Analytical mechanics

475 Pages·2018·2.33 MB·English

by  Nivaldo A

AnalyticalMechanics
Analytical mechanics is the foundation of many areas of theoretical physics, including
quantumtheoryandstatisticalmechanics,andhaswide-rangingapplicationsinengineer-
ing and celestial mechanics. This introduction to the basic principles and methods of
analytical mechanics covers Lagrangian and Hamiltonian dynamics, rigid bodies, small
oscillations, canonical transformations and Hamilton-Jacobi theory. This fully up-to-date
textbookincludesdetailedmathematicalappendicesandaddressesanumberofadvanced
topics, some of them of a geometric or topological character. These include Bertrand’s
theorem,proofthatactionisleast,spontaneoussymmetrybreakdown,constrainedHamil-
tonian systems, non-integrability criteria, KAM theory, classical field theory, Lyapunov
functions,geometricphasesandPoissonmanifolds.Providingworkedexamples,end-of-
chapterproblemsanddiscussionofongoingresearchinthefield,itissuitableforadvanced
undergraduatestudentsandgraduatestudentsstudyinganalyticalmechanics.
NivaldoA.LemosisAssociateProfessorofPhysicsatFederalFluminenseUniversity,Brazil.
HewaspreviouslyavisitingscholarattheMassachusettsInstituteofTechnology.Hismain
researchareasarequantumcosmology,quantumfieldtheoryandtheteachingofclassical
mechanics.

Analytical Mechanics
NIVALDO A. LEMOS
FluminenseFederalUniversity
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Informationonthistitle:www.cambridge.org/9781108416580
DOI:10.1017/9781108241489
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Englishtranslation©CambridgeUniversityPress2018
RevisedandenlargedtranslationofMecaˆnicaAnal´ıticaby
NivaldoA.Lemosoriginallypublishedin
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Names:Lemos,NivaldoA.,1952–author.
Title:Analyticalmechanics/NivaldoA.Lemos,FluminenseFederalUniversity.
Othertitles:Mecaˆnicaanal´ıtica.English
Description:Cambridge,UnitedKingdom;NewYork,NY:Cambridge
UniversityPress,2018.|Includesbibliographicalreferencesandindex.
Identifiers:LCCN2018011108|ISBN9781108416580(hb)|ISBN1108416586(hb)
Subjects:LCSH:Mechanics,Analytic.
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Tomywife
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andmydaughters
Cintia,LuizaandBeatriz

