13) is obtained.
obtained.
particles,
PROBLEMS
PROBLEMS

PROBLEM 1. Find the deflection
deflection of
of a freely falling
falling body
body from
from the vertical caused by
by the

Earth's rotation,
rotation, assuming
assuming the
the angular
angular velocity
velocity of
of this
this rotation
rotation to
to be
be small.
small.
Earth's

SOLUTION, In
In a gravitational
gravitational field
field U
U = -mg»
-mg· r,r, where
where g is the
the gravity acceleration
acceleration
SOLUTION.
vector; neglecting
neglecting the centrifugal
centrifugal force in
in equation
equation (39.9) as containing the square of
of Q,
.Q, we
vector;
the equation of
of motion
have the

if = 2v x$2+g.

(1)
(1)

may be solved by
by successive approximations.
approximations. To
To do so, we put
put v == Vt
+v2,
This equation may
VI -I-v2,
Vtis the
the solution of
of the
the equation i>'t
g, i.e. vi
VI == gt
+vo (Vo
(vo being the
the initial
initial velocity).
where V1iS
vi = g,
gt -1-v0
velocity).
~ vi
VI+
v2 in
in (1)
(1) and
and retaining
retaining only Vt
on the
right, we have
have for
V2 the
Substituting v ==
-I-V2
VI on
the right,
for vi
the equation
+2vo xS2. Integration
V2 == 2v1x$2
2vtx.Q =
= 2tgxs2
2tgx.Q+2voxn.
Integration gives
gives
V•2

+v0=t+%8i=+§i28 XS-Z"*"f2V0 x$2,

(2)
(2)

Vl"here h is
is the initial radius vector
vector of
of the
the particle.
particle.
where
Let the
the z-axis
z-axis be
be vertically
vertically upwards,
upwards, and
and the
the x-axis
x-axis towards
towards the
the pole;
pole; then
then go
gz == bgy11 == 0,
Let
g, =
= --g;
-g; QUO=.=
!l~- = Q cos
cos.\,
f.ly =
= 0, Q, = Q sin
sin.\,
where ,\
is
A, Do
A, where
A is the latitude (which for
for definitetake to
to be north).
north). Putting vo
Vo = 0 in
in (2),
(2), we find
= 0, y == -=§t"g9
-!t3gf.l cos
cos.\.
ness we take
find xx ==
A, Substitu12 of!
of the time of
offall
(2hfg) gives
gives finally x = 0, y = - §(2h
!(2h .fg)a
fg)3 "2
gf.l cos .\,
A, the negative
tion of
fall tt z~ V (Zhlg)
value indicating
indicating an
an eastward
eastwa1d deflection.
deflection.
blue

n

v

n.

n

--

the deflection from
from coplanarity of
of the
of a particle thrown
PROBLEM 2. Determine the
the path of
from the
the Earth's surface with velocity vo.
Vo-

.

SOLUTION.
Let the
the xz-plane be such as to
to .contain
the velocity
S
OLUT1ON. Let
contain the
ve l city vo. The initial altitude
= 0. The lateral deviation isis given by
by (2),
(2), Problem 1:
1: y = -§t3gQ1+t4)-(Qxvoz-Qzvoz)
-lt 3g!lz+t2 (!lxvoz-!lzvoz)
h =
2
or, substituting
substituting the time
time of
of flight tt ,..,~
~ 2902/g,
2voz/g, y =
4voz (lvoz!lx-t•ox!lz)/g2 •
or,
_-= 40012(&v02Q1-t»0xQ:)!g2.

PROB
PROBLEM
Determine the
the eHlect
effect of
of the
the Earth's
Earth's rotation
rotation on
on small
small oscillations
oscillations of
of aa pendulum
pendulum
LEM 3. Determine
(the problem
problem of
of Foucault's
Foucault's pendulum).
pendulum).
(the

SoLuTION. Neglecting
Neglecting the
the vertical
vertical displacement
displacement of
of the
the pendulum,
pendulum, as
as being
being aa
SOLUTION.

quantity
quantity

nd or
der of
of the
the seco
second
order
of smallness,
smallness, we
we can
can regard
regard the
motion as
as taking
taking place
place in
in the
the horizontal
horizontal
the motion
of
2
xy-p!ane
Omitting terms
terms in
In Q2
f.12,! we
we have
have the
the equations
equations of
of motion
motion 55+
+ <02.uv
w 2x =
= 2Q2y•,
2!l,y, 55
ji + w
x
.Iv-plan€-.. omitting
W2y
__
_2Q,x,
where
cu
is
= -2f.lzX, whero; w
the frequency
of oscillation
oscillation of
of the
the pendulum
pendulum ifif the
the Earth's
Earth's rotation
rotation .i_iss
frequency of
•
- thethe second
neglected· mm
Multtplyrng
second equation
equation by
by ii and
and adding,
adding, we
we obta
obtain
single equation
equation
neglect€d°
tlPlylrlg the
in aa single

-

i;;

x

+ y

Motion of
of aa Rigid
Rigid Body
Ilfotion

130

..
.
Et'+2iflzt'+w
+2iQzé + t' =
equation
equation

or
or

is

2
w2§

§39

0 for the complex quantity §{; =
= x-I-fy.
x+iy. For
For Q;
flz ~ w,
the solution of
of this
We the

ft'

=
= exp(
exp( -zfl2f)
-lflzl) [-41
[AI exp(z.w£}
exp(iwt) +42
+A2 clip(
<'Xp( -zu:t)]
-i "'t)]

x¥+I.J°
+iy = (No-I-iyo)
(xo+iyo) exp(-iQzt),
exp( -iflzl),

where
the functions
functions xo(r),
xo(t), ;v0(f)
yo(t) give
give the
the path
path of
of the
the pendulum
pendulum when
the Earth's
Earth's rotation
rotation is
is
where the
when the
vertical with
neglected. The
The effect
effect of
of this
this rotation
rotation is
is therefore
therefore to
to turn
turn the
the path
path about
about the venical
neglected.
velocity
n •.
angular ve
Iocity Qz.

\

CHAPTER
C
HAPTER

VII
V
II

THE
CANO!\ICAL
EQUATIOI'\St
T
HE C
A N O N I C A L EQUATIONST
§40. Hamilton's equations
THE formulation
formulation of
of the
the laws of
of mechanics in terms
terms of
of the
the Lagrangian, and
and
of
from it, presupposes
of Lagrange's equations
equations derived
derived from
presupposes that the mechanical
mechanical
of a system
system is described
described by specifying
specifying its generalised
generalised co-ordinates
co-ordinates and
and
state of
velocities.
not the only
only possible
pos~ible mode
mode of
of description,
description, however.
however. A
velocities. This is not
number of
of advantages,
advantages, especially
especially in the study of
of certain general
general problems of
of
number
mechanics,
mechanics, attach to a description in terms
terms of
of the generalised
generalised co-ordinates
of the system. The question therefore
therefore arises
arises of
of the form
form of
of
and momenta of
and
the
the equations of
of motion corresponding to that formulation
formulation of
of mechanics.
The passage
passage from one
one set
set of
of independent
independent variables
variables to another
another can
can be
The
by means
means of
of what
what is called in mathematics
mathematics Legendre's transformation.
transformation.
effected by
effected
In
In the present case this transformation
transformation is as follows. The
The total differential
of the Lagrangian
Lagrangian as a function
function of
of co-ordinates
co-ordinates and
and velocities
velocities is
IS
of

dL

=

2

2-

A

22: pPi d§'i+.1§
dqt + 2: Pi do,
dqi,

This
This expression
e».l>ression may be written
written

dL =
=

_

aL
AL d
""
cL do; -I- ""
cL
L-n-dqi+
L-.. dqj.
Q'.
iz 0cqi
i
ClJi
Qc
i EQ;
z

(40.1)

since the derivatives
derivatives 3LlEQi
cLfcqi are, by
by definition,
definition, the generalised
generalised momenta,
momenta, and
and
since
OlJiz == 155
Pi by Lagrange's
Lagr~nge's equations.
equations. Writing
\\"riting the second
second term
term in (40.1)
(40.1) as
L l3Q
8oLf
21% do;
°-) -EQ;
"2:.pi
dqi =
= d(ZPi§!©
d(:£Pitli):Sqt dh,
dpi, taking
taking the
the differential
differential d(2pig5i)
d('"22Pitli) to
to the
the left-hand
obtain from (40.1)
( 40.1) ·
reversing the signs, we obtain
side, and reversing

d(_2Pitli-L)
_2Pi dqi-idqi+ ~ Qi
tli dpi.
dpi.
do
Paid - L) = -- 2155
'Q

The argument
argument of
of the differential
differential is the energy
energy of the system
system (of,
(cf. §6);
The
expressed in terms
terms of
of co-ordinates
co-ordinates and
and momenta,
momenta, it is called
called the
the Hamilton's
expressed
Hamilton?
fzmctzbn or Hamiltonian
function
of the system
system::
Ha1m'Itom•a1z of

2 PiQ1:-LPitli- L.

H(p,
q, 0
t) =
=
He), Q,

(40.2)
(40.2)

T
4,i
z

nlay find
find useful the
the following
following table showing certain differences between the
Tt The reader may
in this book and
and that which is generally used in
in the
the English
English literature.
literature.
nomenclature used in

Here
Here
Principle of
of least
least action
action
Principle
I\1aupertuis'
principle
D/Iaupertuis' principle
Action
ACt1OIli .

_

Abbreviated
action
Abbreviated action

-Trans/
- T t d n s f aattors.
ors,

ElseKhere
El'.¢ezr.'here

Hamilton's principle
principle
Hamilton's

Principle of
of least
least action
Principle
{ Mau
1\!Iaupertuis'
principle
pertuis' principle
Hamilton's principal
principal function
function
H:unilton's
Action
ACTION
131

132

§40

The
The Canonical
Canonical Equations
Equations

From the equation in differentials
From

Z

<1H
p dpi+§
dH = -- LPi
dqi+ 2 Qi
qi dpi,

(40.3)
(40.3)

independent variables
Yariables are
arc the co-ordinates
co-ordinates and
and momenta,
momenta, we
in which the independent
have the equations
have

131 = - 3Hf113Q;.

q = EH, PA,

(40.4)
(40.4)

the required
required equations
equations of
of n*otion
n:otion in the
the variables
variables P
p and
and q, and
and
These are the
They form a set
set of
of Zs
2s first-order
first-order differential
differential
called Hamilton's
Hamzltons equations. "l`Iley
are called
equations
Pali) and
the 2s unknown
unknown functions
functions Pi(t)
and Q@('))
qi(t), replacing
replacing the s secondequations for the
Lagrangian treatment. Because
Because of
of their
their simplicity and
and
order equations in the Lagrangian
of form,
form, they
they are
arc also called canonical equations,
equations.
symmetry of
symmetry
The total
time
derivative
of
the
Hamiltonian
is
total
dcrivatiYe of the Hamiltonian

a=H
DH
_ +
-QT
Er
dt = ct
dH
dH

-- =

Substitution
Substitution of
of Qi
qi' and
and

cancel,
cancel, and
and SO
so

--

H . L oH
EH .
L---;:--<}i
cH
+ --pi·
+
Z cq1EQ; Z ascpi1 .
3

Q*
I

:.

Pi from
from equations
equations (40.4)
(40.4) shows
shows that the
the last
last two
two terms
dH,:ldr
dH/dt =
= UI,fct.
2-Ilya.

(40.5)

In particular,
Hamiltonian does not
depend explicitly
explicitly on time, then
then
In
particular, if the Hamiltonian
not depend
dHfdt =
= 0, and we have the law of conservation
conservation of energy.
energy.
dHldt
the dynamical
dynamical variables
Yariables q, Qq or
or q, p,
p, the Lagrangian
Lagrangian and the
the
As "ell
vi ell as the
Hamiltonian involve
involve various
various parameters which
which relate
relate to the
the properties of
of the
the
Hamiltonian
external forces
forces on it. Let
Let AA be one
one such
mechanical system
system itself, or to the external
mechanical
parameter.
Regarding it as a variable, we have instead of
of (40.1)
parameter. Regarding
( 40.3) becomes
and (40.3)
Hence
Hence

dL

dll
dH

)p··
dq·+
)p.
dq··+(cVc>..)
d>..
+. fY
j. p, (in,
L....,l
l.
_ Pi:
, t do,
t +(E*Lj3)l) do,

=

=

l

-

,

'<' p
"5_
Pi dqa+
dqi + 'T
> Qiqi dpidp1- (£L/ffm)
(cL,tc>..) do.
d>...

€»\)p,q =
(EH
(ell.. c,\)p,q
=

-

(aL/e-,,,,
(cLfcA.}iJ,q,

(40.6)

which relates
relates the derivatives
deriYatives of
of the Lagrangian
Lagrangian and
and the Hamiltonian
Hamiltonian with
which
The sufiixcs
suffixes to the
the derivatives
dcriv::~.tives show the
the quantities
respect to the
the parameter A. The
respect
which are to be kept
kept constant
constant in the differentiation.
di!Terentiation.
which
result can be put
put in another
::~.nothcr way. Let
Let the Lagrangian
Lagrar.gian be of
of the
the form
This result
= L0+L',
Lo +L', where L' is a snail
smail correction
correction to the function Le.
Lo. Then
Then the
L =
H' in the hamiltonian
Hamiltonian H -==· H
L'
corresponding addition H'
H0+H'
0 + H' is related to L'
by
by
.
4-0.7
(40.7)
(lf')p,q=
--(L')riq·
(Ink Q(
)
(H ) PJ?
i
.
o
t
'm
1
I
3
o
ic@d that,
(-l-0.1)
we didl
not
It may
) e noticed
I10t
tralusf
int(.). .·(40.3),
Iraq be
(-H•_ }-1) into
(40t".• ), dcpen(
~11at ' linn transforming
.•
_
deence
Il g
1
t1mc-dcPen
UGH
1ble Cxpl
mcluclc 21
a term
term m
in dt
dt to
to take:
take IlCCOLIIlt
account of In
a pos:4
poss1b.c
cxp 1lclt
ICit unclncludc
I

§41
§41

The Routhian
Routhiafz

133

of the
the Lagrangian,
Lagrangian, since
since the
the time would there be only a parameter which
of
would
Analogously to
would not
not be
be involved
involved in
in the
the transformation.
transformation. Analogously
to formula
formula (40.6),
( 40.6),
the
the Partial
partial time
time derivatives
derivatives of
of LL and
and H
H are
are related
related by
by

L
(oHfct)p,q == - (3L/3I)é.Q(oLfot)q,q·
(3H13I)z>,q

(40.8)

PROBLEMS
PROBLEMS
Hamiltonian for a single
single particle in Cartesian,
Canesian, cylindrical
cylindrical and
PROBLEM 1. Find the Hamiltonian
spherical Co-ordinates.
spherical
SoLUTION. In
In Cartesian co-ordinates x, y, z,
:::,
SOLUTION.
H =

1

gm (1)12 +f>y2+;>=-)+£»(x, so. 2);

in cylindrical co-ordinates r,r, ¢,
</>, :cg
z,
in

H
H ==

in
in spherical
sp,herical co-ordinates
co-ordinates r,r, 0,
8, go,
</>,

H

=

z.

o

(

'u

2)

11 ( 2 PA
,
P-1>2
2m Pa'
p,2+
7
+pz2) -l-[=(r,
+
U(r, ¢>,
</>, 2);
z);
+
+P2
2m
r2

(

)

pe2
P-1>2
)
1 (
P62
P¢»2
2+ -2+
-2- p,2+
+ r2 . "B +U(r,
</>).•
m Pr
r
sm2m
,.2
r2 sin'-35
+U(r 0.8, ¢>)
J

PROBLEM 2. Find
for a particle in
Find the
the Hamiltonian
Hamiltonian for
in a uniformly rotating frame of
of reference.
reference.

Sou;noN. Expressing
Expressing the velocity v in
in the energy
energy (39.11) in
in terms
terms of
of the momentum
momentum p
SOLl:T1ON.
by
by (39.l0),
(39.10), we have H
H =
= P2/2m-Qp2f2m-n• rxp+U.
rxp+U.
PROBLEM 3.
3.
PROBLEM

Find the
the Hamiltonian
Hamiltonian for
for aa system
system comprising
comprising one
one particle
particle of
of mass
mass M
M and
and nn
Find

SOLUTION.

energy E
E is obtained from
found in
in §13, Problem,
Problem, by
by
The energy
from the Lagrangian found

of mass m,
m, excluding the motion of
of the centre
centre of
of mass (see §13, Problem).
Problem).
panicles each of
changing the
the sign
sign of
of U.
U. The
The generalised
generalised momenta
momenta are
are
changing
Pa
= aL;av,
cLfova
pa =

"' (m 2 fp.)
= Meg
mva-

Hence

LPa = m

L

Va
Va

= (mM
(mMfp.)
lp)

Substitution in
in
Substitution

gives
E gives

Va
Va

L

2
aa

- (""2q"/11)
(nm 2 fp.)
Va,•
Va

2

vi,
v •.

.L

Va
Va

palm + (1lM
= Pafm
(1/M))§P=
Pc•'

1
1
H=
II
= -2m

up
'""'
L
a

Pa

2

l

+ -1
+

21~·1
2M

(2~ Pap))2++U.U.
2

(

a

§41
§41.. The
The Routhian
Routhian
some cases it is convenient,
convenient, in changing
changing to new variables,
var'iables, to replace
In some
only
only som
some,
and
not
of
generalised
velocities
momenta.
not
all,
of
the
generalised
velocities
by
momenta.
The transe, and
formation is entirely
entirely similar
similar to that
that given in §40.
§40.
formation
To simplify
simplify the formulae, let us at first suppose that
that there
only two
two
To
there are only
g
q,
g,
q,
bles
co-ordin<:tcs gq and
and 5,
g, say,
say, and
and transf
transform
variables
~' cj, ~ to
co-ordinates
orm from the varia
qq,, g,
w
p,
§,
here p.
g, p, g, where
p. is
is th
thee generalised
generalised momentum corresponding to the coordinate Qq.
ordinate

134

The
The Canonical
Canonical Equations
Equations

The differential of
of the
the Lagrangian L(q,
L(q, §,
~' Q,
tj, 5)
~) is

whence
whence

§42

dL
3L/ 3 q.) do
3L/13g) do(
(31,/é'§) d§+(8L/8§)
dL TO
. ((oLfoq)
dq+(oLfoq)
dtj+(oLfc~)
d~+(cLfcg) df
dt
3L/35) do(
3L/35) do,
= 13
p do
dq+p
dq+(oLfo~)
d~+(oLfot)
dt,
=
+p do(

d(L-pq)
pdq-q
dp+(cLfc~
d~+(cLfct)
d(
d(L
-is?) == 16
dq- Q dz)
+ (@L/@§) df
+ (8L/85) df.
we define
define the
the Rozdhiarz
Routhian as
as
IfIf we
R(q, P»
p, E,
~' t) =
= PQptj- L,
R(Q»

(41.1)

in which
the velocity
velocity q'
tj is expressed
expressed in terms
terms of the
the momentum
momentum P
p by means
which the
of
p =
of the
the equation p
= EL/8q.,
eLf ctj, then its differential
differential is
Hence

oR
dR =
=

aLlaf) do-(
3 Llafl if.
-p
-P dQ+é
dq+ti do-(
dp-(oLfo~)
M-(oLfog)
d(

q4 .= =ala
RP,
oRfop,
L/35 == --oRfo~,
3 R Ian,
aoLfo~

3oRfoq,
/ q3 ,
} 5= =- R
•
p
r8L../3§ =
3 R /3§.
oLJot
= --oRfo(

Substituting these
these equations
equations in
in the
the Lagrangian
Lagrangian for the
the co-ordiuate
co-ordinate
SubstitutiNg

(

dt J{
3?
33 ) =Jf.
3cR
R

II

dd (oR)
3R

(41.2)
(41.3)
(41.4)

~, we have
g,

(41.5)

Thus the
the Routhian
Routhian is a Hamiltonian
Han1iltonian with
with respect
respect to the
the co-ordinate
co-ordinate Qq
(equations (41.3))
(41.3}) and
and a Lagrangian with
with respect
respect to the
the co-ordinate f~ (equation
(equations
( 41.5}).
the general
general definition the
the energy
energy of the
the system
system is
According to the
'4k5"oding
II

L /as+~§ a1,/35-L
z>Q+5 aL./35-L,
E = q4 aoLfoq+t
oLfot-L =
= N+t
oLJo~-L.

