Classical Mechanics

610 Pages·2006·6.42 MB·English

CLASSICAL MECHANICS
Gregory’s Classical Mechanics is a major new textbook for undergraduates in mathe-
matics and physics. It is a thorough, self-contained and highly readable account of a
subject many students find difficult. The author’s clear and systematic style promotes a
good understanding of the subject: each concept is motivated and illustrated by worked
examples,whileproblemsetsprovideplentyofpracticeforunderstandingandtechnique.
Computer assisted problems, some suitable for projects, are also included. The book is
structured to make learning the subject easy; there is a natural progression from core
topics to more advanced ones and hard topics are treated with particular care. A theme
of the book is the importance of conservation principles. These appear first in vectorial
mechanicswheretheyareprovedandappliedtoproblemsolving. Theyreappearinana-
lytical mechanics, where they are shown to be related to symmetries of the Lagrangian,
culminatinginNoether’stheorem.
•Suitableforawiderangeofundergraduatemechanicscoursesgiveninmathematicsand
physicsdepartments: nopriorknowledgeofthesubjectisassumed
•Profuselyillustratedandthoroughlyclass-tested,withacleardirectstylethatmakesthe
subjecteasytounderstand: allconceptsaremotivatedandillustratedbythemanyworked
examplesincluded
• Good, accurately-set problems, with answers in the book: computer assisted problems
and projects are also provided. Model solutions for problems available to teachers from
www.cambridge.org/Gregory
The author
Douglas Gregory is Professor of Mathematics at the University of Manchester. He is a
researcher of international standing in the field of elasticity, and has held visiting posi-
tions at New York University, the University of British Columbia, and the University of
Washington. He is highly regarded as a teacher of applied mathematics: this, his first
book,istheproductofmanyyearsofteachingexperience.
Bloodyinstructions,which,beingtaught,
Returntoplagueth’inventor.
SHAKESPEARE,Macbeth,actI,sc.7
Front Cover The photograph on the front cover shows Mimas,
one of the many moons of Saturn; the huge crater was formed
by an impact. Mimas takes 22 hours 37 minutes to orbit Sat-
urn,theradiusofitsorbitbeing185,500kilometers.Afterreading
Chapter 7, you will be able to estimate the mass of Saturn from
thisdata!
CLASSICAL MECHANICS
AN UNDERGRADUATE TEXT
R. DOUGLAS GREGORY
UniversityofManchester
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge cb2 2ru, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Informationo nthi stitle :www.cambri dge.org/9780521826785
© Cambridge University Press 2006
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2006
isbn-13 978-0-511-16097-4 eBook (EBL)
isbn-10 0-511-16097-6
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isbn-13 978-0-521-82678-5 hardback
isbn-10 0-521-82678-0 hardback
isbn-13 978-0-521-53409-3
isbn-10 0-521-53409-7
Cambridge University Press has no responsibility for the persistence or accuracy of urls
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Contents
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..xi
1Newtonianmechanicsofasingleparticle1
1Thealgebraandcalculusofvectors3
1.1Vectorsandvectorquantities.... ....................3
1.2Linearoperations: a+bandλa . . . . . . . . . . . . . . . . . . . ..5
1.3Thescalarproduct a·b . . . . . . . . . . . . . . . . . . . . . . . . ..10
1.4Thevectorproduct a×b . . . . . . . . . . . . . . . . . . . . . . . ..13
1.5Tripleproducts........... ....................15
1.6Vectorfunctionsofascalarvariable. . . . . . . . . . . . . . . . . . ..16
1.7Tangentandnormalvectorstoacurve. . . . . . . . . . . . . . . . . ..18
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..22
2Velocity,accelerationandscalarangularvelocity25
2.1Straightlinemotionofaparticle. . . . . . . . . . . . . . . . . . . . ..25
