Classical mechanics

499 Pages·2004·2.66 MB·English

by  Tom W B Kibble

C M
lassical echanics
5th Edition

C
lassical
M
echanics
5th Edition
Tom W.B. Kibble
Frank H. Berkshire
Imperial College London
Imperial College Press
ICP
Published by
Imperial College Press
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Library of Congress Cataloging-in-Publication Data
Kibble, T. W. B.
Classical mechanics / Tom W. B. Kibble, Frank H. Berkshire, -- 5th ed.
p. cm.
Includes bibliographical references and index.
ISBN 1860944248 -- ISBN 1860944353 (pbk).
1. Mechanics, Analytic. I. Berkshire, F. H. (Frank H.). II. Title
QA805 .K5 2004
531'.01'515--dc 22 2004044010
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2004 by Imperial College Press
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Printed in Singapore.
To Anne and Rosie
vi
Preface
Thisbook,basedoncoursesgiventophysicsandappliedmathematicsstu-
dents at Imperial College, deals with the mechanics of particles and rigid
bodies. It is designed for students with some previous acquaintance with
theelementaryconceptsofmechanics,butthebookstartsfromfirstprinci-
ples,andlittledetailedknowledgeisassumed. Anessentialprerequisiteisa
reasonablefamiliaritywithdifferentialandintegralcalculus,includingpar-
tial differentiation. However, no prior knowledge of differential equations
should be needed. Vectors are used from the start in the text itself; the
necessarydefinitionsandmathematicalresultsarecollectedinanappendix.
Classicalmechanicsisaveryoldsubject. Its basicprincipleshavebeen
known since the time of Newton, when they were formulated in the Prin-
cipia, and the mathematical structure reached its mature form with the
worksofLagrangeinthelateeighteenthcenturyandHamiltoninthenine-
teenth. Remarkably enough, within the last few decades the subject has
once again become the focus of very active fundamental research. Some
of the most modern mathematical tools have been brought to bear on the
problem of understanding the qualitative features of dynamics, in particu-
larthetransitionbetweenregularandturbulentorchaoticbehaviour. The
fourthedition ofthe book wasextended to include newchaptersproviding
abriefintroductiontothisexcitingwork. Inthisfifthedition,thematerial
issomewhatexpanded,inparticulartocontrastcontinuousanddiscretebe-
haviours. Wehavealsotakentheopportunitytorevisetheearlierchapters,
giving more emphasis to specific examples workedout in more detail.
Manyofthemostfascinatingrecentdiscoveriesaboutthenatureofthe
Earth and its surroundings — particularly since the launching of artificial
satellites — are direct applications of classicalmechanics. Severalof these
are discussed in the following chapters, or in problems.
vii
viii ClassicalMechanics
For physicists, however, the real importance of classical mechanics lies
notsomuchinthevastrangeofitsapplicationsasinitsroleasthebaseon
which the whole pyramid of modern physics has been erected. This book,
therefore,emphasizesthose aspects ofthe subjectwhichareof importance
inquantummechanicsandrelativity—particularlytheconservationlaws,
which in one form or another play a central role in all physicaltheories.
For applied mathematicians, the methods of classical mechanics have
evolved into a much broader theory of dynamical systems with many ap-
plications well outside of physics, for example to biologicalsystems.
The first five chapters areprimarily concernedwith the mechanics of a
singleparticle,andChapter6,whichcouldbeomittedwithoutsubstantially
affecting the remaining chapters, deals with potential theory. Systems of
particles arediscussed in Chapters 7 and 8, and rigidbodies in Chapter 9.
ThepowerfulmethodsofLagrangeareintroducedatanearlystage,andin
simplecontexts,anddevelopedmorefully inChapters10and11. Chapter
12 contains a discussion of Hamiltonian mechanics, emphasizing the rela-
tionship between symmetries and conservation laws — a subject directly
relevant to the most modern developments in physics. It also provides the
basis for the later treatment of order and chaos in Hamiltonian systems in
Chapter14. Thisfollowstheintroductiontothegeometricaldescriptionof
continuous dynamical systems in Chapter 13, which includes a discussion
ofvariousnon-mechanicsapplications. Appendices fromthe fourthedition
on (A) Vectors, (B) Conics, (C) Phase-plane analysis near critical points
aresupplemented hereby a newappendix (D) Discrete dynamicalsystems
— maps.
Thewritingofthefirsteditionofthisbookowedmuchtotheadviceand
encouragement of the late Professor P.T. Matthews. Several readers have
alsohelpedbypointingouterrors,particularlyintheanswerstoproblems.
We are grateful to the following for permission to reproduce copyright
material: Springer–VerlagandProfessorOscarE.LanfordIIIforFig.13.20;
Cambridge University Press for Fig. 13.22 and, together with Professors
G.L. Baker and J.P. Gollub, for Fig. D.5; Institute of Physics Publishing
Limited and Professor M.V. Berry for Figs. 14.11, 14.12 and 14.13; the
RoyalSociety and ProfessorsW.P. Reinhardtand I.Dana for the figurein
the answer to Appendix D, Problem 13.
T.W.B. Kibble
F.H. Berkshire
Imperial College London
Contents
Preface vii
Useful Constants and Units xv
List of Symbols xvii
1. Introduction 1
1.1 Space and Time . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Newton’s Laws . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 The Concepts of Mass and Force . . . . . . . . . . . . . . 10
1.4 External Forces . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2. Linear Motion 17
2.1 Conservative Forces; Conservationof Energy . . . . . . . 17
2.2 Motion near Equilibrium; the Harmonic Oscillator . . . . 20
2.3 Complex Representation . . . . . . . . . . . . . . . . . . 24
2.4 The Lawof Conservationof Energy . . . . . . . . . . . . 25
2.5 The Damped Oscillator . . . . . . . . . . . . . . . . . . . 27
2.6 Oscillator under Simple PeriodicForce . . . . . . . . . . 30
2.7 General Periodic Force . . . . . . . . . . . . . . . . . . . 34
2.8 Impulsive Forces; the Green’s Function Method . . . . . 37
2.9 Collision Problems. . . . . . . . . . . . . . . . . . . . . . 39
2.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
ix