Contents
Preface pagexi
1 LagrangianDynamics 1
1.1 PrinciplesofNewtonianMechanics 1
1.2 Constraints 7
1.3 VirtualDisplacementsandd’Alembert’sPrinciple 11
1.4 GeneralisedCoordinatesandLagrange’sEquations 17
1.5 ApplicationsofLagrange’sEquations 24
1.6 GeneralisedPotentialsandDissipationFunction 28
1.7 CentralForcesandBertrand’sTheorem 31
Problems 37
2 Hamilton’sVariationalPrinciple 42
2.1 RudimentsoftheCalculusofVariations 42
2.2 VariationalNotation 49
2.3 Hamilton’sPrincipleandLagrange’sEquations 50
2.4 Hamilton’sPrincipleintheNon-HolonomicCase 56
2.5 SymmetryPropertiesandConservationLaws 65
2.6 ConservationofEnergy 71
2.7 Noether’sTheorem 73
Problems 78
3 KinematicsofRotationalMotion 86
3.1 OrthogonalTransformations 86
3.2 PossibleDisplacementsofaRigidBody 92
3.3 EulerAngles 95
3.4 InfinitesimalRotationsandAngularVelocity 96
3.5 RotationGroupandInfinitesimalGenerators 102
3.6 DynamicsinNon-InertialReferenceFrames 103
Problems 109
4 DynamicsofRigidBodies 112
4.1 AngularMomentumandInertiaTensor 112
4.2 MathematicalInterlude:TensorsandDyadics 114
4.3 MomentsandProductsofInertia 119
vii
viii Contents
4.4 KineticEnergyandParallelAxisTheorem 120
4.5 DiagonalisationoftheInertiaTensor 122
4.6 SymmetriesandPrincipalAxesofInertia 126
4.7 RollingCoin 129
4.8 Euler’sEquationsandFreeRotation 131
4.9 SymmetricTopwithOnePointFixed 139
Problems 145
5 SmallOscillations 149
5.1 One-DimensionalCase 149
5.2 AnomalousCase:QuarticOscillator 154
5.3 StationaryMotionandSmallOscillations 161
5.4 SmallOscillations:GeneralCase 163
5.5 NormalModesofVibration 165
5.6 NormalCoordinates 170
5.7 MathematicalSupplement 176
Problems 178
6 RelativisticMechanics 183
6.1 LorentzTransformations 183
6.2 LightConeandCausality 188
6.3 VectorsandTensors 191
6.4 TensorFields 194
6.5 PhysicalLawsinCovariantForm 196
6.6 RelativisticDynamics 199
6.7 RelativisticCollisions 204
6.8 RelativisticDynamicsinLagrangianForm 207
6.9 ActionataDistanceinSpecialRelativity 209
Problems 211
7 HamiltonianDynamics 215
7.1 Hamilton’sCanonicalEquations 215
7.2 SymmetriesandConservationLaws 220
7.3 TheVirialTheorem 221
7.4 RelativisticHamiltonianFormulation 224
7.5 Hamilton’sEquationsinVariationalForm 226
7.6 TimeasaCanonicalVariable 228
7.7 ThePrincipleofMaupertuis 234
Problems 237
8 CanonicalTransformations 242
8.1 CanonicalTransformationsandGeneratingFunctions 242
8.2 CanonicityandLagrangeBrackets 248
ix Contents
8.3 SymplecticNotation 250
8.4 PoissonBrackets 254
8.5 InfinitesimalCanonicalTransformations 258
8.6 AngularMomentumPoissonBrackets 262
8.7 LieSeriesandFiniteCanonicalTransformations 264
8.8 TheoremsofLiouvilleandPoincare´ 268
8.9 ConstrainedHamiltonianSystems 272
Problems 281
9 TheHamilton-JacobiTheory 288
9.1 TheHamilton-JacobiEquation 288
9.2 One-DimensionalExamples 291
9.3 SeparationofVariables 294
9.4 IncompletenessoftheTheory:PointChargeinDipoleField 299
9.5 ActionasaFunctionoftheCoordinates 303
9.6 Action-AngleVariables 306
9.7 IntegrableSystems:TheLiouville-ArnoldTheorem 312
9.8 Non-integrabilityCriteria 316
9.9 Integrability,Chaos,DeterminismandErgodicity 318
9.10 PreludetotheKAMTheorem 320
9.11 ActionVariablesintheKeplerProblem 323
9.12 AdiabaticInvariants 325
9.13 Hamilton-JacobiTheoryandQuantumMechanics 329
Problems 332
10 HamiltonianPerturbationTheory 338
10.1 StatementoftheProblem 338
10.2 GeneratingFunctionMethod 341
10.3 OneDegreeofFreedom 342
10.4 SeveralDegreesofFreedom 345
10.5 TheKolmogorov-Arnold-MoserTheory 348
10.6 Stability:EternalorLongTerm 353
10.7 LieSeriesMethod 354
Problems 358
11 ClassicalFieldTheory 360
11.1 LagrangianFieldTheory 360
11.2 RelativisticFieldTheories 365
11.3 FunctionalDerivatives 367
11.4 HamiltonianFieldTheory 371
11.5 SymmetriesoftheActionandNoether’sTheorem 374
11.6 SolitaryWavesandSolitons 378
11.7 ConstrainedFields 381
Problems 385