2.2Generalmotionofaparticle. . . . . . . . . . . . . . . . . . . . . . ..28
2.3Particlemotioninpolarco-ordinates. . . . . . . . . . . . . . . . . ..32
2.4Rigidbodyrotatingaboutafixedaxis...................36
2.5Rigidbodyinplanarmotion.... ....................38
2.6Referenceframesinrelativemotion. . . . . . . . . . . . . . . . . . ..40
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..43
3Newton’slawsofmotionandthelawofgravitation50
3.1Newton’slawsofmotion. . . . . . . . . . . . . . . . . . . . . . . . ..50
3.2Inertialframesandthelawofinertia. . . . . . . . . . . . . . . . . . ..52
3.3Thelawofmutualinteraction;massandforce...............54
3.4Thelawofmultipleinteractions.. ....................57
3.5Centreofmass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..58
vi Contents
3.6Thelawofgravitation. . . . . . . . . . . . . . . . . . . . . . . . . ..59
3.7Gravitationbyadistributionofmass. . . . . . . . . . . . . . . . . ..60
3.8Theprincipleofequivalenceand g . . . . . . . . . . . . . . . . . . ..67
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..71
4Problemsinparticledynamics73
4.1Rectilinearmotioninaforcefield.................. ...74
4.2Constrainedrectilinearmotion.................... ...78
4.3Motionthrougharesistingmedium................. ...82
4.4Projectiles......... ..................... ...88
4.5Circularmotion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..92
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..98
5Linearoscillations105
5.1Bodyonaspring. . . . . . . . . . . . . . . . . . . . . . . . . . . . ..105
5.2Classicalsimpleharmonicmotion. . . . . . . . . . . . . . . . . . . ..107
5.3Dampedsimpleharmonicmotion. . . . . . . . . . . . . . . . . . . ..109
5.4Driven(forced)motion. . . . . . . . . . . . . . . . . . . . . . . . . ..112
5.5Asimpleseismograph. . . . . . . . . . . . . . . . . . . . . . . . . ..120
5.6Coupledoscillationsandnormalmodes............... ...121
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..126
6Energyconservation131
6.1Theenergyprinciple. . . . . . . . . . . . . . . . . . . . . . . . . . ..131
6.2Energyconservationinrectilinearmotion.............. ...133
6.3Generalfeaturesofrectilinearmotion................ ...136
6.4Energyconservationinaconservativefield . . . . . . . . . . . . . . ..140
6.5Energyconservationinconstrainedmotion............. ...145
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..151
7Orbitsinacentralfield155
7.1Theone-bodyproblem–Newton’sequations............ ...157
7.2Generalnatureoforbitalmotion. . . . . . . . . . . . . . . . . . . . ..159
7.3Thepathequation. . . . . . . . . . . . . . . . . . . . . . . . . . . ..164
7.4Nearlycircularorbits. . . . . . . . . . . . . . . . . . . . . . . . . . ..167
7.5Theattractiveinversesquarefield . . . . . . . . . . . . . . . . . . . ..170
7.6Spacetravel–Hohmanntransferorbits . . . . . . . . . . . . . . . . ..177
7.7Therepulsiveinversesquarefield . . . . . . . . . . . . . . . . . . . ..179
7.8Rutherfordscattering. . . . . . . . . . . . . . . . . . . . . . . . . . ..179
AppendixAThegeometryofconics . . . . . . . . . . . . . . . . . . . . ..184
AppendixBTheHohmannorbitisoptimal............... ...186
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..188
Contents vii
8Non-linearoscillationsandphasespace194
8.1Periodicnon-linearoscillations.. ....................194
8.2 Thephaseplane((x ,x )–plane) . . . . . . . . . . . . . . . . . . . ..199
1 2
8.3Thephaseplaneindynamics( (x,v)–plane). . . . . . . . . . . . . . ..202
8.4Poincar e´-Bendixsontheorem:limitcycles.................205
8.5Drivennon-linearoscillations... ....................211
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..214
2Multi-particlesystems219
9Theenergyprinciple221
9.1Configurationsanddegreesoffreedom..................221
9.2Theenergyprincipleforasystem . . . . . . . . . . . . . . . . . . . ..223