In terms of
of the Routhian
Routhian it is
In

R./8g,
E :=: RR-5~ aoRJct,
E

(41.6)

as we find by substituting
substituting (41.1) and (41.4).
generalisation of
of the
the above formulae
formulae to the
the case of
of several
several co-ordinates
co-ordinates
The generalisation
gq and -3~ is evident.
evident.
of the
the Routhian may
may be convenient,
convenient, in particular, when
when some of
of
The use of
the co-ordinates
co-ordinates q are cyclic, they
they donut
do.not appear
appear
the co-ordinates
co-ordinates are cyclic. If
the
If the
in the
the Lagrangian,
nor therefore
therefore in the
the Routhian, so that
that the
the latter
latter is a funcLagrangian, nor
of P,
p, f~ and
and ( The
The momenta
momenta P
p corresponding
corresponding to cyclic co-ordinates
co-ordinates are
tion of
tion
the second
second equation
equation (41.3), which in this
this sense
constant, as follows also from the
constant,
contains no new information.
information. \fVhen
When the
the momenta
momenta pp are replaced
their
contains
replaced by their
given constant
(d/dt) 3R(P,
f, 5)./ag =
oR(p, ~'~)jog
= 3R(P,
oR(p, ~'f, rf)/3§
~)fo~
constant values, equations
equations (41.5)
(41.5)(dfdt)
become
equations containing
containing only
only the
the co-ordinates
co-ordinates §,
~, so that
that the cyclic
cyclic cobecome equations
equations are solved
solved for the
the funcentirely eliminated.
eliminated. If
ordinates are entirely
If these equations
~(t}, substitution of
of the
the latter on
on the
the right-hand sides of the
the equations
tions go),
Qq =
§, £)J13P gives the functions
direct integration.
= 3R(p,
oR(p, ~'~)fop
functions q(t)
q(t) by direct
integration.

£.

P
R O B LE M
PROBLEM
Find the
the Routhian
Routhian for
a symmetrical
symmetrical top
~op in an
an external
external Held
field U(<;l>,
U(t/>, 0),
8), eliminating
eliminating the?
the CYClic
cychc
Find
for a
co-ordinate
co-ordinate so'
1/J (where
(where .;.,
1/J, go,
</>, B
0 are
are Eulerian
Eulerian angles).
angles).

Poisson brackets

§42

135

SOLUTION.
The Lagrangian
+<£ cos 0)2--U(g6,
see
SoLUTION.
Lagrangian is L == =}I'1(02-I-£52
!/'l(ll 2 +.,6 2 Sin29).l.%13(,L
sin 2 8)+!/a(~+.,bcos
8) 2 -U(,P, 0);
8); see
Routhian is
§35, Problem 1. The Rollthian

2
R =_
= wb-L
pq;/;-L == Pa"
pq- -pa#
-p<V'f. cos 0-é1'1(62+¢2
8-!1' 1(/}2+.,62 Sin20)
sin 28)+U(,P,
8);
+vi, of;
213
2/a
the
the First
first term
term is
is aa constant
constant and
and may
may be
be omitted.
omitted.
-

_

brackets
§42. Poisson brackets
Let ff (P,
(p, q, I)
t) be some function of co-ordinates, momenta and time. Its
total time derivative is

of
of
df - 3:
of
di = Tt++
EZ'

as
L

Z(
IC
k

of
(of..
of.)
go
+
P ·
oqkqk+ 3/%
op!k
k
39%
'

9

I

Substitution
je given by Hamilton's equations (4-0.4)
Substitution of the values of QA:
tik and
ana Pk
(40.4)
the expression
leads to the
dffdt
= 5f/32t+[H»f]»
offot+[H,f],
(42.1)
(42.1)
dfldr =
OH
aH
where
oH of
of
oH of)
(42.2)
[H,f
]
(
[H,f] =
opk oqk - oqk opk ·
III

L 3Pa
E(
k

a

f).

3QA:

3 9 3 Pa

expression is called
called the
the Poisson bracket of
of the quantities
quantities H and f.
f.
This expression
Those functions of
of the
the dynamical
dynamical variables
variables which
which remain
remain constant during
during
the
the motion
motion of
of the system
system are, as we know, called integrals of
of the motion.
We
(42.1) that
that the
the condition
condition for the
the quantity
quantity ffto
an integral
integral of
\Ve see from (42.1)
to be an
(dffdt =
= 0) can
can be written
the motion
motion (of/dt
the
written
k

offot+ [H,f]
[H,f] =
= 0.
o.
3f/3t-|-

(42.3)
(42.3)

[H,f] =
= 0,
0,
[HJ]

(42.4)
(42.4)

If
integral of
of the
the motion
motion is not
not explicitly
explicitly dependent on
on the
the time,
time, then
If the integral

i.e.
i.e . .the Poisson
Poisson bracket
bracket of the
the integral
integral and
and the
the Hamiltonian
Hamiltonian must be zero.
zero.
For any two quantities f and
and g, the
the Poisson
Poisson bracket
defined analogously
analogously
For
bracket is defined
to
):
to (42.2
( 42.2):
oif o
oif 3g
a
of
38
of
. •
[j,g]
(42.5)
[L
gl =
opk 3%
oqk
oqk opk
k
31%
39k
3P1¢
III

__ ___!!_)·
2""(-___!!__
f(

Poisson bracket
bracket has
has the
the following
following properties, which
which are easily
easily derived
derived
The Poisson
from its definition.
definition.
If
the two functions
functions are interchanged,
interchanged, the
tlie bracket
bracket changes
changes sign; if one of
If the
the functions
functions is a constant c, the
the bracket
zero::
the
bracket is zero
(42.6)
[f,g]=
-[g,f],
] : : -[§»f]»
[Le
_\}so
Al
so

[f,
[f, c] == 0.
0.
[/I+!z,g]
= [f1»
[/l,g]+[/2,g],
[fl
-I-fz, g] =:
g] + [f2» gl,
[ftfz, g] =f1[f2»
= /1 [/2, g] +f
+/2[/It
g].
2 Lf1»§][f1f2»

I

(42.7)
(42.'7)
(42.8}
(42.8)

1-

Taking the
the partial
partial derivative
derivative of
of (42.5) with
Taking
with respect
respect to
to time,
time,

o g] = [of
og]
of ,g ]+ [Jo
31:_ .
'5}-[f»
21[/, 8]
Tt'g
f,
213:
ii
8

[

[

obtain
we obtain

(42.9)

(42.10)
(42.10)

136

§42

The Canonical
Canonical Equations
Equations
The

If
one of
of the
the functions f and g is one of the
the momenta
momenta or
or co-ordinates,
co-ordinates, the
If one
the
Poisson bracket
reduces to a partial
derivative::
Poisson
bracket reduces
partial derivative

[f, Qk]
qk] := 3f/3P1,¢,
off op~:,
[f,
Pk] = - air
offoqk.
[f, Pa]
f Qk-

(42.11)
(42.11)
(42.12)
(42.12)

Formula (42.11), for example, may be obtained by putting g = go
qk in (42.5)
(42.5};,
the sum
term, since
go, and 3%/3P1
sum reduces
reduces to a single term,
since 3%/391
oqkfoqz =
= Skz
oqkfopz = 0. Putting
Pi we have,
have, in partipartiting in (42.11)
(42.11) and (42.12)
(42.12) the
the function
function f equal
equal to
to Qi
qi and
and Pf
cular,
cular,
The relation
The

[Qin go] = 0, [Pia Pa] = 0,

8ur-

is Qk]

[!, kg,
[g,h]]+[g,
[h,f]]+[h, [Lal]
[f,g]]
if,
h]]+[ [h»f]]+[h»

=

1

(42.13)

(42.14)
(42.14)

0,

known as Jacobi's
identity, holds between
between the Poisson
Poisson brackets
formed from
known
]acobi's identity,
brackets formed
j, g and
and h. To
To prove
prove it, we first note the
functions Jf,
three functions
the following result.
definition (42.5),
( 42.5}, the Poisson
Poisson bracket
bracket [f, g]
g] is a bilinear
According
According to the definition
bilinear
homogeneous
f and g. Hence
off
Hence the bracket
homogeneous function
function of the first derivatives of
[h, [f,
homogeneous function
[J, g]],
g ]], for example,
example, is a linear
linear homogeneous
function of the
the second
second
off
and g. The
The left-hand
left-hand side of equation
equation (42.14)
{42.14) is therefore
therefore a
derivatives of
derivatives
f and
functions
linear homogeneous
homogeneous function of the second
second derivatives
linear
derivatives of all three functions
f,
j, g and
and h. Let
Let us collect
collect the
the terms involving
involving the
the second derivatives of
of f.
f.
The first bracket
contains no such terms, since
since it involves
involves only
only the first
The
bracket contains
sum of the second
second and
and third
third brackets
brackets may be symboliderivatives of f. The sum
derivatives
of the
the linear
linear differential
operators D1 and D2, defined
defined by
written in terms of
cally written
differential operators
D1{4>)
=
[g,
4>],
D2(4>)
=
[h,
4>]Then
D1(¢>) = [g, ¢], D2(¢>) = [h, ¢>]. Then

[g, lh,f11+lh,
[h,f]]+[h, [f,g]]
[f,g]]
[g,

=

Z

[g, [h»f]][h,f]]-[h,
[g,J]]
Le,
[h, 1gvf]]

Dl[D2(f)]-D2[D1(f)]
D1[D2(f)]~D2[D1(f)]
=
= (D1D2-D2D1)f.
{D1D2- D~1)f.

=

1

It is easy
easy to see that this
this combination
combination of linear
linear differential operators cannot
cannot
It
involve the
differential
the second
second derivatives
derivatives off. The general
general form of the
the linear
linear differential
operators is
3
8
D1
D2
ex---,
7
'Of
3xk
3x1:

E

;

2:

k

k

where go
xg, ..••. Then
gk and 'Up
'YJk are arbitrary functions of the variables xl,
XI, x2,

D D_ =
D1D2
1

9

2

L

32
3v21
3()
07]l
g YJZ;:. ()2. .++ Et
ek'
gak - - - ,y
oxkoxz ICE
o.xz
k,l km3xk3x1
k,l oxk
Xi
am

2

Z

a

' "lk
o- v -++ L' 'YJkoez- -0 ,r
D2D1
D~1 =
= L
'YJkgz
fz3Xk3X1
me 3xk
0
0
0
'lcJ
336;
k,l
Xk Xz k,l
Xk Xz
kJ

the difference
difference of these,
these,
and the

322

3€¢

I

§42

Poisson brackets
»--

3171
""'
(
6:
Fu
EL (

8§z
0

)3

k--_
OTJz -TJk
336/6
Hx; 1
OXk 7? 3xk
OXk oxz

II

D1D2-DzD1
D1D2
D2D1 =

137

k.Z·
kJ'

.__

~)_!__,

is again
again an operator
operator involving
involving only single
single differentiations.
differentiations. Thus the
the terms in
the
f on the
the second
second derivatives
derivatives of
off
the left-hand
left-hand side
side of
of equation
equation (42.14)
(42.14} cancel
cancel
and,
and, since
since the same
san1e is of course
course true
true off
of g and h, the
the whole
whole expression
expression is identiidentically
cally zero.
bracket is that, if
f and g are two
An important
important property of the
the Poisson
Poisson bracket
iff
integrals
bracket is likewise an integral
Poisson bracket
integral of
of the
the
integrals of
of the
the motion, their Poisson
motion::
motion
g] =
= constant.
constant.
(42.15)
[f, g]
(42.15)

f and g do not depend
Thiss is Poisson's theorem.
theorem. The
The proof
proof is very simple
simple if
iff
depend
Thi
explicitly
on
the
time.
Putting
h
=
H
in
]jacobi's
identity,
explicitly
=
Jacobi's identity, we obtain
obtain
[H, [f,§]]-I[j,g]] + [f,
H]] + [g, [H,f]]
[H,j]] =:
= 0.
0.
[H,
Ln [g,
Le, H]]-I-[g,
g]] =
[H,g]
= 0 and [H,f]
[H,f] =
= 0, then
then [H, [f,
[f,g]]
= 0, which is the
Hence, if [H,
g] =

required result.
required
If the
If
the integrals fj' and
and g of
of the
the motion
motion are explicitly
explicitly time-dependent, we
put, from (42.1),
( 42.1 ),
30
d

dt[f,g]
[f,g]].
§[f,
3] = Jt[f,g]+[H,
LL gl +[H, DO
gl]=

Usjng
Usjng formula (42.10)
(42.1 0) and
and expressing
expressing the
the bracket
bracket [H,
[H, [f,
[j, g]]
g ]] in terms of
of two
others by means
means of ]acobi's
Jacobi's identity,
identity, we find

:t[f,
=
d [f g]] =
do
g

. _ _

1

1 11
3/
1 df~
1[ ~ ,g]1 + 1L[:t. ~~ ],
[

11

[

38
~
,g] + [1. ~~]H]]- [g, 1H,f1]
[H,j]]
-me[f, lg,[g, Hu-[3,
3
3: ,g

of

3:

_|..

+[H,j],g]
-l-[H,f],g

11

dr »g +

+ [f,
f,

dg
do

38

~~ -l-[H,g]
+[H,g]]

3:

(42.16)

which evidently
evidently proves Poisson's theorem.
Of
Of course,
course, Poisson's theorem
theorem does
does not
not always supply further
further integrals
integrals of
of
the
the motion,
motion, since there
there are only
only 2s2s-11 of these (s
(s being
being the
the number
number of degrees
degrees
of
of freedom),
freedom). In
In some cases
cases the
the result is trivial,
trivial, the
the Poisson
Poisson bracket
bracket being a
constant
In
other
cases
the
integral
obtained
is
simbly
a
function
of
____
the
simply a
of the
the orioriconstant. In
ginal
If neither
possibilities occurs,
neither of these two
two possibilities
occurs, however,
however,
ginal integrals
integrals f_f and g. If
then the
the Poisson
Poisson bracket
bracket is a further
further integral
integral of
of the
the motion.
motion.
- e

_

-

-

__.__.__

e..J . . _ _ _

PROBLEMS

PROBLEM 1.
1. Determine
Determine the
the Poisson
Poisson brackets
brackets formed
from the
the Cartesian
Cartesian components
components of
of
PROBLEM
formed from

the momentum
momentum pp and
and the
the angular
angular momentum
mop1entum M
M
the

= rr xX pp

of aa particle.
particle.
of

SOLUTION.
SoLUTION. Formula
Formula (42.12)
(42.12) gives [Mm
[Mz, Pa]
P11l == -8M3l8y
-oMz/oy = -8(.vp»--2pu)l3;v
-o(yp,-zp 11 )/(}y == *Pn
-p,,
and similarly
similarly [M,,
[Mz, pa]
Pz] ==
= 0, [MfI
[Mz, PA]
P•l ==
= Pup 11 • The
The remaining
remaining brackets are obtained by
by cyclically
permuting the
the suffixes
suffixes x,
x, y,
y, z.
z.
permuting
:=

138

Canonical Equations
Equations
The Canonical
The

§43

PROBLEM 2.
2. Determine
Determine the
the Poisson
Poisson brackets
brackets formed
the components
components of
of M.
M.
PROBLEM
formed from
from the

=

...
°-:=

Adirect
formula (42.5)
A direct calculation
calculation from
from formula
(42.5) gives
gives [M¢,
[Mz, My]
M 11 ] =~
= --MY,
-M,, [MU»
[M11 , My]
Mz]
[M,, Mz] == -Mu-M~~.
1Mz»M1]

SOLUUON.
SOLUTION.

-Mz,
"Mr»

Since
different particles
Since the
the momenta
momenta and
and co-ordinates
co-ordinates of
of different
particles are
are mutually
mutually independent
independent variables,
variables,
itit is easy to
to see that the
the formulae derived
derived in
in Problems 1 and
and 2 are valid
valid also for the
the total
total
momentum
momentum and
and angular momentum
momentum of
of any system of
of particles.
particles.

PROBLEM 3. Show that [¢>.
[</>, Mg]
M,] == 0, where ¢,
4> is
is any
any function,
function, spherically symmetrical

about
about the
the origin,
origin, of
of the
the co-ordinates
co-ordinates and
and momentum
momentum of
of aa particle.
particle.

SOLUTION.
SoLUTION. Such a function
function 56
</> can
can depend
depend on
on the components of
of the
the vectors rr and
and pp only
through the
the combinations r2,
r 2, p2,
p 2, rr•
p. Hence
Hence
through
° p.

ot/>
--

346
ar
as

84>
as
ot/>
ct/> • p,
_ o(r2)
2r-|2
• r+ o(p • r) · p,

=

'=

a(r2)

8(p- r)

\

»

and similarly for
ot/>fop. The required
required relation
relation may be
be verified
"by direct calculation from
and
for 8gf>lElp.
veriBed'by
from
formula (42.5),
formulae for
formula
(42.5), using
using these
these formulae
for the
the partial
partial derivatives.
derivatives.

PROBLEM 4. Show that
that [f,
[f, Ma]
Mz] -=
= f*xn,
f•xn, where
is aa vector
vector function
of the
the co-ordinates
co-ordinates
PROBLEM
where ff is
function of
and momentum of
of a particle,
particle, and
and n is
is a unit vector parallel
parallel to
to the z-axis.
z-axis.
and

SOLUTION. An
An arbitrary vector
vector f(r,
f(r, p)
p) may be written as f =
= rqtu
rt/>1 -I-DQ62-I-r
+Pt/>2+r xpgb3,
XPt/>a, where
qf>1,962,
t/>1, t/>2, ¢>a
t/>3 are scalar
scalar functions.
functions. The required
required relation may be
be verified by
by direct calculation
calculation

from
formula of
from formulae (42.9),
(42.9), (42.11),
(42.11), (42.12)
(42.12) and
and the
the formula
of Problem
Problem 3.

The action
action as
as aa function
function of
of the
the co-ordinates
co-ordinates
§43. The
In formulating the principle
principle of least action,
action, we have considered
considered the integral
integral
II

C/J

£2

£1

Ld.

(43
.1)
(43.1)

taken along
along a path
between two given positions gM
q<ll and
and q<2>
the system
system
taken
path between
2 ( 2 ) which the
t1 and 12.
t2. In
In varying
varying the
the action,
action, we compared the
the
given instants
instants I1
occupies at given
values of this
this integral
integral for neighboring
neighbouring paths with
the same
same values
values of q(t1)
q(t 1)
values
with the
and q(t2).
q(t 2 ). Only
Only one of
of these paths corresponds
corresponds to the
the actual motion,
motion, namely
namely
the path
path for which
which the integral
integral S has its minimum
minimum value.
the
Let us now consider
consider another aspect
aspect of the
concept of action, regarding
regarding S
Let
the concept
quantity characterising the
the motion along the
the actual path, and
and compare
as a quantity
the values of
of S for paths having
having a common
common beginning
beginning at q(t1) =
= qU),
qU>, but
the
passing
different points
points at time
time to.
t2. In
In other words, we consider
consider the
the
passing through different
action integral for the
the true path as a function
function of the
the co-ordinates
co-ordinates at the
the upper
action
limit of
of integration.
limit
The change
change in the action
action from one path
path to a neighbouring
neighbouring path
given
The
path is given
expression (2.5)
(2.5)::
(if there is one degree of freedom) by the expression

3L

84

ss = [ o~ sq]t +
8S
oq
t1
£1
39
£22

d 3L
3I
J(
:L~
)sq dt.
dr.
f( 09oq. dodt oi:3oqq. )3g
t2

32

¢1
tl

D

'

-

Since the
the paths of actual
actual motion
motion satisfy Lagrange's equations, the
the integral
Since
SS is zero. In
In the first term we put 8q(t1)
Sq(t1) =
= 0, and
and denote
denote the value of
in 8S

I

The action
action as
as aa fzmdibn
function of
of the
the so-ordinates
co-ordinates
The

§43

139

3q(Z2)
L/ 3 q' by P,
Sq(t2) by 8g
Sq simply.
simply. Replacing 3oLfoq
p, we have finally 8S
SS = pSq
p8q or, in
general case of any number
number of degrees
degrees of
of freedom,
freedom,
the general

Z Pi89i»

8S

(43.2)

the partial derivatives
derivatives of the
the action
action with
From this relation
relation it follows that the
with
respect to the
the co-ordinates are equal
equal to the corresponding
corresponding momenta
momenta::
respect

slag,

Pi-

(43.3)

L.

(43.4)

The
regarded as an explicit
The action
action may
may similarly
similarly be regarded
explicit function
function of time,
time, by
t 1 and
and at a given point gm,
q!l>, and
considering paths starting at a given instant £1
2
ending at a given point 9q<( 2 )> at various
partial derivative
various times I2
t2 =
= z.
t. The
The partial
derivative
oSfot thus obtained
obtained may be found by an appropriate
appropriate variation
the Integra
integral.
38/3:
variation of the
l,
It is simpler,
simpler, however, to use formula
fommla (43.3),
( 43.3), proceeding
It
proceeding as follows.
From the
definition of the
the definition
the action,
action, its
its total time
time derivative
derivative along
along the path
path is

dSfdt
dsldr

=:
:

Next, regarding
the sense
Next,
regarding S as a function
function of
of co-ordinates
co-ordinates and
and time,
time, in the
sense described
cribed above, and using formula
formula (43.3), we have

dS
oS
dS
as
= . +
dt = J!+
it

as.
23%
.
LAg; = it +ZP¢€I=tLP'4'·
'

fI

oS
oq/'

oSfot =
= L-219i42i
L-L,p,qi or
A comparison gives 3S/3:
oSfot
=
as
far =

*

oS
ot

_1_

¢

f

-H.

