Modern Classical Mechanics

710 Pages·2021·20.965 MB·English

by  T. M. Helliwell, V. V. Sahakian

ModernClassicalMechanics
Inthismodernanddistinctivetextbook,HelliwellandSahakianpresentclassicalmechan-
icsasathrivingandcontemporaryfieldwithstrongconnectionstocutting-edgeresearch
topics in physics. Each part of the book concludes with a capstone chapter describing
various key topics in quantum mechanics, general relativity, and other areas of modern
physics, clearly demonstrating how they relate to advanced classical mechanics, and
enabling students to appreciate the central importance of classical mechanics within
contemporary fields of research. Numerous and detailed examples are interleaved with
theoreticalcontent,illustratingabstractconceptsmoreconcretely.Extensiveproblemsets
at the end of each chapter further reinforce students’ understanding of key concepts, and
provide opportunities for assessment or self-testing. A detailed online solutions manual
andlectureslidesaccompanythetextforinstructors.Oftenaflexibleapproachisrequired
when teaching advanced classical mechanics, and, to facilitate this, the authors have
outlined several paths instructors and students can follow through the book, depending
onbackgroundknowledgeandthelengthoftheircourse.
T.M.Helliwell taught at Harvey Mudd College for more than 40 years, as well as serving
in several administrative roles, including Physics Department Chair. He received two
National Science Foundation fellowships, was awarded the Henry T. Mudd Prize for
servicetothecollegein1997,andservedasaconsultantattheJetPropulsionLaboratory.
He is the author or co-author of more than 40 research papers on general relativity,
cosmology,andquantumtheory,andtheauthoroftwotextbooksonspecialrelativity.
V.V.SahakianisProfessorofTheoreticalPhysicsatHarveyMuddCollege.Hehasreceived
twoNationalScienceFoundationfellowships,andauthoredorco-authoredmorethan25
research papers on string theory, cosmology, and quantum gravity. His research focuses
on understanding the small-scale structure of space, often studying black holes and
exploringnewframeworksthatextendtheStandardModelofparticlephysicsandstandard
inflationarycosmology.

Modern Classical Mechanics
T. M. HELLIWELL
HarveyMuddCollege,California
V. V. SAHAKIAN
HarveyMuddCollege,California
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DOI:10.1017/9781108874687
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Names:Helliwell,T.M.(ThomasM.),1936–author.|Sahakian,V.V.(VatcheV.),author.
Title:Modernclassicalmechanics/T.M.Helliwell,HarveyMuddCollege,
California,V.V.Sahakian,HarveyMuddCollege,California.
Description:Cambridge,UnitedKingdom;NewYork,NY:Cambridge
UniversityPress,[2021]|Includesbibliographicalreferencesandindex.
Identifiers:LCCN2020039628(print)|LCCN2020039629(ebook)
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DedicatedtoBonnie,Gaia,andYuliya

Contents
Preface pagexiii
AcknowledgmentsandCredits xvii
NotationandConventions xviii
UsefulRelations xix
PartI
1 NewtonianParticleMechanics 3
1.1 InertialFramesandtheGalileanTransformation 3
1.2 Newton’sLawsofMotion 5
1.3 One-DimensionalMotion:DragForces 9
1.4 OscillationinOne-DimensionalMotion 13
1.5 Resonance 18
1.6 MotioninTwoorThreeDimensions 23
1.7 SystemsofParticles 25
1.8 ConservationLaws 28
1.9 Collisions 45
1.10 ForcesofNature 47
1.11 Summary 48
Problems 50
2 Relativity 63
2.1 Einstein’sPostulatesandtheLorentzTransformation 63
2.2 RelativisticKinematics 71
2.3 RelativisticDynamics 79
2.4 Summary 107
Problems 107
3 TheVariationalPrinciple 119
3.1 Fermat’sPrinciple 119
3.2 TheCalculusofVariations 120
3.3 Geodesics 128
3.4 Brachistochrone 131
3.5 SeveralDependentVariables 136
3.6 MechanicsfromaVariationalPrinciple 137
vii
viii Contents
3.7 MotioninaUniformGravitationalField 139
3.8 ArbitraryPotentialEnergies 145
3.9 Summary 146
Problems 147
4 LagrangianMechanics 153
4.1 NonconservativeForces 153
4.2 ForcesofConstraintandGeneralizedCoordinates 154
4.3 Hamilton’sPrinciple 155
4.4 GeneralizedMomentaandCyclicCoordinates 161
4.5 SystemsofParticles 168
4.6 TheHamiltonian 175
4.7 WhenisH (cid:2)= E? 179
4.8 TheMoralofConstraints 181
4.9 SmallOscillationsaboutEquilibrium 183
4.10 Recap 185
Problems 187
5 FromClassicaltoQuantumandBack 197
5.1 ClassicalWaves 197
5.2 Two-SlitExperimentsandQuantumMechanics 205
5.3 FeynmanSum-over-Paths 208
5.4 HeliumAtomsandtheTwoSlits,Revisited 211
5.5 TheEmergenceoftheClassicalTrajectory 217
5.6 WhyHamilton’sPrinciple? 222
5.7 TheJacobiAction 223
5.8 Summary 226
Problems 228
PartII
6 ConstraintsandSymmetries 233
6.1 ContactForces 233
6.2 SymmetriesandConservationLaws:APreview 245
6.3 CyclicCoordinatesandGeneralizedMomenta 246
6.4 ALessStraightforwardExample 248
6.5 InfinitesimalTransformations 250
6.6 Symmetry 253
6.7 Noether’sTheorem 255
6.8 SomeCommentsonSymmetries 266
6.9 Summary 267
Problems 267