9.3Energyconservationforasystem . . . . . . . . . . . . . . . . . . . ..225
9.4Kineticenergyofarigidbody... ....................233
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..241
10Thelinearmomentumprinciple245
10.1Linearmomentum. . . . . . . . . . . . . . . . . . . . . . . . . . . ..245
10.2Thelinearmomentumprinciple. . . . . . . . . . . . . . . . . . . . ..246
10.3Motionofthecentreofmass. . . . . . . . . . . . . . . . . . . . . . ..247
10.4Conservationoflinearmomentum. . . . . . . . . . . . . . . . . . . ..250
10.5Rocketmotion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..251
10.6Collisiontheory.......... ....................255
10.7Collisionprocessesinthezero-momentumframe.............259
10.8Thetwo-bodyproblem....... ....................264
10.9Two-bodyscattering........ ....................269
10.10Integrablemechanicalsystems. . . . . . . . . . . . . . . . . . . . . ..273
AppendixAModellingbodiesbyparticles...................277
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..279
11Theangularmomentumprinciple286
11.1Themomentofaforce. . . . . . . . . . . . . . . . . . . . . . . . . ..286
11.2Angularmomentum........ ....................289
11.3Angularmomentumofarigidbody....................292
11.4Theangularmomentumprinciple. ....................294
11.5Conservationofangularmomentum....................298
11.6Planarrigidbodymotion...... ....................306
11.7Rigidbodystaticsinthreedimensions...................313
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..317
viii Contents
3Analyticalmechanics321
12Lagrange’sequationsandconservationprinciples323
12.1Constraintsandconstraintforces . . . . . . . . . . . . . . . . . . . . ..323
12.2Generalisedcoordinates. . . . . . . . . . . . . . . . . . . . . . . . ..325
12.3Configurationspace( q–space).................... ...330
12.4D’Alembert’sprinciple. . . . . . . . . . . . . . . . . . . . . . . . . ..333
12.5Lagrange’sequations. . . . . . . . . . . . . . . . . . . . . . . . . . ..335
12.6Systemswithmovingconstraints. . . . . . . . . . . . . . . . . . . ..344
12.7TheLagrangian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..348
12.8Theenergyfunction h . . . . . . . . . . . . . . . . . . . . . . . . . ..351
12.9Generalisedmomenta. . . . . . . . . . . . . . . . . . . . . . . . . ..354
12.10Symmetryandconservationprinciples. . . . . . . . . . . . . . . . . ..356
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..361
13ThecalculusofvariationsandHamilton’sprinciple366
13.1Sometypicalminimisationproblems................ ...367
13.2TheEuler–Lagrangeequation. . . . . . . . . . . . . . . . . . . . . ..369
13.3Variationalprinciples. . . . . . . . . . . . . . . . . . . . . . . . . . ..380
13.4Hamilton’sprinciple.... ..................... ...383
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..388
14Hamilton’sequationsandphasespace393
14.1SystemsoffirstorderODEs..................... ...393
14.2Legendretransforms. . . . . . . . . . . . . . . . . . . . . . . . . . ..396
14.3Hamilton’sequations.... ..................... ...400
14.4Hamiltonianphasespace(( q, p)–space)............... ...406
14.5Liouville’stheoremandrecurrence................. ...408
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..413
4Furthertopics419
15Thegeneraltheoryofsmalloscillations421
15.1Stableequilibriumandsmalloscillations.............. ...421
15.2Theapproximateformsof T andV . . . . . . . . . . . . . . . . . . ..425
15.3Thegeneraltheoryofnormalmodes................. ...429
15.4Existencetheoryfornormalmodes. . . . . . . . . . . . . . . . . . ..433
15.5Sometypicalnormalmodeproblems................ ...436
15.6Orthogonalityofnormalmodes................... ...444
15.7Generalsmalloscillations. ..................... ...447
15.8Normalcoordinates. . . . . . . . . . . . . . . . . . . . . . . . . . ..448
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..452