Formulae (43.3) and (43.5) may be represented
represented by the expression
expression
Formulae

dS

=

LPi doz-Hd!
dqi-H dt
Zn
i

5)
(43.5)
(43.
(43.6)

the total diiterential
differential of
of the
the action
action as a function
function of co-ordinates and
and. time
for the
the upper limit of
of integration
integration in (43.1).
(43.1 ). Let
Let us now
now suppose
suppose that
that the
the coat the
the beginning
of the
the motion,
motion, as well as at the
the end,
end,
ordinates (and
(and time)
time) at the
ordinates
beginning of
variable. It
It is evident
evident that
that the
the corresponding
corresponding change
change in S will be given
given
are variable.
expressions (43.6) for the beginning
by the difference of the expressions
beginning and end of the
path, i.e.

dS =

Z ple) dqi(2)_

H(2) dz(2)-

Z P1(1) dq(1)+

H(1).dI(1}I

(43.7)
(43.7)

relation shows that, whatever
whatever the
the external forces
forces on the
the system
system during
This relation
during
its motion, its
its Iinal
final state cannot be an arbitrary function
function of
of its initial
initial state
state;,
its
thos
e
y
mot
onl
the expression
expression on the
the right-hand
only those motions
are possible
possible for which
which the
ions are
eq
side
of
ua
tio
n
(43
si~e
?f
equation
(
43.
7)
perfect
differential.
the
existence
perfect
diHerentia
is
a
l.
Thus
the
existence
of the
.7)
. .
le
of
as
le
cip
t
ac
prin
tion, quite apart from any particular form
p~mciple of least action,
form of the
the Lagranggian,
i l l , impo
1I?P_oses
ses ce
rtain restrictions
certain
restrictions on Me
the range of
of possible
possible motions.
motions. IIn
n partiis possible
it
r,
cula
ct;lar, It IS possible to
to derive
derive aa number
number of
of general
general properties,
properties, independent
independent
of the
the external
external fields,
fields, for
for beams
beams of
of particles
particles diverging
diverging from
from given
given points
points in
in
of

Thee Canonical
Cano11ical Equations
Equations
Th

140

§44

part of the
act of
Thee study
study of
of these
these properties forms
forms a part
the subj
subject
of geometrical
space. Th
opIicls.']°
optics."!

It
It is of
of interest to note that IIantilton's
Hamilton's equations
equations can be formally derived
<;lerived
from the condition of minimum
action
in
the
form
minimum
=

IHZ
<2: Pi
Pi dqgdq,- H
H dz),
dt),

II

S
s

(43.8)
(43.8)

1i

which
( 43.6), if the
the co-ordinates
co-ordinates and momenta
momenta are varied indeindewhich follows from (43.6),
pendently. Again assuming
pendently.
assuming for simplicity
simplicity that there is only one co-ordinate
co-ordinate
momentum, we write the variation of the action as
and momentum,

8S
ss ==

aH,'aq)sq df-(@Hlapp;
I [8z>
[Sp d<z+1>
dq+p d8Q-(
dSq-(oHfoq)Sq
dt-(oHfop)Sp dr].
dt].

I 81>{dQ
Sp{dq(oHfop) dt}+
[poq]- f 8q{d
Sq{dp+(ol-Ifoq)
dt}.
1>+(3H/ 3 Q) dr}.
- (aftlap)
dr} + [1259]

the second
second term
term gives
integration by parts in the
An integration
II

SS
ss =

At the
the limits
limits of integration
integration we must
must put 89
Sq =
= 0, SO
so that
that the
the integrated
integrated term
term
is zero. The
The remaining expression
expression can be zero only if the
the two integrands
integrands
vanish
vanish separately,
separately, since the
the variations
variations 8p
Sp and 8g
Sq are independent
independent and arbitrary
arbitrary::
dg
dr, do
( aHi r q) dr, which, after division by dz,
dq =
= (3HlElP)
(oHfop)dt,
dp =
= --(ol-Ifoq)dt,
dt, are
Han1ilton's equations.
equations.
Hamilton's
§44.
§44. Maupertuis' principle

motion of a mechanical
mechanic.1.l system
system is entirely
entirely determined
determined by the
the principle
The motion
of least
least action:
action: by solving
solving the
the equations
equations of motion
motion which
which follow from that
that
principle, we can
can find both the
the form of the
the path
path and the
the position
position on the
the path
principle,
path
function of
of time.
as a function
If
the problem
problem is the
the more
nH>re restricted
determining only
only the
the path,
If the
restricted one of determining
without reference
without
reference to time, a simplified
simplified form of
of the
the principle of least
least action
action
\Ve assume
assume that
that the
the Lagrangian,
Lagrangian, and therefore
therefore the
the Hamiltonmay be used. We
not involve
involve the
the time
time explicitly,
explicitly, so that the
the energy of the
the system
system is
ian, do not
H(p, Q)
q) ::= E =
= constant. According to the principle of least action,
conserved: H(P,
the
the variation of
of the
the action, for given
given initial and
and final co-ordinates and
and times
and t, say), is zero. If, however,
however, we allow a variation
variation of the
the final time
time t,
(to and
Z,
the initial
initial and final co-ordinates remaining
remaining fixed, we have (cf.(43.7))
(cf.(43.7))
the

(44.1)

8S+ESt = 0.
8S+E8t

(4
4.2)
(44.2)

II

SS = -l-ISt.
8S
H8t.

We now
now compare,
compare, not
not all virtual motions
motions of the
the system,
system, but only
only those
the law of
of conservation
conservation of energy.
energy. For
For such paths we can
can
which satisfy
satisfy the
which
constant E, which
which gives
replace H in (44.1) by a constant
Tt

975

See The
The Classical
Classical Theory
Theory of
of Fields,
Fields, Chapter
Chapter 7,
7, Pergamon
Pergamon Press,
Press, OxfoI'd
Oxford 1975.
1
·
See

§44

141
141

Maupertuis'
Mazzpertzzis' principle

II

C/J

Writing the action
action in the form (43.8) and again replacing
replacing H byE,
Writing
by E, w_e
we have

ii

The first term
term in this
this expression,
expression,
The

(44.3)

2Pi dflz,
dq,,
iJ;PA

(44.4}
(44.4)

i

II

So=
So

dQ¢-E(r- to),

sometimes called
called the
the abbreviated action.
is sometimes
Substituting (44.3) in (442),
(44.2), we find that
that
Substituting

.SSo
_8S0

0.

=

(44.5)

the abbreviated action
action has a minimum
with respect
respect to all paths which
which
Thus the
minimum with
the law of conservation
conservation of
of energy
mergy and pass through the
the final point
satisfy the
satisfy
point
instant. In
In ..order
the momenta
momenta
at any instant.
order to use such a variational principle,
principl e, the
(and
be expressed
(and so the
the whole
whole integrand
integrand in (4-4.4))
(44.4)) must
must be
expressed in terms of the
the
co-ordinates q and their
their differentials
differentials dg.
dq. To
To do this, we use the definition
co-ordinates
definition of
n1on1entun1::
momentum
3
0 ( q ,dq)
(44.6)
Pt
Pi = -.L
oqi
dt
3%

_

L(

nefgY1
and the
the la
law
conservation
nservat10n 0off energy:
w 0off co
'
\

(

)

II

e
do
EE(q.
E.
9' ~~)
dt =E.

)

1

(44.7)

the differential
differential do
dt in terms of the
the co-ordinates q and
and their differendifferenExpressing the
tials
dq
by
means
of
(44.7)
and
substituting
in
(44.6},
we
have
the
momenta
tials do
means of
and substituting
(44.6),
momenta
variational
prinin terms
terms of q and do,
dq, with the
the energy E as a parameter.
The
prinparameter. The
ciple so obtained
obtained determines
determines the
the path
path of the
the system,
system, and is usually
usually called
ciple
Maupertuis'
although its precise
formulation is due to EULER
It/Iaupertuis' principle, although
precise formulation
EULER and
LAGRANGE.
LAGRANGE.

g

above calculations
calculations may
may be carried out
out explicitly
explicitly when
the Lagrangian
Lagrangian
The above
when the
difference of the kinetic and potential
energies::
takes its usual form (5.5) as the difference
potential energies

i

L

The momenta are

= 12

Pt
Pe:

and
and the
the energy
energy is

E

Th
e last
The
last equ
equation
gives
ation give
s

do

=__

=

::=

2 "i1¢(Q)Qi:1#aik(q}qiq~r.- U(q).
U(Q)~

i,k

3oLf
I*'/IQi
oqi

=

2k ¢1u¢(9)Q1¢,
aik(q)qk>
Z

l L ¢w¢(9)¥?¢¥?1¢
aik(q)cjtcik +
+ U(9)V(q).
%
i,k

x/ [E du : dif dQkl2(-E' U)] ;

(44.8)
(44.8)

142

The Canonical
Canonical Equa
Equations
The
from*

substituting
substituting this
this in

LPi dif
dq,
L
Z
;Pi
i,k
=

i

abbreviated action
action::
we find the abbreviated

So
SO

=

=°.

i,k

dqk
/:
09

aik-(Inv-"

dr
dt

§44

dqi,
dpi,

I v'[[ 22(E2.i,k ark deli
dq, dqk]·
d9A:l
( E - U)
u)Z
/

i,k

aik

(44.9)

»

In particular,
particular, for a single
single particle
kinetic energy
energy is T =
= ! m(dl/dt)2,
m(dlfdt)2,
In
particle the kinetic
where
the mass of the
the particle
particle and dl
dl an element
element of its path;
the variational
where m is the
path , the
principle which
determines the
the path
principle
which determines
path is

un

I

81/[2m(ES v'[2m(E- U)] dz
dl =
= 0,

(44.10)
(44.10}

where
between two given
where the
the integral
integral is taken
taken between
given points in space.
space. This form is
due to
to ]ACOBI,
}ACOBI.
motion of the
the particle,
= 0, and (44.10) gives the
the trivial result
In free motion
particle, U =
result
8S Jdl =
particle moves along the
between the
= 0, i.e. the
the particle
the shortest
shortest path
path between
the two
given points,
points, i.e. in a straight line.
given
Let us return
return now to the
the expression
expression (44.3) for the action
action and
and vary it with
Let
respect to the parameter
parameter E. We have

f

SS
8S

= -

as
cSo
0

oE
0E
n

substituting in (44.2),
(44.2), we obtain
obtain
substituting

I

SE- (t- t0 )SE- ESt·
8E-(z-z0)8E-E8z,
'

8.90/3E
oSofoE == rt-- 10.
to.

When
the abbreviated
abbreviated action has the form (4-4.9),
(44.9}, this
this gives
When the
V[20ik

do dqk/2(E- un =

t"tOa

(44.11)
(44.12)
(4:4.12)

which is just
the integral
integral of equation
equation (44.8).
(44.8). Together with
the equation
equation of
which
just the
with the
the
path,
it
entirely
determines
the
motion.
the
the
PROBLEM
RO B LE M
P

the differential equation of
of the
the path from the
the variational princip1e
Derive the
principle (44.10).

f

NU
-J{H au·
l1r
dl-v(E-V~·d8r}.
d
l-VE-U
-ds
or
2v(E-U)
dl
as
N
IE-U)
(
Zi
r

SOLUTION. Effecting the
the variation, we
SOLUTION.
we have

88Jv(E-U)dl=
V ( E - U ) dl =

•

Sr

dr

8rIn the
the second
second term
term we
we have
have used
used the
the fact
that M2
d[2 = dry
dr2 and
and therefore
therefore dld6l
dl d8l :=_ dt#
dr· d
dllrIn
fact that
Integrating this
this term
term by
by parts
parts and
and then
then equating
equating to
to zero
zero the
the coefficient
coefficient of
of Sr
8r in
in the
the integfafld,
integrand,
Integrating
we obtain
obtain the
the differential
differential equation
equation of
of the
the path:
path:
we
=

3U/af.
-oU/or.

II

d[[ ~ f < dr]
1

d

2vCE-U>- v(E-U)21/(E-U)
dl
dl
dl

-

§45

143

Canonical
Canonical transformations
transformations

Expanding
Expanding the
the derivative
derivative on
on the
the left-hand
left-hand side
side and
and putting
putting the
the force FF -=
= -8Ulélr
-oUfar gives
gives

d2r/dl2
d2rfdl 2 = [F-(F[F -(F • t)t]/2(E- U),
U),

where
dr/dl is
is a unit
unii: vector tangential to
to the
the path. The difference F-(FF -(F • t)t
t)t is the comwhere tt = dr/dl
ponent
from
ponent Fn
Fn of
of the
the force
force normal
normal to
to the
the path.
path. The
The derivative
derivative d2r/dl
d2rfd.l 22 = dt/dl
dt/dl is
is known
known from
diilerential
differential geometry
geometry to
to be
be n/R, where
where R
R is
is the
the radius
radius of
of curvature
curvature of
of the
the path
path and
and nn the
the unit
unit
vector
vector along
along the
the principal
principal normal.
normal. Replacing
Replacing EE- U
U by
by 1}rqz;2,
!ntv2, we have
have (mo2/R)n
(t7(1J 2/R)n == F",
Fn, in
in
agreement with
with the
the familar
familar expression
expression for
the normal
normal acceleration
acceleration in
in motion
motion in
in aa curved
curved
agreement
for the
path.

§45. Canonical transformations
§45.

The
restriction ,
The choice of
of the
the generaliSed
generalised co-ordinates q is subject to no restriction;
they may
may be any
any s quantities
quantities which
which uniquely
the position
the system
system
they
uniquely define the
position of the
space. The
The formal appearance
appearance of
of Lagrange's
Lagrange's equations
equations (2.6) does not
in space.
on this choice,
choice, and
and in that~sense
that-sense the
the equations may
may be said
said to
to be
depend on
transformation from the
the co-ordinates
co-ordinates go,
q1 , 92,
q2 , •••
invariant with
invariant
with respect
respect to a transformation
any other independent quantities QUO,
Q1, Q2,
Q2, ..... The new co-ordinates Q
Q are
to any
functions of q, and
and we
~e shall
shall assume
assume that
that they
they may explicitly
explicitly depend
depend on
on the
the
functions
time,- i.e. that the
the transformation is of the
the form
t1me,-

g, = Q£(Q»
Q,(q, I)
t)
Qi

(45.1}
(45.1)

"-:

II

(sometimes called a point transformation).
(sometimes
are unchanged by
the transformation (45.1),
(45.1),
by the
Since Lagrange's equations are
(40.4) are also unchanged. The latter equations, howhowHamilton's equations (40.4)
allow a much
much wider
range of transformations.
transformations. This is, of course,
course,
ever, in fact allow
ever,
wider range
the Hamiltonian
Han1iltonian treatment the
the momenta Pp are variables
variables indebecause
because in the
pendent of and
and on
on an
an equal footing
footing with
with the
the co-ordinates q. Hence the
the transformation may
may be extended to include
include all the
the 2s independent variables
p
formation
variables P
and
and q:
(45.2}
Pi up, q, r).
Q --e Q¢(}9, q, £)»
(45.2)

I
II

enlargement of
of the class of possible
transfom1ations is one of the it
imThis enlargement
possible transformations
portant
advantages of the
the Hamiltonian treatment.
portent advantages
The
equations
of
motion
do, not, however,
however, retain
retain their canonical
canonical form
The
motion dOII'lOt,
under all transformations
transformations of the
the form (45.2).
( 45.2). Let
Let us derive
derive the
the conditions
conditions
under
satisfied if the equations
equations of motion
motion in the new variables P, Q
which
which must be satisfied
Q
to be of the
the form
are to
.3)
(45.3)
(45
3H'/8Q,P
Qf = a/f'lam,

Han1iltonian H'(P,Q).
H'(P,Q). Among
Among these
these transformations,
transformations, there
there is a
with some Hamiltonian
particularly
important class called canonical
carwnicaltransformations.
particularly important
transformations .
The formulae for canonical
canonical transformations
transformations can be obtained
obtained as follows. It
The
It
been shown
shown at the
the end
end of §43 that
that Hamilton's equations
equations can
can be derived
derived
has been
the principle of least
least action
action in the
the form
from the
from

(; Pi dq£- H dt)

0,

II

8

(45.4)
(45.4)

144

§4s
§45

The
The Canonical
Canonical Equations
Equations

in which the
the variation is applied to
to all the
the co-ordinates and
and momenta indeIf the
Q also
the new
new variables
variables P
P and
and Q
also satisfy
satisfy Hamilton's equations,
pendently. If
the
the principle of
of least
least action
action

a;
f<2
S

i

Q - H ' dr)
Pi
dQ,-H'
dt) =
= 0
Pe d

(45.5)

must
must hold.
hold. The two forms (45.4) and
and (45.5)
(45.5) are certainly
certainly equivalent
equivalent if their
their
integrands are the same
same apart
apart from the total
total differential
differential of
of some
some function F
F
of
between the
of co-ordinates,
co~ordinates, momenta
momenta and
and time. The difference
difference between
the two integrals
integrals
then a constant, namely the
the difference
difference of
of the
the values of
of F
F at the
the limits of
of
is then
integration~,
variation. Thus we will take
integration, which
which does not
not affect the variation.

Et
g - HH dz
2 Pt ddq,dt =
= 2 Pt do£-H'
dQ,- H' do+dF.
dt + dF.

Transformations
Transformations which satisfy this
this condition
condition are said to be canonical.'l'
canonical.t Each
canonical transformation
transformation is characterised
characterised by a particular
particular function
function F,
canonical
F , called the
generating function
of the transformation.
f i c t i o n of
Writing
Writing this
this relation
relation as

DF
dF =
=

that
we see
see that
we

Pi=
oFfoq,,
Pi = 3Flaqi,

2 Pi d9¢dq,- 2 p
Pi d'Q¢+(H'-H)
d·Q,+(H' -H) do,
dt,
Pi = -aFlaQ1,

H'

(45.6)

=
H+ aFlal;
= H+oFJot;

(45.7)

here
here it is assumed
assumed that
that the
the generating
generating function
function is given as a function
function of the
the
co-ordinates and the time: F
F == F(q, Q,
t). When F
F is known,
old and new co-ordinates
Q, Z).
formulae (45.7)
(45. 7) give the
the relation
relation between
p, q and P, Q
the new
formulae
between P,
Q as well as the
Hamiltonian.
It
convenient to express
express the
the generating
generating function
function not
not in terms
terms of the
It may be convenient
and Q but
but in terms
terms of
of the old co-ordinates
co-ordinates q and
and the
the new momenta
momenta
variables q and
variables
P. To
To derive
derive the
the formulae for canonical
canonical transformations
transformations in this
this case, we must
P.
appropriate Legendre's
Legendre's transformation
transformation in (45.6),
(45.6), rewriting
rewriting it as
effect the appropriate

2

2

d(F+§
Pt d9,+
d(F+ 2 HQ.)
PtQt) =
= LPt
dqt+ 2 Q, dP1zJI(H'-H)
dPt+(H' -H) dr.
dt.

argument of
of the differential
differential on the
the left-hand
left-hand side, expressed
expressed in terms
terms of
of
The argument
function <I>(q,
t), say. There
Then!
the variables q and P, is a new generating function
<D(q, P, Z),

3(1)/aqi,

II

Pi

Q; = 3<1D/3P1,

H+
H' =
= H
+ 5®/31.
oci>Jot.

.'

(45.8)

can similarly
similarly obtain the
the formulae
formulae for canonical
canonical transformations
transformations inWe can
depend On
on the variables P
p and Q, or
generating functions
functions which depend
volving generating
p
P and P.

t The canonical form
form of
of the
the equations of
of motion is
is preserved
1`
transformations but also by
by transformations in
in which*
which'the
integrands in
in
the intcgrlands
constant factor.
factor. An example
example is the
the transformation P,
P, = aap,
Q,
=
q,
H'
p l l Q,
q,, H'

t

not only
only by
by the canonical
not
(45.4) and (45.5) differ by a
= aH,
aH, with any constant
constant a.
a.

eratili function
the gen
generatinl:{
function is
is (D
<I> = "f,-(q,
~J;(q, t)P£,
t)P 1 , where the
the/;
are arbitrary functions,
functions, WC
we
ff are
I IfIf the
obtain aa transformation
transformation in
in which
the new
new co-ordinatcs
co-ordinates are
are Q,
= J,(q, r),
t), i.e.
i.e. are exprco:sed
obtain
which the
Q: =f§(q,

in terms
terms of
of the
the old
old co-ordinates
co-ordinates only
only (and
(and not
not the
the momenta).
momenta). This
This is
is aa point
point tl-ansformat1on,
transformation,
in
and is of
of course
course aa particular
particular canonical
canonical transforrnanon.
transformation.
and

§45

Canonical
Canonical transformations
transformations

145

The relation
relation between
between the
the two
two Hamiltonians
Hamiltonians is always of
of the
the same
same form
form::
The
the difference H'
- H is the partial
partial derivative
derivative of the
the generating
generating function
functi9n with
the
H' respect
time. In
In particular, if the generating
generating function
function is independent
independent of
respect to time.
then H'
H' =
= H, Le.
i.e. the new Hamiltonian
Hamiltonian is obtained
obtained by simply
simply substitutsubstituttime, then
their values in terms of the
the new variables P, Q.
Q.
ing for p, qq in H their
The wide range of the canonical
canonical transformations
transformations in the
the Hamiltonian
Hamiltonian treatThe
treatdeprives the
the generalised
generalised co-ordinates and
and momenta of
of a considerable
considerable
ment deprives
part
original meaning.
transformations (45.2) relate
relate each
part of their original
meaning. Since the transformations
of the
the quantities P, Q to both the
the co-ordinates qq and
and the
the momenta P,
p, the
of
the
variables
variables Q are no longer
longer purely spatial co-ordinates, and
and the
the distinction
between Q and P
P becomes essentially
essentially one of
of nomenclature.
nomenclature. This is very
between
clearly seen, for example, from the transformations
transformationt Q, =
= P¢,
p,, IP¢
P, =
= - iq~,,
i,
which
obviously does not
the canonical
canonical form of
of the equations
equations and
and
which obviously
not affect t•he
simpl,y to calling
calling the
the co-ordinates
co-"ordinates momenta
momenta and
and vice versa.
amounts simply
On account of
of this arbitrariness of
of nomenclature, the
the variables P
p and
and q in
in
On
the Hamiltonian treatment
tJ:eatment are often called
called simply canonically
canonically conjugate
the
The conditions
conditions relating
relating such
such quantities can
Cl'!n be expressed
expressed in terms
quantities. The
of Poisson
brackets. To
To do this,
this, we shall first prove a general
general theorem
theorem on the
Poisson brackets.
invariance of Poisson
Poisson brackets
brackets with
respect to
to canonical
canonical transformations.
transformations.
invariance
with respect
Let
f, Gina
bracket, for two
Let [[!,
g]p,q be the
the Poisson
Poisson bracket,
two quantities
quantities f and
and g, in which
the differentiation
differentiation is with
with respect
to the
the variables
variables Pp and
and q, and
and [[!,
g]P,Q
that
the
respect to
f, g]
P,Q that
which the
the differentiation
differentiation is with
with respect
toP
Q. Then
in which
respect to
P and Q.

[f,g]p,q
=
If»
.glam =

[f,g]P,Q·
If»
§]p,Q-

(45.9)
(45.9)

truth of
of this
this statement can
can be seen
seen by direct
direct calculation,
calculation, using the
The truth
the formulae
of the
the canonical
canonical transformation.
transformation. It
It can
can also be demonstrated
demonstrated by the
mulae of
following argument.
of all, it may
noticed that the
the time
time appears as a parameter in the
First of
may be noticed
canonical transformations
transformations (45.7) and
and (45.8). It
It is therefore
sufficient to prove
canonical
therefore sufficient
prove
(45. 9) for quantities
quantities which
which do not
not depend
depend explicitly
explicitly on time. Let
Let us now
(45.9)
regard g as the
the Hamiltonian
Hamiltonian of some fictitious
fictitious system.
system. Then, by
formally regard
(42.1 ), if,
[J, g]m
g]p,q =
= -of/dz.
- dfldt. The
The derivative dj;dt
depend only on
formula (42.l),
df,*'dt can depend
the properties of
of the
the motion
motion of
of the
the fictitious
fictitious system, and
and not
on the
the
not on
the particular
particular
choice of variables. Hence the Poisson bracket
bracket [[!,
g]
is
unaltered
the
f,
unaltered by the
passage from
from one
one set
set of
of canonical
canonical variables
variables to
to another.
passage
and (45.9) give
Formulae (42.13) and
Formulae
[Q~,, Qk]p,q
= 0:
0, [Pia
[Pt, Prim
Pk]p,q =
= 02
0, [Pia
[P~,, Qkina
Qk]p,q =
= Bik»
Stk·
[Qb
Qk]1i*,q =

(45.10)

These are the conditions,
conditions, written
written in terms
terms of
of Poisson
Poisson brackets, which
which must
must
These
satisfied by the
the new variables
variables if the
the transformation
transformation P,
p, q ->
--+ P, Q is canonical.
canonical.
be satisfied
It is of
of interest
interest to
to observe
observe that the
the change
change in the
the quantities
quantities p,
during the
It
P, q during
motion may itself
itself be regarded
regarded as a series
senes of
of canonical
canonical transformations. The
motion
p, be the values
the canonical
meaning of
of this statement is as follows.
follows. Let
Let it,
qe. Pt
values of the
canonical
meaning
erating fun
Whose gen
generating
function
is F
F == Zq¢Qf!:.q<Qt.
Tt WhoSe
ction is

The
The Canonical
Canonical Equations
Equations

146

§46

variables
time Z, and Q¢+-n
Par their values
time 1-1-7.
variables at timet,
qt+.,., Pt+r
values at another
another time
t +-r. The latter
are some
parameter) :
of the
the former (and
(and involve 1'T as a parameter):
some functions
functions of

Pt+T
=.p(qe, Pe.
t,-T).
PH-1 =.P(Q¢,
Pa Z,-T).
If
formulae are regarded
regarded as a transformation from the
the variables
variables go
qe, P:
Pt
If these formulae
en this
to Qz+
qt+n
Pt+n
then
this transformation is canonical.
canonical. This is evident
evident from the
to
,, th
<r» Pa
expression
erential of
expression dS
dS =
= E(P£+*1'dQ£+1
~(Pthdqt+ 7 -Pfdélt)
-ptdqt) -(HH,
-(Ht+ 7 -H£)dt
-Re)dt for the dif
differential
of the
action S(qI+.,.
S(qt+T, q:.
qt, I,
t, T),
-r), taken along
along the the
true path, passing
passing through the points
points q,
qt
and
and q¢+,
qt+T at times
times tr and
and t -I-7'
+T for a given
given 1'T (cf. (43.7)). A comparison
comparison of
of this
formula with (45.6) shows that
that -S
-Sis
generating function
function of
of the transformula
is the generating
formation.
qt+T =
= Q(Q¢,
q(qe, Pt,
qt+'r
Ps,

t,T),
Z,7'),

§46. Liouville's theorem
For the. geometrical
geometrical interpretation of
of mechanical
mechanical phenomena,
u~e is often
often
For
phenomena, use
of phase space.
space. This is a space of 2s dimensions,
dimensions, whose co-ordinate
co-ordinate axes
made of
to the
the s generalised
generalised co-ordinates and
and s momenta of
of the
the system
system
correspond to
Each
point
in
phase
space
corresponds
to
a
definite
state
the
concerned.
concerned. Each point
phase space corresponds
definite state of the
\Vhen the
the system
system moves,
moves, the
the point representing
representing it describes
describes a curve
curve
system. When
system.
the Phase
phase path.
·
called the
The
The product of differentials
differentials dl"
dr =
= dq1
dq1 ... dQsdP1
dq 8 dpt ... dpi
dp 8 may be regarded
regarded
space. Let
Let us now consider
consider the
the integral
integral
element of
of volume
volume in phase
as an element
phase space.
phase space,
fj` dl"
dr taken
taken over
over some
some region
region of
of phase
space, and
and representing
representing the
the volume
volume of
region. We shall
shall show that this
this integral
integral is invariant with
with respect
respect to
that region.
canonical transformations;
transformations; that
that is, if the
the variables
variables P,
p, q are replaced
replaced by
by
canonical
P, Q
Q by a canonical
canonical transformation,
transformation, then
then the
the volumes
volumes of
of the
the corresponding
corresponding
P,
regions of
Q are equal
of the spaces of p, q and P, Q
equal::
'

...do.
t ... If dQ1---dQs
J...f
···fdQ1
dql···do.
dqs do.
dPl···
dPs = I
dQl···dQ dp....dp..
dPt···dP
=

(46.1)
(46.1)

8•

8

I

c o l '

II

'CJ

of variables
variables in a multiple
multiple integral
integral is effected
effected by the
the
The transformation of
... f.i dQ1
dQt ... do,
dQ 8 dpi
dPt ... dpi
dP8 =
= f...
f ... f Ddql
Ddq1 ... das
dq 8 do;
dp1 ... dP,
dp 8 ,
formula fj...
where
where
o(Qt. ... , Qs,
Pt.
, PS)
Ps)
Qs»_Pi,
3(Q1,
D=-:..:....__...:...::._
_..._
_
(46.2)
•
3
...
,
qs,
Pt
•
...
,
Ps)
o(qt.
Qs: 111, - - I
3(91»
I

is the
bian of
the Jaco
Jacobian
of the
the transformation. The proof
of (46.1)
(46.1) therefore amounts
proof of
to proving that
that the Jacobian
Jacobian of
of every
every canonical
canonical transformation is unity:
to
unity :
= 1.
D=

(46.3)

We shall
shall use a well-known
of Jacobian
Jacobians whereby
whereby they
they can
can be
well-known property of
treated somewhat
somewhat like fractions.
fractions. "Dividing numerator and
and denominator" by
treated
by
o(qt. ...,
... , q8,
q8 , P1,
Pt, ...,
... , PS)9
P 8 ), we obtain
3(91.
=

!|

D
D

o(Qt, ... , Qs:
Qs. P1,
Pt, ...
3(Q1,
- H r, Ps)
t o o '

o(qt. ... , Q83
qs, P13
Pt. ...
Ps)
3(q1a
- " 3, P8)

I

o(qt, ... , Q89
qs, 111,
Pt. ... , Ps).
p)
3(Q1,
c(qt, ... , qs, PI,
Pt. ... , Ps)
5'(91,
t o o '

°°°)

psy

in
pear in
of Jacobian
Jacobians is that, w
when
the same
same quantities ap
appear
Another property of
hen the
l
ab
rl
er va
the partial
partial differentials,
differentials, the Jacobian
reduces to
to one in few
fewer
variables,
es,
both the
J acobian reduces
° l o a

I

1

1

\l

l

§47

i

The HamiltonHamilton-Jacobi
equation
The
nulacobi equation

1
I

I

147

which these
these repeated
repeated quantities are regarded
regarded as constant in carrying out
out
in which
the
the differentiations. Hence

D

=

l:

o(Qt, ... ,. Qs)
/{ 3(f>1»
o(pt, --->.Ps)
... , Ps) }
. }
3(Q1»
(46.4)
{ o(qt. ... , qs) P=cons~ant
o(Pl, ... , Ps) q=constant
q=constant.•
3(Q1»
Qs) P=constant
3(P1,

II

D

g . . ,

°°°r

The
The ]acobian
Jacobian in the
the numerator is, by definition,
definition, a determinant of order
order s
whose element
nth column
the
element in
in the
the ith
ith row and kth
column is 3Qil3qk.
oQi/ oqk. Representing
Representing the
canonical transformation
transformation in terms of the generating
generating function
function <D(q,
<ll(q, P)
P) as in
canonical
2
(45.8), we have 3Q¢lBqk
oQ~,foqk =
= 32(I>
o <l>foqkoP~,.
In the same way we find that
that the
/aqk3P¢- In
2
in-element
ik-element of
of the
the determinant in the
the denominator
denominator of (46.4) is 3211>/3q£3P;¢.
o <tlfoq,oPk.
means that
that the
the two determinants
determinants differ only
only by
the interchange
interchange of
of rows
This means
bY the
and columns,
columns; they
they are therefore
therefore equal,
equal, so that the
the ratio
equal to
and
ratio (46.4) is equal
completes the
the proof.
unity. This completes
Let us now
now suppose that each
each point in the
the region
considered
Let
region of phase space considered
moves in the course
course of time in accordance
accordance with
with the
the equations
equations of motion
motion of the
the
mechanical
system. The region
region as a whole therefore moves also, but
volume
mechanical system.
but its volume
unchanged::
remains unchanged

.

Jf dl*
dr = constant.
constant.

(46.5)

=

This result
result,J' known
known as
as Lion-ville's
Liouville's theorem,
theorem, follows
follows at
at once
once from
the invariance
invariance
This
from the
of th
thee volume
volume in phase
phase space under
canonical transformations
thee
of
under canonical
transformations and from th
showed at the
fact that
that the
the Chan
change
in p
and 0q during the
the motion
motion may,
the end
end
__-___ge in
N and
m ay, as we sshowed
fact

of §45, be regarded as a canonical
canonical transformation.
of
In an entirely similar
similar manner
manner the
the integrals
integrals
In

UH; dpi dpi dank dpi,

H § di Q i d P f »

tvék

1

in which
which the
the integration
integration is over manifolds
manifolds of two,
two, four, etc. dimensions
dimensions in
to be
be invariant.
invariant.
phase
space, may be sh
shown
phase space,
own to
§47. Th
The
Hamilton-Jacobi equation
equation
§4'7.
e Hamilton-Jacobi
action has been
considered as a function of co-ordinates
co-ordinates and
and
In §43 the action
been considered
the partial
derivative with
with respect
and it has been
shown that the
time, and
been shown
partial derivative
respect to time
of this function
function S(q, Z)
t) is related
related to
to the Hamiltonian
Hamiltonian by
of

35'/3t+H(q,p,
oSfot+H(q,p, I)
t) =
= 0,
o,

and
and its partial derivatives with respect to
to the
the co-ordinates are
are the
the momenta.
momenta pp in the
the Hamiltonian
Hamiltonian by the
derivatives
Accordingly
replacing the
the momenta
Accordingly replacing
the derivatives
oSjoq, we have the
the equation
equation
35'/Elg,

35'

H

as

85'

t)

- r =0
oS_ .|_
+H(q., ...
qs;5 ..oS,' ... , cS;
• • , Qs

ot

(47.1)
(47.
1)

oql
oqs
st be
tisfied by
which
~ust
be sa
~ati~fied
by the
the function
function S(q, t). This
This First-order
first-order partial
partial
which mu
diffe
rential equa
tion is
equatwn
Is calle
calledd the
the HamiltonHamilton-jacobi
equation.
differential
-facobi equation.
_

y

I

I

,

lib'

3

148

J

§47

The
The Canonical
Canonical Equations
Equations

Lagrange's
and the
the canonical equations, the
Like Lagrange
's equations and
the Hamilton]acobi
basis of
the basis
of a general
general method of
of integrating the
the equations
equations
Jacobi equation is the

ns
motion:
of motio

Before
the fact that every
every firstBefore describing this
this method, we should recall
recall the
differential equation has a solution depending on
on an arbitrary
order partial differential
function; such
such a solution
solution is called
called the
the general integral
integral of the
the equation. In
In
function;
mechanical applications,
mechanical
applications, the
the general
general integral
integral of the
the Hamilton-]acobi
Hamilton-Jacobi equation
equation
than a complete integral, which
which contains
contains as many
many independent
is less important than
arbitrary constants as there are independent variables.
variables.
The
The independent variables
variables in the
the Hamilton-]acobi
Hamilton-Jacobi equation are the
the time
and the
the co-ordinates.
co-ordinates. For
For a system
system with
degrees of
of freedom,
freedom, therefore,
and
with s degrees
therefore, a
complete integral
integral of this
this equation must
must contain
contain s-l-1
s + 1 arbitrary
arbitrary constants.
constants.
complete
Since
the function S enters
enters the equation
equation only through its derivatives,
derivatives, one
Since the
of
of these constants is additive,
additive, so that a complete
complete integral of
of the
the Hamilton}acobi equation is
·
]acobi
n

= f(I,
f(t, Q1,
qt, ...,
... , Qs,
q8 ;
S =

01:1, ...,
••• ,
d1,

a: 8 )+A,
a8)+A,

(47.2)

constant.•
constant

(47.3)

where
Ol:lJ ...,
••• , as
01: 8 and
and A are arbitrary constants.T
constants. t
where al,
Let us now
now ascertain
ascertain the
the relation between
of the
the
Let
between a complete integral of
Hamilton-J acob_i equation
equation and
and the
the solution
solution of the
the equations
equations of
of motion
motion which
which
Hamilton-jacobi
To do this, we effect a canonical
canonical transformation
transformation from the
the
is of interest. To
variables q, P
p to
to new variables, taking
taking the function
function f (r,
(t, q; a)
01:) as the
variables
function, and
and the
the quantities a01:1,
01:2, ...,
••• , as
01: 8 as the
the new
new momenta.
generating function,
t , Ag,
Let the
the new
new co-ordinates
co-ordinates be 51,
{31, 52,
{32, ...,
•.. , {3
Since the generating
generating function
Let
Be.8 • Since
on the
the old co-ordinates and
and the
new momenta,
momenta, we use formulae
formulae
depends on
the new
(45.8): PI
f/3% ii
Elfl3a4, H' =
H + atlas. But since the
p, =
= Ojfoqt,,
f3i =
= ojfo<Xf,
= H+Offot.
the function
function f
satisf
ies the
satisfies
the Hamilton-]acobi
Hamilton-Jacobi equation, we see that
that the
the new
new Hamiltonian is
= H
+Ofjot == H
+ oS jot == 0. Hence
Hence the canonical
canonical equations
equations in
H+8S/3:
zero: H' =
H+22f/8:
the new
new variables
variables are Di
<Xi =
= 0, 5¢
/3 1 =
= 0, whence
the
whence

f3t =
Ba

II

II

01: 1 = constant,
*If

of the
the s equations
equations 3f/3a¢
ojjo01:1 =
= 8¢,
{31, the s co-ordinates
co-ordinates qq can be expressed
expressed
means of
By means
terms of the time and
and the 2s constants
constants a01: and
and {3.
the general
general
in terms
B. This gives the
integral of
of the
the equations
equations of
of motion.
integral
motion.

the general
general integral
integral of
of the
the Hamilton-Jacobi
Hamilton-Jacobi equation
equation is
is not
not needed
needed here,
here, we
t Although
Although the
we
may show
show how
how itit can
can be
be found
found from
a complete
complete integral.
integ~:al. To
To do
do this,
this, we regard
regard A
A as
as an
an arbiarbifrom a
trary
the remaining
- . ' J, Qs,
trary function
function of
of the
remaining constants'
constants: S == f(r,
f(t, quo,
q1 , ...
q,; d 1l ,, "•••' :, m) -I-A(a1, ° " l as), ReReplacing
placing the
the if by
by functions
functions of
of co-ordinates
co-ordinates and time given by
by the s conditions
conditions 3S/8al == 0,
0,
we
we obtain
obtain the
the general
general integral
integral in
in terms
terms of
of the
the arbitrary
arbitrary function
For, when
when the
the
function A(a1, ...,,a.).
Ols)- For,

S

at

function S
S is
is obtained
obtained in
in this
this manner,
manner, we
have
function
we have

ag
as
as ( as
as) 2:(( BS
as) ncrxk
aq~ = ( 3/i
aq~ )a"' + k Earp
oak )Q
o Fig;
Eqt
Elect
q

a a.) +A(ar, ... , a,)_
ast aac
A(a1, ...

cs)
cq, )~
"'.
=-( 3qi
=

(

ES

a

•
S(r,
Q,
· · (3S/3Ql)¢z
("'Sf"'
· f y th
H arn1."I ton-J aco
b"1i e
·
S(t,
q; f"')
quantities
c;
uq1)u satisfy
saus
t hee Harn
equauon,
t h e function
h efore
sat"
The quantities
The
quati·on, smce
1lton-Jacob
tis
oefore sa
th
since
fu
f
.
Th
.
"S
'C!q•
t tif
er
•s f yy)
the
is assumed
asstuned to
to be
be aa complete
complete integral
mtegral of
o that
that equation.
equauon. The
e quantities
quanuues 35:
v 1 3q4
is

..

.

A

'i
'i
I

Separation of
of the
the variables
variables
Separation
•

§48

149

the solution of the
the problem of the
the motion
motion of a mechanical
mechanical system
system by
by
Thus the
the
the Hamilton-Jacobi
Hamilton-Jacobi method
method proceeds
proceeds as follows. From the
the Hamiltonian,
Hamiltonian,
we form
the Hamilton-]acobi
Hamilton-Jacobi equation, and
and find its complete
complete integral
integral (4'7.2.).
(47.2).
form the
Differentiating this
this with
with respect
the arbitrary
arbitrary constants a01: and
and equating
Differentiating
respect to the
the derivatives
derivatives to new
new constants B,
{3, we obtain Ss algebraic
algebraic equations
the

asian = Br,

(47.4)

whose solution
solution gives the
the co-ordinates
co-ordinates q as functions
functions of
of time and
and of the
the 2s
whose
arbitrary
arbitrary constants.
constants. The
The momentzi
momenta as functions
functions of time
time may then be found
from
the equations PA
p1 =
= 35'/Zigi.
oSJoq~,.
from the
If
we
an
incomplete
If have incomplete integral
integral of
of the
the Hamilton-]acobi
Hamilton-Jacobi equation,
equation, dependon fewer
fewer than s arb
arbitrary
cannot give the
the general
general integral
integral
ing on
ing
itrary constants, it cannot
of the
motion, but
but it can
the equations
equations of
of motion,
can be used to simplify
simplify the
the finding
finding of
of the
the
general
general integral.
integral. For
For example, if a function
function S
S involving one arbitrary
arbitrary conknown, the
the relation
relation 35'/3
oSjo01:
= constant gives one equation between
a =
stant a01: is known,
Q1,
qt. ...,
... , Qs
q8 and
and t.t.
Hamilton-Jacobi equation
equation takes a somewhat
somewhat simpler
simpler form if the
the funcThe Hamilton-]acobi
tion H does not
not involve the
the time explicitly,
explicitly, i.e. if the
the system
system is conservative.
conservative.
tion
The time-dependence
time-dependence of the
the action
action is given
given by a term - Et:
The
Et :

-

-

II

S = S0(q)
Et
So(q)-Et

(47.5)

(see §44),
~44), and
and substitution
substitution in
in (47.1) gives for the
the abbreviated
abbreviated action
action S0(q)
So(q)
the Hamilton-]acobi
Hamilton-Jacobi equation in the
the form
the

Ol]l

1 . ¢ '

als
oqs

=

E.

II

. 3S0
330)
H(l]l, ...
qs; 8q'
oSo,
Hq1,
-»-», Qs,
§» ..., oSo)

(47.6))
(47.6

§48. Separation
Separation of
of the
the variables
In a number
number of
of important
important cases, a complete
complete integral
integral of the
the HamiltonIn
the variables",
variables", a name given
given to
to
Jacobi equation can be found by "separating the
the
the following
following method.
Let us assume
assume that some
some co-ordinate, QUO
q1 say, and
and the
the corresponding
Let
derivative
23 Sl3q11 appear
appear in the Hamilton-]acobi
Hamilton-Jacobi equation
equation only in some
derivative oSfoq
combination <;6(919
cfo(qh 38/391)
oSJoq1) which
which does not
not involve the
the other co-ordinates,
co-ordinates, time,
combination
time,
derivatives, i.e. the
the equation is of the
the form
form
or derivatives,

--at·

{

as
as)}

ۤ;)}

whe
re Qi
otes all
where
qc den
denotes
all the
the co-ordinates
co-ordinates except
except
We seek
ution in
We
seek aa sol
solution
in the
the for
form
of
a
sum:•
m of sum
=

0'
0,

(48.1)

q1.
quo.

S'(q,, I)t) +
+ S1(q1)
S1(q1);;
S'(q¢»

II
to

S

=

II

as
as as
as (

(D
~Ȣ>(s
<ll ii,
q,, 2,
t, 3qg
oq, 9' -3:
cfo 21»
lJl, oq1

(48.2)

I

§48

sons
The Canonical
Canonical Equa
Equations
The

150
150

( dqJ}N

substituting this in equation
equation (48.1),
(48.1 ), we obtain
obtain
D

J

1

s

1

dS1
ds;

=

0.
o.

(48.3)

II

oS' 3.5"
oS'
3.5"

<11q,,, t,, oq, , ot , 4> ( 9qh <D{
Q i ! 3Qi
3:
1 dQ1

Let
Let us suppose
suppose that
that the solution
solution (48.2)
(48.2) has been
been found.
found. Then, when
when it is
substituted in equation (48.3), the latter must
become
an
identity,
must become
identity, valid (in
particular) for any
When Q1
any value of
of the co-ordinate
co-ordinate Q1q1 . When
q1 changes, only
only the
(48.3) is an identity, ¢>
4> must be a
function ¢4> is affected, and so, if equation (48.3)
constant.
constant. Thus equation
equation (48.3)
(48.3) gives the two equations
equations

4>(% dS1/dQ1)
dS1Idq1) =
= 1.1,
01:1.
(ba,

(48.4)
(48.4)

/acq,,
6D{q¢,
go, oS'
ES'/81,
<I>{q,, t, 8.5"
oS' 1
1ct; at}
01:1} =
= 0,
o,

(48.5)

where 011
where
01: 1 is an arbitrary constant. The first
first of these is an ordinary differential
differential
equation, and
obtained from it by simple
simple integration.
integration.
and the
the function
function $1(91)
S1(q 1) is obtained
The remaining
remaining partial
partial differential
differential equation
equation (48.5)
(48.5) involves
involves fewer independent
variables.
variables.
If
can successively
successively separate
separate in this
co-ordinates and
and the
If we can
this way all the
the Ss co-ordinates
the
time,
time, the
the finding of
of a complete
complete integral
integral of
of the Hamilton-]acobi
Hamilton-} acobi equation is
reduced to
to quadratures. For
For a conservative
conservative system
system we have
to
reduced
have in practice to
separate
separate only
only s variables
variables (the
(the co-ordinates)
co-ordinates) in equation (4'7.6),
(47.6), and
and when
when this
complete the
the required
required integral is
separation is complete

=

II

S
S

)z
L
01:1, 01:2, ... , a8)-Ol:s}- E(a1,
E(01:1, ... , as
Ol:s)t,
Zk Sk(qk;
So(qks ala
a2)

k

°°*r

---»

1

(48.6)

where each
each of the
the functions
functions So
S k depends on
on only
only one co-ordinate; the
the energy
energy
where
the arbitrary
arbitrary constants a01:1,
... , as,
01: 8 , is obtained by
by substituting
E, as a function of the
t , ...,
So =
= 259¢
~Sk in equation (47.6).
SO
the separation
separation of a cyclic variable.
variable. A cyclic
cyclic co-ordinate
co-ordinate
particular case is the
A particular
q1 does not
not appear
appear explicitly
explicitly in the
the Hamiltonian,
Hamiltonian, nor
therefore in the
the Hamiltonnor therefore
91
]acobi
reduces to 35'/Zig1
Jacobi equation. The
The function
function ¢>(Q1J
cl>(qt. 35'/391)
oSioq1) reduces
oSioq 1 simply,
simply, and
and
simply $1
s1 =
= a1q1,
Ol:1qt, so that
from equation (48.4) we have simply

S

=

S'(q£, I)-|-0£1q1-

(48.7)

II

The constant
constant <11
01: 1 is just
the constant value of the
the momentum
PI = HS
oSioq1
The
just the
momentum P1
/391
to the
the cyclic
cyclic co-ordinate.
corresponding to
The
appearance
of
the term -Et
- Et for a conservative
conservative system
system
The appearance of the time in the
corresponds to
to the
the separation of
of the
the "cyclic variable" L
t.
the cases previously
considered of
of the
the simplification
simplification of the
the integraThus all the
previously considered
of the
the equations
equations of
of motion
motion by the use of cyclic variables
variables are embraced
embraced
tion of
tion
by
the method of
of separating the
the variables in
in the
the Hamilton-_Iacobi
Hamilton-Jacobi equation.
by the
To those cases are added others in
in which the
the variables can
can be separated even
To
ntly
eque
Hamilton-Jacobi
treatment is cons
conse'!uently
not cyclic. The Hamiltonthough they are not
Jacobi treatment
thee most
most powerful
powerful method
method of
of Ending
finding the
the general
general integral
integral of
of the
the equatIons
equattons of
of
th
motion.
motion.

Separation of
of the variables

§48

151

To
To make the variables separable in the Hamilton-]acobi
Hamilton-Jacobi equation the
co-ordinates must
must be appropriately chosen.
chosen. We shall
shall consider
consider some
some examples
examples
co-ordinates
of
may be of
of separating the
the variables
variables in different
different co-ordinates,
co-ordinates, which
which may
physical interest
interest in connection
connection with
with problems
the motion
motion of
of a particle
physical
problems of the
particle in
various external Fields.
fields.
various

co-ordinates. In these
these co-ordinates
co-ordinates (r, 6,
0, ¢),
cfo), the Hamiltonian
Hamiltonian is
(1) Spherical
Spherical co-ordinates.
=

II

m

H

)

(pt

2
11 (
P¢2
U ,
Pi
P.l
2
_|,_
Pg
- Pr2+-2-+
+
-|- U(r, 0,
-2
2
. 2(J ) +
H cfo),
2
7
72
m
r
r sstn
1n26
2m

and the
the variables
variables can
can be separated if
and

)

)

b(O)
c(cfo)
t(¢)
U
( t +
. .
U =
= a
a(r)+""'"2+
,
rr2
r2 sinzfl
sin20 .
t2

a(r), b(O),
c(cfo) are arbitrary
arbitrary functions.
functions. The
The last term
term in this
this expression
expression
a(1r),
b(6), c(¢)
unlikely to
to be of
of physical
and we shall
shall therefore take
for U is unlikely
physical interest, and

w
where
here

U

a(r)+b(O)fr2.
= a(r)-|12(6)/r2.

I

In this
this case the
the Hamilton-]acobi
Hamilton-Jacobi equation for the
the function $0
So is
In

[( )

( or )

(48.8)

( )

1 3300 22
1
3.90 22
1
3.90 22
_I_(oS ) +
+a(r)+-1-[(oSo)
+Zmb(O)]
+
1
(oSo)
=E.
E.
-I-2mb(6) +
a(r) +
()(}
sin2(J 395
ocfo
2m or
2mr2 36
2mr2 sin26

Since the
the co-ordinate
co-ordinate cPt, is eye
cyclic,
seek a solution in the
form $0
So
lie, we seek
the form
Since
= P¢95
S1(f) + S2(0), obtaining
the equations
p9cfo ++S1(r)
obtaining for the
the functioNs
functiorts S1(r)
S 1(r) ands
andS 2(9)
2(0)the
equations

~~2 )2-I-2mb(0)+
) 2 + 2mb(O) + s~::(J . =
sin26
(T
ds

P¢2

{3,

II

(

1

E.
)
finally
Integration gives finally
Ss = - Et +p¢¢
- zmbw)
-p/sin2@]
MB
+ P9cP +
+I
v[f3Zmb(O)Pifsin20] d6
d(J +
+

-(

=

-

=

II

1 ds; 2
_1_( dSt )\a(r)+
+ + B{3 .
2mr2
2m dr
a(r) 2m2'2

+ fv{2t»[Ejv{2m[E-a(r)]-{3fr2}
dr.
+
000]-m/2} dt.

(48.9)

The arbitrary
arbitrary constants
constants in (48.9) are P¢»
p9 , B
{3 and
and E;
E; on differentiating
differentiating with
with
The

and equating the
the results
results to
to other constants, we have
the
respect to these and
have the
general
solution
of
the
equations
of
motion.
general
of the equations
motion.
co-ordinates. The
The passage from cylindrical
cylindrical co-ordinates
co-ordinates
(2) Parabolic co-ordinates.
denoted by p,
p, ¢,
cfo, z) to parabolic
co-ordinates g,
~. 11,
7], ¢
cP is effected by the
(here denoted
parabolic co-ordinates
form
ulae
formulae
(48.10)
z = ME-12),
v(~7J>·
Pp = V(§"0)»
(48-10)

e

co-ordinates 6 and
and 17
7J take
take values
values from
from 00 to
to oo,
oo; the
the surfaces
of constant
constant
surfaces of
The co-ordinates
f and
and 7J
are
easily
seen
to
be
two
families
of
paraboloids
of
revolution,
'U are Caslly seen to be two families of paraboloids of revolution, with
with

152

Thi!> Canonical
Canonical Equations
Equations
The

§48

the
written,
the z-axis as the
the axis of symmetry. The
The equations
equations (48.10)
(48.10) can
can also be written,
in terms
terms of

r =

V(2»*2+p2)

=

%(§+°J)

(48.11)

"2
7} =
= r-z.

(48.12)

the radius in spherical co-ordinates), as
(i.e. the

§~ =
r-I-z,
= r+z,

Let
Let us now
now derive
derive the
the Lagrangian
Lagrangian of
of a particle in
in the
the co-ordinates gg, "Q,
7], is.
cfo.
Differentiating the
the expressions
expressions (48.10)
(48.10) with
respect to time
time and substituting
Differentiating
with respect
in
i11 the
the Lagrangian
Lagrangian in cylindrical
cylindrical co-ordinates
co-ordinates

L =
= %f"(p""+p2~;52+22)!m(p-2+p2~2+z2)- U(f>,
U(p, ¢.
cfo, 2),
z),
L

\

obtain
rain
we ob

L

)

£2 132
+ "1 +%m§~a<£2-UQ, .), sl>)-

%ttt(§+'0)( g

(48.13)
(48.13)

The momenta
momenta are P;
P€ = 1m(s+»)£/§,p,
!m(g+'l})gfg,p"' = 1t~(§+))»a/m
!m(g+'l})i]/'I},P¢> == mgrfl>,
and
fn§m&, and
Hamiltonian is
the Hamiltonian

2 §Pt+Wq2

H

m

§+"2

2

+ P¢ -I-U(§, v],¢).
2m§-q
r

(48.14)

J

The physically
physically interesting
interesting cases of
of separable
separable variables
variables in these
these co-ordinates
co-ordinates
The
to a potential
potential energy
energy of
of the form
correspond to
correspond

Ag) + b('q)

U
The equation
equation for
So is
is
The
for SO

2

"1(§+"2)

5+v7

[t(-) (-)1
350
3§

a(r+z)-|~b(r- z)
2,

330

2

872

2

+

Ztnév (-)
3¢>
1

3S0

2

+

.

(48.1
(48.15)
5)

a(§) ~I-I2("l)

E.

§+"0

The
a term
The cyclic
cyclic co-ordinate
co-ordinate cPf, can
can be
be separated
separated as
~sa
term P¢¢P¢>cP· Multiplying
Multiplying the
the equaequaby m(§
m(g + 11)
7}) and
and rearranging,
rearranging, we
then have
have
tion by
tion
we then

2t(5?
~)

380
os0)22

~ (-

*
"7-I-3

(-i?)

3500)22
p2
(oS
p2
)-m£
+ma(~)-mEg+_.l.._+27} +mb(7])-mE7]+_.l.._
+ma(§)-mE§+2g+2q
2
b
"
(
m
~
~ +
~
272

Putting
Putting S0
So == P¢'I5
P¢>cP +81(5)
+ S1 (~) + S2(vl),
S2( 7]), we
we obtain
obtain the
the two
two equations
equations

ds
P¢
dS1)22
p2
2t ( ~ -l~ma(§)-mE§+
+ma(g)-mE~+ ~2 = B,
{3,
go
dE
2§

)

2)2

)

2

dS
p
27] ( +mb(71)-mE71
+__!!_ = -~,
ds- 2+mb("2)#150 +-B»
272
d7}
27)

do

=

•

0.

§48

of the 'variables
variables
Separation of

a/1
IJ[

integration of which gives finally
integration

S
S

=

M

-Ez+p
-Et+p9cfo+

+

¢

la mE

f3
18
2§

E+--~-

in
fmE+

--B

"16*(°2)

Zn 272
the arbitrary
arbitrary constants
constants are P¢»
p9 , {3B and E.
Here the

153

1

ma(fl
t"¢*(§)
P¢2
~
do +
2§
4§2

_ at] dv»

(48.16)

4"02

7], ¢,,
cfo, defined by
g, 11,

co-ordinates. These
These are
(3) Elliptic
Elli}>tzlc co-ordinates.

p =
= <tv[(a-1)(1-02)],
uy[(~2-1)(1-7]2)],
= ug7].
(48.17)
USU(48.17)
P
zz =
The constant oa is a parameter of
of the transformation. The co-ordinate f~ takes
to to,
oo, and
and -q7} from -0I
- 1 to +
+ 1. The
The definitions
definitions which
values from 11 to
which are geometrically
terms of
pa to points
of the distances
distances T1
r1 and
and r2
metrically clearest
clearestt are obtained
obtained in terms
A1
oz VI
[(z-0)2+ p2],
A 1 and A2 on the z-axis for which z = i±u:
r1 = v[(z-u)2+p2],
to
[(z-I- o)2-I-p2]. Substitution
r2 =
=v[(z+u)2+p2].
Substitution of (48.17) gives
u(~-7]),
== 4£-"0)»
<5
~ =
= (r2-I-r1)/20,
(r2 + rt)/2u,

r1
VI

_

r2 ::= §'(§+'0),
a{g+7]),
T2
\

(48.18)

"7
7} = (r2-r1)/Zo.
(r2- rl)/2u.

the Lagrangian
Lagrangian from
from cylindrical
cylindrical to
to elliptic co-ordinates, we
Transforming the
find

§2
#2
L = %f~(§2-v2)( §2- 1-+ 1-02) +
+|%o2(§2- 1)(1 - U(§, '11 (6)-

#FW

The Hamiltonian is therefore

( g2~

1

(48.19)

1
1
1
2 +
"
"22)P*22 + ( £2 __ 1'1 +
L
(1-'YJ2)p.l+
1
+
+ 1-v22)P¢
2"w2(§2-792) [Q 1)Pg2 (
(48.20)
+
U(5, "2»
+ U(~.
'YJ• ¢>)cfo).
The physically
physically interesting cases of
of separable variables
variables correspond to a
potential energy
1
,
. .
H
H = 2ma2(;2_'YJ2)

[c~2-1)p€2+

_|_

{(

~'YJ2)Pl]

)+b(

)},

V2-H1
r2-r1
__ 02
U --.=
= a(§)-I-b("2)
a(fl+b('YJ) =~(a(
r2+r1) +b( r2-r1 )}.
U
~2 -7}2
r1r2
2u
2u
£2-'02
- V172
20'
20'

1

(48.21)

where
and b(-Q)
b( 'YJ) are
are arbitrary
arbitrary functions.
The result
result of
of separating
separating the
the
where a(~)
Ag) and
functions. The
variables in the
the Hamilton--Jacobi
Hamilton-Jacobi equation
equation is
is
variables

S

=

::

-E!+1J¢<;'>+
-Et+p9cfo+

IJ
L/[[

up g
f3-- 22mu2a(~)
. M' la( )
2mo2E-pB'
2mu2E+
~g2- 1

J
in

P¢2.

(§2--1)2

8-I-2mo2b(v2)

dg+

P¢2

+ J[zmu2E- ~+::~('YJ)_,la
((1 ~"'22)2]
"22)2 d7].
7}

7}

(48.22)
(48.22)

e sur
faces of
A1 and
f Th
'J'he
surfa~es
of constant
constant tfare
the ellipsoids
ellipsoids 22/02£2+p"/v2(t'-*l)
z2fa1f2+p2/a2(f!~1) '=
= 1,
1, of
of which
which A1
are the
As
of constant
constant 11
"' are
are the
the hyperboloids
hyperboloids z2/0*3q2
z2fai.,S-p2fa2(1-.,t)
= 1,
1,
of
--pa/o2(1 -172) =
al
Alt8fe
faces
are the
the foci
foe•;; the
the sur
surfaces
O with foci A1 and An.
also
with foci
Aa.

s

154
154

§49

The Canonical
Canonical Equations
Equations
The
P
ROBLEMS
PROBLEMS

PtmiiLE:\1 I1.
Fmd a complete
complete integral of the Hamilton*]acobi
Hamilton-Jacobi equation
equation for
for motion of
of a
P1<<MLEM
. l"md
field U = a'r
rx'r -- Fx
Fz (a combination of
of a Coulomb Held
field and
and a uniform Field),
field), and
and
particle in aa Held
fmd
a conserved
consern~d function
function of
of the
the co-ordinates
co-ordinates and
and momenta
momenta that
that is
is specific
specific to
to this
this motion.
motion.
find a

+

=lFry2.
SOLUTION.
SoLUTION. The
Tht Field
field is of the type
type (4815),
(-1-8.15), with 11(6)
a(~) = a
rx -- 5Fg2,
~F~ 2 , b(17)
b(q) = a
rx
~Fq 2 •
The
The complete
complete integral
integral of
of the
the Hamilton--AIacobi
Hamilton-Jacobi equation
equation is
is given
given by
by (48.16)
( 48.16) with
with these
these functions
functions
a(§)
my). To
a(~) and
and b(
I>( q).
To determine
determine the
the significance
significance of
of the
the constant
constant 8.
P. we
we write
write the
the equations
equations
zgpf -ii nm(§)
Iufzlf
| P2l°f = p.
?.fpf
ma(~)
ml:t +
+ ~P~tt
{3,

"!.1/p;f

~t
-t N!]J(1})
mb(,/) --

91)15

H|I1.1!
mf:,,

+ ~p~!lJ
P:,!1g ' - B.
{3.

Snbtractmg,
p,11 == PS/01]
Suhtractmg, and expressing the momenta P:
p~ == PSl83
l1Sfill; and
and p,
l'Sfl1'1 in
in terms
terms of
of the
the
1
1
momenta pp == FSh"p
l1SI< p and
and p__
p= == fn.S'#'0z
l S/llz in
in cylindrical co-ordinates,
co-ordinates, we obtain
obtain after
after al
a simple
simple
momenta
calculation
calculation

B

-

=

or.

¢r"v

"\

r

+ P-£(:p
HI

Ppz) +

PE,

up"

u: Fp

re

|

The
The expression
expression in
in the
the brackets
brackets is
is an
an integral
integral of
of the
the motion
motion that
that is
is specific
specific to
to the
the pure
pure Coulomb
Coulomb
field (the ;;;-component
of the vector
\"ector (15.1'7)).
(15.17)).
Held
x-component of

Problem 1, but
but for a Held
field U
PROBLEM 2. The same as Problem
field of
of two
two Fixed
fixed points at a distance Zo
2a apart).
apart).
Held

= arxl1l/rr 11 -l+ a2]r2
rx 2 /r 2

(the Coulomb

+

SOLUTION. This
Eeldis of the type(48.2I),with a(§) =
b(11) = ((rx,a , -- a2)r7..»'0.
Thisfieldisofthetype(48.21),witha(~)
= (al
(rx 1 + a2)C,.0,
rx 2 )~ 1 a,b(q)
rx 2 )qfa.
actionS(~.
q, (0,
lfl, I)
t) is obtained
obtained by
by substituting these expressions in
in (48.22). The signiF1signifiThe action
go my,

cance of
of the
the constant
constant
cance

is
8p is

found in
in aa manner
manner similar
similar to
to that
that in
in Problem
Problem 1,
1; in
in this
this case
case it
it
found

expresses the
the conservation
conservation ooff the
the quantity
quantity
expresses

B{3

=

a~ (P~

+

~~)

,118
•11" = (r
(r xx

- .118
_lP +
+ 2mo(a1
2ma(a1 cos
co~ 61
01 ++ a-»
a~ cos 6..).
0~) .

. n

pf
oF

p2::- + P8p-I_l + *Pa
p_
11

and
and

0 1 and
and 02
0 2 are
are the
the angles
angles shown
shown in
in Fig.
Fig. 55.
55.
01

02

§49. Adiabatic
Adiabatic invariants

20'

7?-»2'pP-=P,»»

q

FIG.
55
FIG. 55

Let
Let us consider
consider a mechanical
mechanical system
system executing
executing a finite motion
motion in one
one dimencharacterised by some parameter
properties of
sion and characterised
sion
parameter AA which specifies the properties
the system
system or of the
the external
external field in which it is placed,T
placed, t and let us suppose
suppose that
that
AA varie
variess slow
slowly
(adiabatically) with time as the result of
of some
some external action
action;,
ly (adiabatically)
only slightly
slightly during
during the
by a "slow" variation we mean one in which AA varies only
period T of
of the motion
motion::
period

T dA/dz
<L\fdt <A.
~.A.

(49
.1)
(49.1)

To simplify the
the formulae, we
we assume that there is only
only one
one such pararr1eter,
but all
all the
the
'I't To
parameter, but
for any
any number of
of parameters.
results remain valid for

Adiabatic in
invariants
zrariants

§49

15
1555

If
system would be closed
closed and would
If AA. were constant, the system
would execute a
strictly periodic
periodic motion
motion with a constant
constant energy
energy E and a fixed period
T(E).
strictly
period T(E).
When the parameter
parameter AA. is variable,
variable, the system
system is not
not closed
closed and
and its energy is
not conserved.
conserved. However,
However, since
since AA. is assumed
assumed to vary only slowly, the rate
rate of
change E of
of the energy
energy will also be small.
small. If
rate is averaged
averaged over
over the
change
If this rate
period T and the "rapid" oscillations
oscillations of
of its value are thereby
thereby smoothed
smoothed out,
period
the resulting
resulting value E
E determines
determines the rate of
of steady
steady slow
sl9w variation
variation of
of the
energy of
of the system,
system, and this rate
rate will be proportional
of change
change i).
energy
proportional to the rate of
of
parameter. In other
quantity E, taken in
of the parameter.
other words,
words, the slowly varying
varying quantity
hehave as some function
function of A.
A.. The
The dependence
dependence of E on AA.
this sense, will behave
expressed as the constancy
constancy of
of some combination
combination of
of E and A.
A.. This
can be expressed
quantity, which
which remains
remains constant
constant during
during the motion of
of a system
system with
with slowly
slowly
quantity,
varying parameters,
adiabatic invariant.
varying
parameters, is called an adiabatic
z'n'variant.
Let
Let H(q, P;
p; A)
A.} be the Hamiltonian
Hamiltonian of
of the system,
system, which
which depends
depends on the
parameter A.
A.. According
(40.5), the rate
rate of
of change
change of the energy
energy
According to formula (40.5),
of the system
system is
of

Et'

dF
oH
oHdA.
@H
aH
dl
--==-=--

ot
6:

dt
dr

oA. dt.
<3/1

(49.2)
(49.2)

The expression
expression on the right depends
depends not only
only on the slowly varying quantity
quantity
The
on the rapidly
rapidly varying
varying quantities
quantities Qq and P.
p. To
To ascertain
ascertain the steady
steady
AA. but
but also on
variation of
of the energy
energy we must, according
according to the above
above discussion,
discussion, average
average
variation
over the period
period of
of the motion.
motion. Since
Since /1
A. and therefore
(49.2) over
therefore AA. vary only
outside the averaging:
averaging:
slowly, we can take i). outside

iJH
cdA. 571
dE
dt
dt as'
dr = dr

a;·

(49.3)

and in the function
function 6Hldl
oHfoA. being
being averaged
averaged we can regard
p, and
regard only q and P,
not A,
A., as variable.
variable. In
In other
other words,
words, the averaging
averaging is taken over
over the motion
not
which would occur
occur if AA. remained
remained constant.
constant.
The averaging
averaging may be explicitly
explicitly written
The

aH
aH =
-

oA.
61

f

J_faH
dt.
aH dr.
T

T

0

oA.
61

e
According to Hamilton's equation
equation Qq =::
= 3HI'3p,
oHiop, or do
dt =
= do
dq_,_ (8H
(oHfop).
The
According
/3z))~ Th
ect
integration
time can therefore be replaced
replaced bY
by one
one with
with resp
respect
integ
ration with respect
respect to time
to the co~o1-dinate,
co-ordinate, with the period
written as
period T written
T
r

=
=

ad:
f dt =5§d9
= f dq + (aHlap);
(oHfop);
T
T

0

(49.4)

Thee Canonical Equations
Equations
Th

156

§49
§49

here the §
f sign
sign denotes
denotes an integration over
over the complete
complete range of
of variation
variation
here
back") of
of the co-ordinate
co-ordinate during
during the period.t
("there and back")
periodft Thus (49.3) becomes
becomes

EE
dE _
dr
dt

do
do,f(aH,/ap)
dA §~ (aHlaa)
(oH,tolt) dqf(oHfop)
.
dr
dt
~§ del(aH,fap)
dqf(oHfop)

-

(49.5)

already been
mentioned, the integrations
integrations in this
this formula
formula must be
As has already
been mentioned,
taken
taken over the path
path for a given
giYen constant
constant value of
of A. Along such
such a path the
constant value E, and the momentum
momentum is a definite
definite function
Hamiltonian has a constant
of
parameters
of the variable
variable co-ordinate gq and of
of the
the two independent constant
constant parameters
E
E and PL
A. Putting therefore
therefore p =
= P(q,
p(q; E, A) and differentiating
differentiating with respect
respect
to AA the equation
equation H(q,P,)l)
H(q,p,A.) =
= E, we have 0Hl(M
iJHfolt +
+ (6H/r3p)(6P/dA)
(oHfop)(opfo>..) =
= 0, or

aHl3A
oHfo>..
EFHl8p
(IHfcp

op
31)

aX

Substituting this
this in the
the numerator of (49.5)
(49.5) and writing
writing the integrand
integrand in the
Substituting
denominator
denominator as 3?)3E,
opfoE, we obtain

EE
dE

dq
ddAHopfo>..)
(3p/3A) do
dt §~ (81)l35)
(cpfoE) do
dq
dz

-

do
dt -

,e ( 3?
op Dr
dE Hpd/\'~
epdA'
.
d =0.
J oE
cit do
dt) dqQ = o.
AE Tt+
d¢+"A
al(

or

I

-

v

-

'

1

.

Finally, this may be written
written as
as
Finally,

dlfdt =
= 0,
dl/dz

whe
re
where

= fof?p dg/21
dqf27T,.,

(49.7)
(49.7)

Ill

"'~4

I

(49.6)

the integral
path for given E and )L
integral being taken over
over the path
A. This shows that,
that, in
the approximation here considered,
considered, II remains constant when the
the parameter AA
varies, i.e. I is an adiabatic invariant.
The quantity
quantity I is a function
function of the energy
energy of the
the system
system (and
(and of the paraThe
paraA.). The
The partial derivative
derivative with respect
energy determines the
the period
meter 1).
respect to energy
period
motion: from (49.4),
of the motion:

__

22n aofI

.·
t h e l110t10r1
motton
o f the
the
Tt IfIf the
of
•
.
•h

" aE
BE

.

=

.

J: ap do
dq
J' aE

= T
=

(49.8)
.

.

· a rotat10n,
·
h co-ord"tnate q is
· an
system is
ts
rotatton,
and the
t e co-ordmate
ts
an angle
a ngle
.
system
and
A
b
k
"
I
.
n"
' ·e.
tion' • 1.et he
e integration
tntegratton with
wtt respect to
to 'I'
mUst bee taken
ta en over a "complete
comp ete rota
rotatto
go must
th

.

otat .
of
A
f rrotation
m 0 lon 95,
'1',
Of
froth
to 21r,
ro 0
to
21T·

§50

Canonical
variables
Can
oneal 'variables

or
OI

6E/6]
aEtai =:
= w,

157
(49.9)

where w
w =-= 2nfT
vibration frequency
frequency of
of the
the system.
system.
21:/T is the vibration
The .integral
,integral (49.7) has a geometrical
geometrical significance
significance in terms
terms of the
the phase
The
phase
path
phase
path of
of the system.
system. In the case considered
considered (one degree
degree of freedom),
freedom), the phase
space
plane) with co-ordinates
two-dimensional space
space (i.e. a plane)
co-ordinates
space reduces
reduces to a two-dimensional
P,
periodic motion
motion is a closed
p, q, and the phase path
path of
of a system executing a periodic
closed
The integral
integral (49.7)
(49. 7) taken round
the area
curve in the plane. The
round this curve is the
integral
enclosed. It
enclosed.
It can be written
written also as the area integral

I =
=

n

IIJJ do)
dp do/Zn.
dqf2n.

(49.10)

example, let us determine
determine the
the adiabatic invariant
invartant for a one-dimenone-dimenAs an example,
Hamiltonian is
sional oscillator.
oscillator. The Hamiltonian
sional

H

:=

%/1m + m2q2,

(49.11)

where w is the eigenfrequency of
of the
the oscillator.
oscillator. The
The equation
equation of
of the phase
where
phase
path is given by the'law
the 'law of conservation
conservation of energy H(P,
H(p, q) =
= E. The
The path
path
2
is an ellipse with
V(2mE) and v(2Efmw
I ( z E / m )),, and its area,
with semi-axes v(2mE)
area, divided
divided
2n, is
by Zn,
II=
= Elmo.
(49.12)
Efw.

adiabatic invariance
invariance of I signifies that, when
when the parameters
parameters of the
the
The adiabatic
oscillator
proportional to the frequency.
oscillator vary
vary slowly, the energy
energy is proportional
frequency.
§50.

Canonical variables

Now
the parameter
that the
the system
system in question
question is
Now let the
parameter iA. be constant, so that
Let us effect a canonical
canonical transformation
transformation of the variables q and P,
p,
closed. Let
the new "momentum". The
The generating
generating function
function is the abbreviated
abbreviated
taking I as the
S0 , expressed
expressed as a function
function of
of q and I. For
For $0
S 0 is defined
defined as the integral
integral
action So
action

S0
Ip(q, E;
S 0 (q, E;
E; 1)
A.) =
= Jp(q,
E; 1)
A.) dg,
dq,

(50.1)

taken for a given
given energy
energy E and parameter
A.. For
For a closed system,
system, however,
however,
taken
parameter A.
I is a function
function of
of the
the energy
energy alone,
alone, and so $0
S 0 can equally
equally well be written as a
function S0(q,
S0 (q, I;
I; A),
A.), and the partial
partial derivative
derivative (GSo
(iJS0 ldg)E
fcq)E is the same as the
the
function
derivative
(iJS
foq)
for
constant
I.
Hence
derivative (6500 I09)I1
constant
Hence
=

iJS0 (q, I; 1)/
A.)jcq,
69,
6S0(Q,

II

Pp

(50.2)

corresponding to the first of
of the formulae (45.8)
(45.8) for a canonical
canonical transformatransformacorresponding
The second of
of these
these formulae
formulae gives the
the new
new "co-ordinate", which
which we
tion. The
denote by ow:r
(50.3)
nu
(50.3)
w =
= 0S0(q,
aS0 (q, I;I; 1)/61.
A.)fci.

The variables
The
variables II an
and
are called
called canonical
canonical 'z,1ariables;
<L·.miables; II is
is called
called the
the action
action
d w are
variable and
and 'w
w the
the t.ngle
angle -variable.
variable.
'variable

158

§50

The Canonical
Canonical Equations
Equations
The

Since
A) does not
Since the generating
generating function
function S0(q,
So(q, I;
I; .A)
not depend
depend explicitly
explicitly on
time, the new Hamiltonian
H' is just
just H expressed
Hamiltonian H'
expressed in terms
terms of
of the new
variables.
H' is the energy
variables. In other
other words, H'
energy EU),
E(l), expressed
expressed as a function
function of
of
the
the action
action variable.
variable. Accordingly,
Accordingly, Hamilton's
Hamilton's equations in canonical
canonical variables
variables
are

w=

I=
I = 0,

dE(l)/dl.
dE(I)ldI.

(50.4)

The first of
of these
these shows
shows that I is constant,
constant, as it should
should be; the
the energy
energy is
The
the second
second equation
equation we see that the
the angle
constant, and I is so too. From the
variable is a linear
linear function of
of time
time::
=

dE
dE
_.r
t
dl
dl

+ constant

= w(I)t
w(l)t

II

w

+ constant;
constant;

(50.5)

it is the
the phase
phase of the oscillations.
oscillations.
The
The action S0(q,
S 0 (q, I)
I) is a many-valued
many-valued function
function of
of the co-ordinates.
co-ordinates. During
each period
period this function increases
increases by

ASo =
~So

22m!,
I,

(50.6))
(50.6

as is evident
evident from (50.1) and the definition
definition of
of I (49.7)
(49.7). During
During the same time
variable increases
increases by
the angle variable

A(630/51) = 6(A.S'0)/61
= ~(oSofol)
o(~So)/ol =

2n.
2'/r.

II

A
w
~w

(50. 7)
(50.7)

one-valued function F(q, P)
p) of
Conversely, if we express q and p, or any one-valued
of the canonical
canonical variables,
variables, then
then they
they remain
them, in terms of
remain unchanged
unchanged when
when
w increases by Zn
2n (with I constant).
constant). That
That is, any one-valued
one-valued function F(q,
p),
nu
F(q, p),
expressed in terms
terms of
of the canonical
canonical variables, is a periodic
periodic function
function of av
w
when expressed
with
with period 2-rr.
2'1T.
The
The equations of
of motion
motion can also be formulated
formulated in canonical
canonical variables
variables for
system that is not
not closed,
closed, in which the parameter
a system
parameter /l). is time-dependent.
The transformation
transformation to these
these variables
variables is again effected by formulae
formulae (50.2),
(50.2),
The
S 0 given by the integral
integral (50.1) and exgenerating function $0
(50.3), with a generating
pressed
terms of
of the variable
variable I given
given by the integral
integral (49.'7).
(49. 7). The
The indefinite
indefinite
pressed in terms
integral (50.1) and the
the definite
definite integral
integral (49.7) are calculated
calculated as ifif the
the paraintegral
parameter A(z)
).(t) had a given fixed value; that
that is, SO
S 0 (Q,
(q, I; 2.(t))
).(t)) is the previous
previous
meter
func
tion
with
function with the constant
constant).A finally replaced
replaced by the specified
specified function
function ).(t).'t
2(t).t
Sinc
e the generating
Since
generating function is now;
like
the
parameter
A.,
an
explicit
now,parameter A,
explicit
function of
of the
the time, the new Hamiltohian
Hamiltonian H'
different from the old one,
function
H' is different
which was the energy
energy E(I).
E(I). According
According to the general
general formulae of the canonical
canonical
which
transformation (45.8), we have
transformation
'
H' = EU;
H'
E(I; A)
A.) +
+ 030
oS lb:
j(Jt
0

II

= EU;
E(I; A)
A.) + As,
Ai,

1't

(50.8)
.8)
(50

·
·
S o tthus
h us determined
d eterrmne
· d is
· not
not the
the true
true
It
however,
SO
lt must
must be
be emphasised,
emphasised,
however, that
t h at the
the fifunction
unctiOn
ts

abbre\"iated action
action for
a system
system \\ith
a time-dependent
time-dependent Hamiltonian.
Hamiltonian.
abbreviated
for a
with a

§51

Aeeuraey of
Accuracy
of conservation
conservation of
of the
the adiabatic
adiabatic invariant
invariant

with the notation
notation

A = (030 la')-2,1;

159
(50.9)

here A must
must be expressed
expressed in terms
terms of
of I and w by (50.3) after
after the differentiation
differentiation
W
ith respect
with
respect to l.
A..
Hamilton's equations
equations now become
Hamilton's
become

(
»(-)
(al

().A) .

OH'
oH'

Z
PA
A.
OW 1,A
J,). ,
32:1
ow
6"
•
AA
aH'
w. = -oH' = w(J·
w(I; A)
+ (o.A)
A.) +
A..
1,
'
'
aoi
al
6oi !w,1
w,J.
I1=

--= - -

J

(50.10)

(50.11)

where w = (0E,I0I)
(oEfol);. is the oscillation
oscillation frequency,
frequency, again ca
calculated
A. were
lcu lated as if /1
constant.
PR
O B L EM
PROBLEM

Write down the
the equations of
of motion in
in canonical
canonical variables
variables for
a harmonic
harmonic oscillator
oscillator (whose
(whose
for a
( 49.11 )) with time-dependent
time-dependent frequency.
frequency.
Hamiltonian is (49.Il))
SOLUTION.
SoLUTION. Since
Since all
all the
the operations
operations in
in (50.1)-(50.3)
(50.1}--(50.3) are
are for
for constant
constant A
A (1
(A being
being in this
this case
case
the
the frequency co itself,
itself), the
the relation
relation of
of q and [J
p to
tow
the same
same form
constant frequency
'IU has the
form as for constant
`
with nu
w =
=cot:
·
cot:
q
q

Hen
Hence
ce

and
and

=

21 .
2E ..
sin
_ f -99 Sm
Stn or:
u: =
stn nu,
u•,
\1
'J NIU)
mw -\t NIL)
mlo

P
p = -y'(2Iwm)
v(2I<"Jm) COS
cos w.

Sn

Ipdq

I PW @w),,.,, do

as (cw)
.I (-)
(''So) = ((~)
)
(`So

.I =

r`w
t<"J

=

q,]
q,I

=

(50.10) and
and (50.11)
(50.11) then
then become
Equations (50.10)

Ii

Fu'
{u· I1
.n

:'.

520

Co;
(w 4•

21560522

.

do

2w.

I
San Zn.
=.. _!_sin

.T'

20
2<·J

2w.

= -I((b
w) COS
-I(<" <'J)
cos 224;
2rc, to'
rl: = (U
w ++ (Tb,
(l;J,2<>J)
Zn) sin Zn.

§51. Accuracy of
of conservation of
of the
the adiabatic invariant
The
The equation
equation of
of motion
motion in the form (50.10)
(50.10) allows a further
further proof
that the
proof that

action
action variable
variable is an adiabatic
adiabatic invariant.
invariant.
The
The function
function S0(Q9
S 0 (q, I,
I;}.)
not a single-valued
single-valued function
function of
of Q:
q: when the
2.) is not
co-ordinate
co-ordinate returns
returns to its original
original value, $0
S 0 increases
increases by an integral
integral multiple
multiple
of 2711.
2nl. The
The derivative (50.9),
(50.9}, however, is single-valued,
single-valued, since the differentiation
tion is at constant
constant I and the increments
increments of 30
S 0 disappear.
disappear. The
The function
function A, like
any single-valued
single-valued function,
function, is a periodic
function when expressed
expressed in terms
periodic function
of
of th
thee angle
angle variable
variable to.
w. The mean
mean value, over
over the period, of
of the derivative
derivative
. l law of
periodic function
c£1jcw
of a periodic
function is zero. Hence,
Hence, on averaging
averaging (50.10) and taking
6/
J.
). outside the
the mean
mean value
value (when
(when 2.}. "aries
only slowly),
slowly), we have
varies only

I

= - (an few )I A = 0,

(51.1)
(51.1)

as
was to
be proved.
pro ved .
to be
as was
The equations
equations of
of Motion
motion (50.10) and
and (50.11)
(50.11) enable
enable us
us to
to consider
consider the
the
The

160

§51
§51

The
The Canonical
Canonical Equations
Equations

accuracy
accuracy with
with which.
which the adiabatic
adiabatic invariant
invariant is conserved.
conserved. The
The question may
may
be stated
parameter 1(1)
_ and l+
stated as follows: let the parameter
A(t) tend
tend to constant
constant limits lA_
A+
I- of
as tz ->
--+ - oo and tz ->
--+ +
+ oo,
oo; given
given the initial
initial (I
(t ->
--+ - oo) value L
of the adiabatic
adiabatic
as
invariant, find the change in it, 11/
AI = l+
I+ - I..
I- as t ->
--+ -|+ oo.
oo.
From (50.10),
From
- _

I.f

=-

oo

I

II

AI
M

aA .
-Adt.
an i dr.
(in
aw

(51.2))
(51.2

-oo

nu, with period
period Zn;
shown above, A is a periodic
periodic function
function of
of w,
2n; let us expand
expand
As shown
it as a Fourier series

A=
A=

'I AI.
L
eilw
Z e""*A/.
oo
00

(~1.3)
(51-3)

II=
= --oo
oo

Since A is real, the expansion
expansion coefficients are such
such that
that A_/
A_ 1 =
= AS.
Aj. Hence
Hence
Since

PA
aA

=

kw
aw

iI izafw AIA,
i[eilw

I1=-oo
=-'oo

=
=-.

L i[eilw
A 1•
22 re Z
z`leilzv A1.
oo
00

I1=1
=1

(51.4))
(51.4

When i is sufficiently
sufficiently small,
small, w
positive (its sign
sign being
same as that
that
When
w' is positive
being the same
of w;
(50.11)), i.e. nu
w is a monotonic
monotonic function
function of
of the time
time t.z. When
of
w ; see (50.11)),
When we
integration over
over w in (51.2),
(51.2), the limits
limits are
change from integration
integration over
over tr to integration
change
unaltered:
unaltered:

I

-IoAdA
~dw.
L1§:l._<}i
do.
AI
D.!=
· 8ow
w ddtdw
tdw
00

|

II

(51.5)

--oo
to

(51.4), we can transform
transform the integral
integral by formally treating
treating w
Substituting (51.4),
nu
complex variable.
variable. We assume
assume that
that the integrand has no singularities
singularities for
as a complex
path of
of integration off the
the real axis into
into the
the upper
and displace
displace the
the path
real w,
nu, and
upper
complex variable.
variable. The
The contour
contour is then
then "caught
"caught up"
up" at the
half-plane of this complex
of the
the integrand, and forms
forms loops
loops round
round them,
them, as shown
shown
singularities of
singularities
singularity nearest
nearest the real axis, i.e.
schematically in Fig. 56. Let
schematically
Let w
zoo0 be the singularity

Qi)

We

FIG.
FIG.

56
56

§51

Accuracy
of conservation of
of the adiabatic invariant
invariant
Accuracy of

161
161

one with the
the smallest
smallest (positive)
(positive) imaginary
imaginary part.
The principal
contributhe one
part. The
principal contribucomes from the neighborhood
neighbourhood of
of this point,
integral (51.5) comes
tion to the integral
point, and
term in the series (51.4) gives a contribution
contribution containing
containing a factor
each term
retaining only the term with
the negative
negative exponent
exponent of
of
exp( -lim
exp(-I
im w
to).
Again retaining
with the
0 ).
•
•
•
I
smallest
smallest magnitude
magmtude (i.e.
(t.e. the term
term with
wtth Il =
= 1),
1), we Mundt
findt
!1I
,....., exp(
exp(im to).
w 0 ).
AI Rx.:
- in

(51.6)

Let to
t 0 be the (complex)
(complex) "instant" corresponding
corresponding to the
the singularity
singularity to:
w0 :
Let
w(t0)
to.
w(t 0 ) =
=w
it0 1 has the same
same order of
of magnitude
magnitude as the
the charac0 • In general, ltol
variation of
of the
the parameters of
of the system.j[
system.! The order of
time 'cr of variation
teristic time
of the exponent
exponent in (51.6)
(51.6) is
magnitude of
magnitude
.

imw
im
to0

wr,...., I/
rfT.
T.

,....,
"*-'(D'C

(51.7)

Since we assume
assume that rr >
}> T, this
this exponent
exponent is large. Thus the
the difference
difference
Since
- I..
I_ decreases
decreases exponentially
exponentially as the rate
rate of variation
variation of
of the
the parameters
lI+
+ parameters
of
of the system
system decreases.ll
decreases. II
To determine
determine w
the first
approximation with respect
respect to Tl-r
Tf-r (i.e.
To
We0 in the
First approximation
1
retaining only
only the term ,...., (T/r)'
(T/r)- 1 in the exponent), we can omit
omit from (50.11)
retaining
the small
small term
term in i:
the
A:
(51.8)
dwldt
dwfdt == co(I,
w(I, ,1(¢)),
A.(t)),

I

i v

and the
the argument
argument I of
of the function
function co(I,
w(I, A.)
taken to have a constant value,
A) is taken
L. Then
say I_.
We0 =-w
=

'•

to

w(I, 1(0)
t;a
A.(t)) ddt;_

(51.9)

the lower
lower limit
limit may be taken
taken as any real value of
of Z,
t, since
since it does not
not affect the
the
the
required
imaginary part
part of 'w0.§
w 0 .§
required imaginary
integral (51.5)
(51.5) with w
from (51.8)
(51.8) (and
(and with one
one term from
from the series
series
The integral
nu from
oAfow) becomes
(51.4) as 6A/020)

tJ.I
AA I ~,.....,r re
e

'\

I ..

iefw
ze•w

Z do •
idw
co(I,
w(I, A)
A.)

(51.10)

Hence we see that
that the singularities
singularities that are in question
question as regards the nearest
nearest
Hence
singularities (poles
(poles and branch
branch points) of the
the functions
functions
to the real axis are the singularities
In special
special cases
cases itit may
may happen
happen that
that the
the expansion
expansion (51.4)
(51.4) does
does not
not include a term
term with
'I't In
we must
example, Problem I1 at
at the
the end
end of
of this
this section),
section); in
in every
every case, we
must take
Il = 11 (see, for
for example,

present in
in the
the series.
series.
ll present
If the
the slowness
slowness of
of variation of
of the
the parameter 2.
A. is expressed
expressed by
by its
its depending on
on tt only
If
through
a ratio
ratio C
~ = tlt'
tf'r: with r1: large,
large, then to
t 0 = 1:60,
1:~ 0 , where 60
~ 0 is a singularity of
of MC)
A.(~) that is
ha
throug
independent of
of r.1:.
independent
) , then not
Note that,
that. ifif the initial
initial and
and Final
final values
values of
of 10)
A.(t) are
are the
the same
same (,1+
(A.+ = L
A._),
not only
II Note
ferto Hue AI
the dif
diffe~nce
AI but
but also
also the difference
difference liE
E+ - E_
E_ of
of the
the Final
final and
and initial energies
energies are
AE = E+
onent a dyea1ggexp
exponentially
s~all: from
from (499),
(49.9), AE
AE = (ro
AI in
in that
that case.
case.
D AI
re deta•l.-d proof of th
§
more
· of
·
·
§ A mo
· ·1 e b proof
A
ese statements,
statements, and
and aa calculation
calcu1attOn
of the
the coefficient
coefficient
o f the
the exponential
exponenttal
of 1h686
of
• (51
(51 -6)'
6)
' is
IS g ve
v -n
y A.• A.
A. Slutsk"
•
ru
by
sk
n
in
in
gi
,
iet Physics
tn
•
In, Sov
Somet
Phys1cs IL..75
JETP
676, 1964,
1964.
Tp 18, 676,
the term
term with
with the
the lowest
lowest value
value of
of
the

It

.

_

-

_

_

162

§52

The
The Canonical
Canonical Equations
Equations

A(t) and 1/w(t). Here it should
should be remembered
conclusion that !11
Ag)
remembered that the conclusion
AI is
exponentially small depends on
on the hypothesis that
that these functions
functiflns have no
real singularities.
PROBLEMS

PROBLEM
Al for a harmonic
harmonic oscillator
PROBLEM 1. Estimate
Estimate ,1/
oscillator with a frequency
frequency that
that varies slowly
slowly according to
from (D_
ro_ =
= (00
w 0 for
fort=
-CO to
to
from
-oo
t

w 2 ==
=
Hz

II

aerJ.r
++ as"

(JJ 2 --,--,--,0)02

0

e"'
I + e"

= \/aw
yqw 0 for
fort=
co (a>
~w 0 ).t
=
t = OO(a
> 00,, a <<w(})-t

OJ+
(0+

).,(a

1)
+
+

parameter 2.
A. the frequency
co itself,
itself, we have
SOLUTION. Taking as the parameter
frequency (U

_i =

_

a

'a

1

II .

~ = 2a e"'
e tc -|·1- a
a - e""
e :xr
CL)

This function
has poles
poles for
e-at=
This
function has
for e
-at =

at
and ee-at=
-a. Calculating
Calculating the
the integral
integral
--1
1 and
-a.

dt,
lfw, dt,

we End
find that
that the
the smallest
smallest value
value of
of in
im to
rc 0 comes
comes from
from one
one of
of the
the poles
poles air
cxt 0 == -log
-log ((-a),
we
-a),

and
and

is
is

im to
u• 0 == to
oJ 0 n,la
n 1·a for
for a >
> I,
l,
=
1/a
= w
l·J0
nva
0n

form
"for
a <1
< I1..

fx

For aa harmonic
harmonic oscillator,
oscillator, A
A ~ sin
sin 211:
2u (see
(see §50,
§SO, Problem),
Problem), so
so that
that the
the series
series (51.3)
(51.3) reduces
reduces
For
to two
two terms
terms (with Il = i 2). Thus, for a harmonic oscillator,
oscillator,
to

±

Al »
e x p((-Z
m to).
:0./
~- -exp
2 iim
«"o).

2. A
particle oscillates
oscillates in
in aa potential
potential well.
well. Determine
Determine how
how its
its energy
energy varies
varies under
under aa
PROBLEM 2.
A particle
-cx.X with
a small
small coelhcient
coefficient aa (x
(x being
being aa Cartesian
Cartesian co-ordinate).
co-ordinate).
frictional l`orcej,=
force f.= -at
frictional
with a
average (25.1
(25.13)
over the
the oscillation
oscillation period,
period, neglecting
neglecting damping
damping in
in the
the first
first
SOLUTION. We
We average
3) over
approximation. Then
Then
approximation.

dE
dE

::1

a

BE_ - - = --a
o f, =
dt
dt

T

i

-

T
T2

a

.ri· dt=- -

U
0

7

f_

Qua
2na

_
_

.i dd1·=
--l!E),
.x
r= l(E),
my
m7

where l(E)
/(E) is
is the
the adiabatic
adiabatic invariant
invariant and
and m the
the mass
mass of
of the
the particle.
particle. Expressing
Expressing the
the oscillation
oscillation period
period
T
in terms
Tin
terms of
of I by (49.8) gives

lntegrating, we
have
Integrating,
we have

f
-id/
_ -dl?

dE
dE

dt
dr

-a!/m.
= *arm.

exp(-at/m)
I/(E)=J(£
(E) =l(E0)
-it/rrz)
0 ) exp(

(1)

This
This formula implicitly determines
determines EU),
E(t). For
For a harmonic oscillator,
oscillator, itit becomes (25.5l.
(25.5). The
The solution
ifaTfm<<
is valid if
aT]m << I1..

§52. Conditionally periodic motion
Let us consider
consider a system with
with any number
number of degrees
degrees of
of freedom,
freedom, executing
executing
Let
mot\on finite
finite in all the co-ordinates, and
and assume
assume that
that the
the variables
variables can
can be
a motion
r,
the Hamilton-]acobi
Hamilton-Jacobi treatment.
treatment. This means
means cha
that,
completely separated in the
completely
. naturt: of the oscillator
.
.
The hannonic
harmonic
is
shown by
by
'I't The
nature o f the oscillator as
shown

is independent
independent of
of the
the energy.
energy.
is

- ,
f r e q u enay
lation frequency
the fact
fact that
that the
the oscil
osci!lanon
the

§52

Conditionally
Conditionally periodic
periodic motion
motion

;L

163

when
when the
the co-ordinates
co-ordinates are appropriately chosen,
chosen, the abbreviated
abbreviated action
action
written in the form
can be written

So
SO

=

Si(qt).
$i(Qi)»

i

=oSofoqt = ds5]dqi,
dStfdqt, each function
function

as
as a sum
sum of
of functions
functions each
each depending on only
only one co-ordinate.
co-ordinate.

once the generalised
Di
Since
generalised momenta
momenta are Pt
St can be written
II

Si

(sz.
(52.1)
1)

i

as0!Iagi =

Pi dpi.

(52.2)

These are many-valued
many-valued functions.
functions. Since
Since the motion
motion is finite, each
each co-ordinate
co-ordinate
can
can take values
values only
only in a finite range.
range. When Qi
qt varies
varies "there and
and back" in this
this
range, the action
action increases
increases by

where
who
re

ASK = AS; = 27715,

(52.3)

Pi

(52.4)

I;

rrJ

dqillZq

=

2:

235k(qk,
I),18Ii,
oSk(qk.I)foh

II

b*S0(q, I)Joii
I)/31;
= oSo(q,

II

wt
We

Ill

variation of Qi
qi just
taken over the variation
the integral being taken
just mentioned Tt
Let us now effect a canonical
canonical transformation
transformation similar
similar to that
that used
used in §50,
Let
for the case of
of a single
single degree
degree of
of freedom.
freedom. The
The new
new variables
variables are "action varivariIt and
and "angle variables"
ables" If
k

(52.5))
(52.5

where the generating function is again
again the
the action expressed as a function
function of
of
where
and the»~
the- lih The
The equations
equations of
of motion
motion in these
these variables
variables are
the co-ordinates and
N9 w,.
w•,. =
i;/ I- == 0,
= é'E(I)I6II,
iJE(I)fiJI, which give
IIt =
= constant,

3E(Ul3Ii]r -|constant.
= [[oE(l)/oit]t
+constant.

(52.6})
(52.6

(52.7)
(52.7)
analogously
to (50.'7),
(50.7), that
that aa variation'
"there and
and back"
back" of
of
We also find, analogous
ly to
variation "there
the co-ordinate
wt::
co-ordinate Qi
qt corresponds
corresponds to a change
change of
of 2'Zn
1T in Wi:
Wt
We:

We

277.

(52.8))
(52.8

other words, the
the quantities Wt(q,
I) are many-valued
many-valued functions
functions of the co
co-In other
WM, 1)
when the
the latter vary
vary and
and return to their
their original values,
values, the
the Wt
ordinates: when
We

should
emphasised, however, that this refers to
to the
the formal
formal variation of
of the
the coco1't ItIt shou
ld be emphasised,
f
ordinate 9•
over the
the whole
whole possible
possible range of
of values,
values, not
not to
to its variation during the
the period o
of
ordinate
QUO over
the
the actual
actual motion
motion as
as in
in the
the case
case of
of motion
motion in
in one
one dimension.
dimension. An
actual Flnite
finite motion
motion of
of aa
An actual
tem with
system
with S€V€1'a
s-:veral1 deg
degrees
of freedom
freedom not
not only is not
not in
in general periodic as a whole, but
rees of
SYS
not
does not evell
even involve
•nvolve 8a periodic
periodic time
time variation
variation of
of each
each co-ordinate
co-ordinate separately
separately (see
(see below).
below).
does

164

§sz
§52

The
The Canonical Equations
Equations

may vary
vary by any
any integral
integral multiple of
of 271.
2'1T. This property may
may also be formulated
formulated
as a property
property of the function
function wi(p, q), expressed
expressed in terms of
of the co-ordinates
and momenta, in the phase
phase space
space of
of the system. Since
Since the Ii,
J,,, expressed
expressed in
and
terms of
of P
p and
and q, are one-valued
one-valued functions,
functions, substitution
substitution of Ii(p,
wi(q, 1)
I)
Ii(/), q) in Wi(f2)
gives a function
function "i(P9
wi(p, q) which
which may vary by any integral
integral multiple
multiple of Zn
2'1T
(including
passing round
(including zero) on passing
round any closed
closed path
space.
path in phase
phase space.
Hence it follows that
that any one-valued
one-valued functions
function t F(p, Q)
q) of
of the state of
of the
system, if expressed
expressed in terms of
of the canonical
canonical variables,
variables, is a periodic function
function
of
period in each
of the angle
angle variables,
variables, and
and its period
each variable
variable is 27rr.
2'1T. It can be expanded
series;;
as a multiple Fourier series
=

Z
L ...l,=-oo
L I1¢1z2~~-z.
Al lcl• exp(i(hw1 -I+ ... ++ laws)}»
lsws)},
l,=-«>
oo
00

1

Ll,=-X>
to

II

F
F

co
oo

CXP{i(l1'Z»U1

/1, l2,
/2, ...,
where h,
... , is
ls are integers. Substituting
Substituting the angle variables as functions
of
of time, we find that the time dependence
dependence of F is given by a sum
sum of
of the form

F
F

=

E
2: ......l,=-«>
2: AI1;2...I.
A1 1 -l.
§
l,~-«> l,=-OO
co

l1=--00

oo
co

1 2 ••

(

exp
exp { it (

N

BE
BE
zlroE
oE)} :
+ Ils+ ... +
1311+

oh

8

ois
3/8
_

(52.9)

this sum
sum is a periodic
function of time, with
with frequency
frequency
Each term in this
periodic function

lim
l 1w 1 + ... + lsc0s,
l,w,,

sum of
of integral
integral multiples
multiples of the fundamental
fundamental frequencies
frequencieS
which is a sum

w,.

'Up

=

1i.
aEfai,..
6E/6

(52.10)
(52.11)

Since the frequencies
frequencies (52.10)
(52.10) are not
not in general
general commensurable,
commensurable, the sum
Since
itself is not
not a periodic
periodic function, nor, in particular, are the co-ordinates q and
itself
momenta P
p of
of the system.
system.
momenta
motion of
of the system is in general
general not
not strictly
strictly periodic
periodic either as a
Thus the motion
whole or in
in any
any co-ordinate. This means
means tha.t, having
having passed
given
whole
passed through a given
not return to that state in a finite time. We
We can say,
say,
the system
system does
does not
state, the
that in
in the
the course of
of a sufficient
sufficient time the system passes
passes arbitrarily
arbitrarily
however, that
however,
given state. For
For this
this reason
reason such
such a motion
motion is said
said to be conditionally
close to the given
periodic.
Periodic.
In certain particular cases, two
two or more
more of the fundamental
fundamental frequencies
In
w,. are commensurable
commensurable for arbitrary
arbitrary values
values of
of the /I,.. This is called
called degeneracy,
degeneracy,
co,
frequencies are commensurable, the
the motion of
of the system is said
said
and if all s frequencies
degenerate. In the latter
latter case the motion
motion is evidently
evidently periodic,
to be completely degenerate.
periodic,
of every particle
particle is closed.
and the path
path of
Rotational co-ordinates
co-ordinates go
</>(see
the second
second footnote
to §49)
§49) are
are not
not in
in one-to-one
one-to-one relation
relation
Tt Rotational
(see the
footnote to
es of
with the
the state
state of
of the
the system,
system, since
since the
the position
position o
off the
the latter
latter is
is the
the same
same for
for all
all Valu
values
of ¢,
</>
with
differing by
by an
an integral
integr-al multiple
multiple of
of 2'rr.
27T. IfIf the
the co-ordinates
co-ordinates qq include
include such
such angles,
angles, therefore,
therefore,
diHleril"lg
which
these can
can appear
appear in
in the
the function
function F(P,
F(p, q) only
only in
in such
such expressions
expressions as
as cos
cos ¢4> and
and sin
sin ¢.
¢>, which
these
are in
in one-to-one
one-to-one relation
relation with
with the
the state
state of
of the
the system.
system.
are

§52

16
1655

Conditionally periodic
periodic motion
motion
Conditionally

The existence
existence of
of degeneracy
degeneracy leads,
leads, first of all, to a reduction
reduction in the number
number
The
independent quantities
quantities If
It on which the energy
energy of the
the system
system depends.
depends.
of independent
If two frequencies
If
frequencies w1
w1 and we
w2 are such
such that
that
N13El3_I1

(52.12)

3El3I/6 - ZU}¢3El3]g.

(52.13)

II

7123E/312,

where al
n1 and
and 722
n2 are integers,
integers, then
then it follows that
that 11
hand
h appear
appear in the energy
energy
where
and 12
only as the sum
n2/1++ n1I2.
sum n2h
nd2.
A very
very important
important property
property of degenerate
degenerate motion
motion is the increase in the
number of
number for a general
of one-valued
one-valued integrals
integrals of the motion
motion over
over their
their number
general
non-degenerate system
system with
with the same number
number of degrees
degrees of
of freedom.
freedom. In the
non-degenerate
latter case, of
of the 2s-2s-11 integrals of
of the motion,
motion, only
only s functions of the state
of the system
system are one-valued,
one-valued; these may be, -for
for example,
example, the
the s quantities
quantities In.
h
of
remaining ssintegrals may
may be written
written as differences
differences
The remaining
- 1 integrals
'Gui

The constancy
constancy of these
these quantities follows immediately
immediately from formula
formula (52.'7),
(52.7),
The
but
but they
they are not
not ohe-valued
one-valued functions
functions of
of the state of
of the system,
system, because
because the
not
one-valued.
angle
variables
are
angle
not
When there is degeneracy,
degeneracy, the
the situation is different.
different. For
For example,
example, the relaWhen
tion
(52.12)
shows
that,
although
the
integral
tion
shows
although the integral

(52.14)

201712 - '*W 27*1

not one-valued,
one-valued~ it
it is so except
except for the
the addition of an arbitrary integral
integral
is not
2'1T. Hence we need
need only
only take a trigonometrical function of this
of 2»rr.
multiple of
obtain a further one-valued
one-valued integral
integral of the
the motion.
quantity to obtain
example of degeneracy
degeneracy is motion
motion in a field U =
= - mfr
cxfr (see Problem).
Problem).
An example
consequently a further one-valued
one-valued integral
integral of the motion
motion (15.17)
There is consequently
besides the two
two (since
(since the
the motion
motion is two-dimensional)
two-dimensional)
peculiar to this field, besides
ordinary one-valued
one-valued integrals,
integrals, the angular
angular momentum
momentum M and
and the energy
energy E,
ordinary
which
hold for motion
motion in any central
central field.
which hold
It
noted that
that the existence
existence of further
further one-valued
one-valued integrals
integrals
It may also be noted
of degenerate
degenerate motions;
motions; they
they allow
allow a complete
complete
leads in turn to another property of
leads
of the
the variables
variables for several
several (and
(and not
only one'l)
onet) choices
choices of
of the coseparation of
not only
ordinates. For
For the quantities Ii
J,, are one-valued
one-valued integrals
integrals of the motion
motion in
ordinates.
allow separation of the variables.
variables. When
When degeneracy
degeneracy occurs,
occurs,
co-ordinates which
which allow
of one-valued
one-valued integrals
integrals exceeds
exceeds s, and
and so the choice of those
those
the number of
11 is no longer
longer unique.
desired I,
which are the desired
unique.
example, we may again mention
mention Keplerian
Keplerian motion,
motion, which
which allows
allows
As an example,
separa
tion of the varia
co-ordinates.
separation
variables
bles in both spherical and parabolic co-ordinates.
re
We igno
ignore
t We

such
such triv
trivial
cl~anges
. t h e co-ordinates
ial cha
nges in
1 n the
co-ord"mates

as QUO'
ql ,
as

=

q1 '(q1 ) , QUO'
q2, =
= q2'(q2)q2'( q2) .
q1'(q1).

166

The Canonical
Canonical Equations
Equations
The

§52

finite motion
motion in one dimension,
dimension, the
In §49 it has been shown that, for finite
an adiabatic
adiabatic invariant. This statement holds
holds also for systems
systems
variable is an
action variable
proved, ih the general
It can
than one degree
degree of
of freedom.
freedom. It
can be proved,
general case,
with more
more than
with
by a dire
direct
generalisation of
of the method
method given
given at the beginning of
of §51.
ct generalisation
For a multi-dimensional system with
with a variable
variable parameter A(z),
A(t), the equaFor
of motion in canonical
canonical variables
variables give
give for the rate
rate of variation
variation of each
each
tions of
action variable
variable Ii
I; an expression analogous
analogous to (50.10):
action

.

aA.
c8A

a a,

]. = - - A ·
I

aw;
W·I ~·

15)
(sz.
(52.15)

= (050
(aS 0 (aA}I.
equation is to be averaged over a time
I3g), This equation
where, as before, A =
but
fundamental periods of
of the system but
interval large
large compared
compared wide
with the fundamental
interval
small compared
compared with the time
time of variation
variation of l(t).
A(t). The
The quantity
quantity .AA is again
small
taken outside the mean
mean value,
value, and
and the derivatives
derivatives 6Aldw,
aAJaw.- are averaged
averaged as
taken
if the motion
motion took place
place at constant A, as a conditionally periodic motion.
motion.
if
nu,-, and the mean
function of the angle variables
variables w;,
mean
periodic function
Then A is a unique periodic
values of its derivatives
derivatives 6A]6w,aAJaw.- are zero.
values
Finally, we-may
we--may briefly discuss
discuss the properties of finite motion
motion of closed
closed
Finally,
where the varisystems with
with s degrees
degrees of
of freedom
freedom in the
the most
most general
general case, where
systems
Hamilton-Jacobi equation are not
not assumed
assumed to be separable.
ables in the Hamilton-]acobi
The fundamental
fundamental property of
of systems with
with separable
separable variables
variables is that
that the
The
integrals of the motion
motion If,
h whose
whose number
number is equal
equal to the number
number of degrees
degrees
integrals
of freedom,
freedom, are one-valued.
one-valued. In the general
general case where
where the variables
variables are not
not
of
however, the
the one-valued integrals of the
the motion include only
only
separable, however,
homogeneity and
a.<d isotropy
isotropy of space
space
those whose constancy
constancy is derived
derived from the homogeneity
those
and time, namely
namely energy,
energy, momentum and
and angular
angular momentum.
and
The phase
system traverses those
those regions
regions of phase space which
which
path of the system
phase path
The
are defined
defined by the given
given constant values
values of
of the one-valued.integrals
one-valued integrals of the
anJ s one-valued
one-valued integrals,
integrals,
motion. For
For a system with separable variables and
motion.
s-dimensional manifold
manifold in phase
space. During a
these conditions define an s-dimensional
phase space.
these
path of
of the system passes arbitrarily close
close to every
every point
sufficient time,
time, the path
sufficient
hypersurface.
on this hypersurface.
In a system where
where the variables
variables are not
not separable,
·separable, however,
however, the number
In
path occupies,
of one-valued
one-valued integrals
integrals is less than
than s, and
and the phase
phase path
occupies, completely
completely
of
manifold of
of more
than s dimensions
dimensions in phase
phase space.
more than
partly, a manifold
or partly,
more than
degenerate systems,
systems, 011
on the other hand,
hand, which
which have more
than s integrals
integrals
In degenerate
of the motion,
motion, the phase
phase path
occupies a manifold
dimensions.
manifold of fewer than s dimensions.
path occupies
of
If
If the Hamiltonian
Hamiltonian of the system
system differs only by small terms from one which
motion are close
variables, then the properties of the motion
allows separation of the variables,
to those of
of a conditionally
conditionally periodic motion, and
and the difference
difference between the
of a much
higher order of smallness
smallness than that of the additional
additional terms in
much higher
two is of
the Hamiltonian.
the

§sz
§52

167

Conditionally
Conditionally Periodic
periodic motion
motion

P
ROBLEM
PROBLEM

U =
Calculate
Calculate the
the action
action variables
variables for elliptic
elliptic motion
motion in a field U
= -a/r.
- afr.

SOLUTION. In
In polar
polar co-otdinates
co-ordinates r,r,
SOLUTION.

l¢
It~> ==

21r

1

the
the plane
plane of
of the
the motion
motion we
we have
have
2"

211

o0

ful

'max

1
If = 2-

TL
T up
I P9 d¢>d</> == M,

go</>in
in

'mln

2

M

d
r
r2
(E+")-m2]

= -M+<rV(#Hl2|E|)-M+av(m/21£1).

7'

Hence
Hence the
the energy,
energy, expressed
expressed in
in terms
terms ooff the
the action
action variables,
variables, is
is EE == -maZ/2(/,»-I-I¢)2.
-ma 2 f2(Ir+I~ 2 • It
It
depen
+I¢, and
fundadepends
only on
on the
the sum
sum IfIr+II/>,
and the
the motion
motion is
is therefore
therefore degenerate,
degenerate; the
the two
two fundads only
mental frequencies
frequencies (in
(in rr and
and in
in 95)
</>) coincide.
coincide.
mental
The parameters
parameters pp and
and ee of
of the
the orbit
orbit (see
(see (1S.4))
(15.4)) are
are related
related to
to IfIr and
and 1¢
II/> by
by
The

p

Z

l¢2

ma

!

£2

e2 =

(

It~>
11- ( It~>+Ir
l¢
I¢+Ir

V

)2•

Since IfIr and
and III/> are
are adiabatic
adiabatic invariants,
invariants, when
when the
the coefficient
coefficient aa or
or the mass
mass m
m
varies sslowl:
Since
varies
jowl\
•
Q#
•
. unchanged,
•
.menslons
•
.inverse
the eccentricity
eccentricity of
of the
the orblt
orbit remains
remains unchanged,
its d1
dimensions
vary in
in Inverse
propor·
the
while its
vary
proportion to
to aa and
and to
to m.
m.
tion
o

O

U

I

IINDEX
NDEX
Acceleration,
eration, 1
Accel
Action, 2, 138f1`.
138ff.
abbreviated, 141
141
abbreviated,
157
variable, 15'7
Additivity of
angular momentum,
momentum, 19
19
angular
energy, 14
14
energy,
integrals of the
the motion,
motion, 13
13
integrals
Lagrangians, 44
Lagrangian,
mass, 1'7
17
mass,
momentUITI, 15
15
momentmn,
Adiabatic invariants,
invariants, 155,
155, 159th.,
159ft., 165
165
Adiabatic
Amplitude, 59
complex, 59
59
complex,
Angle variable,
157
Angle
variable, 157
Angular
momentum, 19ff.
19ff.
Angular momentum,
105ff.
of rigid body, 105H`.
Angular
97f.
Angular velocity, 9'7f.
Area integral,
integral, 31n.
31n.
Area

force, 128
Coriolis force,
109
Couple, 109

Brackets, Poisson,
Poisson, l35ff.
135ff.
Brackets,

Eccentricity, 36
36
Eccentricity,
Eigenfrequencies, 6'7
67
44
Elastic collision, 44

Beats,
Beats,

63

Canonical equations
equations (VII),
(VII), 131ff.
131ff.
Canonical
Canonical
Canonical transformation,
transformation, 143ff.
143ff.
Canonical variables,
variables, 157
157
Canonical
Canonically conjugate
conjugate quantities,
quantities, 145
145
Canonically
Central
Central Held,
field, 21, 30
in, 30H`.
30ff.
motion in,
Centrally symmetric
symmetric Held,
field, 21
21
Centrally
Centre of
of Held,
field, 21
21
Centre
Centre of
of mass,
mass, 17
17
Centre
system,
41
system, 41
Centrifugal force,
128
Centrifugal
force, 128
Centrifugal potential,
potential, 32,
32, 128
128
Centrifugal
Characteristic equation,
equation, 67
Characteristic
Characteristic frequencies,
frequencies, 67
67
Characteristic
system, 8
Closed system,
between particles (IV),
(IV), 41ff.
Collisions between
elastic, 4-4ff.
44ff.
elastic,
Combination frequencies,
frequencies, 85
integral, 148
Complete integral,
164
Conditionally periodic
periodic motion,
motion, 164
Conditionally
Conservation laws (II),
(II), 13H`.
13ff.
Conservation
Conservative systems,
systems, 14
Conservative
Conserved quantities,
quantities, 13
13
Conserved
Constraints, 10
10
Constraints,
equations of,
of, 123
equations
holonomic, 123
123
holonomic,
Co-ordinates, 1
Co-ordinates,
cyclic, 30
30
cyclic,
generalised,
generalised, llff.
f.
normal, 68
68£.
normal,
f.

Cross-section, effective,
effective, for
scattering,
Cross-section,
for scattering,
49ff.
49f'f.
C system,
system, 41
41
Cyclic to-ordinates,
co-ordinates, 30
Cyclic

d' Alembert's principle, 124
d'Alembert's

Damped oscillations,
Oscillations, '74ff.
74ff.
Damped
Damping
Damping
aperiodic, '76
76
aperiodic,
coefficient, '75
coefficient,
75
decrement,
decrement,

75
'75

164£.
Degeneracy, 39,
39, 69,
69, 164f.
Degeneracy,
complete, 164
164
complete,

freedom, 1
Degrees of freedom,
Disintegration of particles, 41ff.
Dispersion-type absorption,
absorption, 79
79
Dispersion-type
Dissipative function,
76f.
Dissipative
function, 76f.
DUITimy suffix, 99n.
Dummy

Elliptic functions,
functions, 118f.
Elliptic
Elliptic integrals,
integrals, 26, 118
Elliptic
Energy, 14,
14, 25f.
25f.
Energy,
Centrifugal, 32, 128
centrifugal,
internal, 17
17
internal,
see Kinetic energy
kinetic, see
see Potential energy
potential, see
Equations of
of motion
motion (I),
(1), 1ff.
Equations
lf.
canonical (VII), 131ff.
canonical
of (III), 2.5ff.
25ff.
integration of
of rigid body,
body, 10'7ff.
107ff.
of
Eulerian angles, 110ff.
Euler's equations, 115, 119
Finite motion, 25
Finite

Force, 99
Force,
generalised, 16
16
generalised,

Foucault's pendulum, 129f.
Foucault¢'s
Frame of
of reference,
reference, 44
Frame
inertial, sf.
Sf.
inertial,
non-inertial,
non -inertial, 126ff.
126ff.

Freedom, degrees
degrees of,
of, 11
Freedom,
Frequency, 59

circular, 59
59
circular,

combination, 85
85
combination,

Friction, 75,
75, 122
122
Friction,

transformation, 6
Galilean transformation,.6
Galilean

Galileo's relativity
relativity princlp1¢»
principle, 66
Galileo's
I11\R
OR

Index

169

Mechanical similarity,
similarity, 22ff.
22ff.
Mechanical

General integral,
integral, 148
148
General
Generalised
Generalised
co-ordinates,
ff.
co-ordinates, 1
1ff.

Molecules, vibrations
of, 70ff.
70ff.
Molecules,
vibrations of,

Moment
of
of force, 108

forces, 16
forces,

momenta,
momenta, 16
velocities, l1ff.
velocities,
f.

of inertia,
inertia, 99ff.
99ff.
of

Principal,
00ff.
principal, l100ff.

Generating function,
function, 144
Generating

Momentum,
Momentum, 15f,
1 Sf.

Hamilton-Jacobi
147ff.
HamiltonnlIacobi equation, 14'7ff.
Hamilton's equations,
equations, 132
Hamilton's
Hamilton's function,
function, 131
Hamilton's
Hamilton's principle, 2ff.
Holonomic constraint, 123
Holonomic

Multi-dn11ensional motion,
motion, l162ff.
Multi-dimensional
62ff.

angular, see Angular
momentum
angular,
Angular momentum
generdised,
generalised, 16
16

Half-width, 79
79
Half-width,
Hamiltonian,
Hamiltonian, 131f.
131£.

Impact parameter, 48
Impact
Inertia
law of, 5
of, 99ff.
99ff.
moments of,
principal,
principal, 100ff.
100ff.
principal axes of, 100
100
principal
tensor, 99
99
tensor,
Inertial frames, 5f.
Sf.
Infinite
Infinite motion, 25
Instantaneous axis, 98
Instantaneous
Integrals
Integrals of
of the
the motion,
motion, 13,
13, 135
135

moment of,
of, see
see Angular
Angular momentum
momentum
moment

Newton's equations,
equations, 99
Newton's

*

Jacobi's identity,
identity, 136
]acobi's

Keple
Kepler's
3Sff.
r's problem, 35ff.
Kepler's second law, 31
Kepler's
Kepler's third law, 23
energy, 8, 15
Kinetic energy,
of rigid body, 98f.
98£.
of

Laboratory system,
system, 41
41
Laboratory
Lagrange's equations,
equations, 3f.
3£.
Lagrange's
Lagrangian, Zff.
2ff.
Lagrangian,

motion, 5
for free motion,
of free
free particle,
particle, 6ff.
6ff.
of
in non-inertial
non-inertial frame,
frame, 127
127
in

for one-dimensional
one-dimensional motion,
motion, 225,
58
for
5 , 58

of rigid
rigid body, 99
of
for small oscillations, 58, 61, 66, 69, 84
of system
system of
of particles, 8ff.
of
of
of two
two bodies, 29

Latus rectum,
rectum, 36
36
Latus

Least action,
action, principle
principle of,
of, 2ff.
2ff.
Least

Legendre's transformation,
transfonnation, 131
131
Legendre's

Liou
ville's theorem,
Liouville's
theorem, 147
147

L system,
system, 41

1\Ta
1\'Iass,
ss, 7
additivity of,
of, 17
17
additivity
Centre
centre of,
of, 17
17
reduced,
uced, 29
red
J.VIathieu's
82n.
hi€u's.c
1uation, 82n.
1vIat
.equation
J.Vtaupcrtu•s principle,
principle, 141
141
nl8upcttuls

.

.

Newton's third
third law,
law, 16
16
Newton's

Nodes, line of, 110
Non-holonomic constraint,
constraint, 123
123
Non-holonomic
Normal co-ordinates,
co-ordinates, 68
68£.
Normal
f.
Normal oscillations, 68
Nutation, 113
113
Nutation,
One-dimensional
One-dimensional motion, 25ff.,
2Sff., 58ff.
Small oscillations
oscillations
Oscillations, see Small
Oscillations,

Oscillator
Oscillator

one-dimensional, 58n.
58n.
one-dimensional,
Rpace, 32,
32, 70
70
space,

Particle,
Par
tile, 1
Pendulums, 11£.,
95,
Pendulums,
l f . , 26, 33ff., 61, 70, 95
102£., 129f.
1o2f.,
compound, 102f.
102£.
compound,
conical,
conical, 34
Foucault's, 129f.
Foucault's,
spherical, 33f.
33£.
spherical,
Perihelion, 36
movement
movement of, 40
Phase, 59
Phase,
path, 146
146
path,
146
space, 146
Point transformation,
transformation, 143
Point
Poisson brackets, 13Sff.
135ff.
Poisson
Poisson's theorem, 137
Poisson's
Polhodes, 117n.
11 7n.
Polhales,
Potential energy,
energy, 8, 15
Potential
centrifugal, 32, 128
centrifugal,
94
effective, 32,
32, 94
effective,
of oscillation,
oscillation, 27ff.
from period of
Potential well, 26, 54f.
54£.
Potential
regular, 107
107
Precession, regular,
Rapidly oscillating Held,
field, motion
motion in,
in, 936
93fl
Rapidly
Reactions, 122
Reduced mass,
mass, 29
29
Reduced
Resonance, 62,
62,
Resonance,

79

in non-linear Oscillations,
os.cillations, 876'87ff.

parametric, 80H`.
80ff.
parametric,

Rest, system
system at,
at, 17
I7
Rest,

of motion,
motion, 99
Reversibility of
Reversibility

170
Rigid bodies, 96

angular momentum
momentum of,
of, 105fl:`.
105ff.
angular
contact, l122ff.
in contact,
22ff.
equations of
of motion
motion of,
of, 107ff.
107ff.
equatlons

motion of (VI), 96fl`.
96ff.
Rolling,
Rolling, 122
Rotator, 101, 106
Rough surface, 122
Routhian,
Routhian, 134f.
134f.
Rutherford's fonnula,
53f.
Rutherford's
formula, 53f.

Scattering,
Scattering, 483.
48ff.
cross-section, effective, 49ff.
Rutherford's formula for, 53f.
55ff.
small-angle, 55ff.
Sectorial velocity,
velocity, 31
Sectorial
of variables, 149ff.
Separation of
Similarity, mechanical, 22ff.
Sliding, 122
Sliding,
Small oscillations,
oscillations, 22, (V) 58ff.
Small
anhannonic, 84ff.
enharmonic,
74ff.
damped, '74ff.
77ff.
forced, 61ff., '7'7ff.
free, 58ff., 65ff.
linear, 84
non-linear, 84fi`.
84ff.
non-linear,
nom1al, 68
nominal,
Smooth surface,
surface, 122

Index
Space
Space

homogeneity of, 5, 15
homogeneity
isotropy of, 5, 18
Space oscillator,
oscillator, 32,
32, '70
70
Space

Time
T
milne
of, 5, 13H.
13ff.
homogeneity of,

isotropy of, 8f.
Sf.
isotropy
Top
Top
asy.;nmetncat, 100,
100, 116ff.
asymmetrical,
ccfastUi9
"fast", 113f.
spherical, 100, 106
symmetrical, 100,
100, 106f.,
1 06f., 111f.
symmetrical,
Torque, 108
Turning points,
points, 25, 32
Turning
Two-body problem, 29
Unifonn field,
field, 10
10
Uniform

Variation, 2, 3
first, 3
First,
Velocity, 1
angular, 9'7f.
97f.
allglllaf,
sectorial, 31
sectoral,
translational,
translational, 97
Virial, 23n.
theorem,
theorem, 23f.
23f.

Well, potential, 26, 54f.