Geometric mechanics: toward a unification of classical physics

606 Pages·2007·3.122 MB·English

by  Richard Talman

Richard Talman
Geometric Mechanics
1807–2007 Knowledge for Generations
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Richard Talman
Geometric Mechanics
Toward a Unification of Classical Physics
Second, Revised and Enlarged Edition

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Printed on acid-free paper
ISBN: 978-3-527-40683-8
V
Contents
Preface XV
Introduction 1
Bibliography 9
1 ReviewofClassicalMechanicsandStringFieldTheory 11
1.1 PreviewandRationale 11
1.2 ReviewofLagrangiansandHamiltonians 13
1.2.1 Hamilton’sEquationsinMultipleDimensions 14
1.3 DerivationoftheLagrangeEquationfromHamilton’sPrinciple 16
1.4 Linear,MultiparticleSystems 18
1.4.1 TheLaplaceTransformMethod 23
1.4.2 DampedandDrivenSimpleHarmonicMotion 24
1.4.3 ConservationofMomentumandEnergy 26
1.5 EffectivePotentialandtheKeplerProblem 26
1.6 MultiparticleSystems 29
1.7 LongitudinalOscillationofaBeadedString 32
1.7.1 MonofrequencyExcitation 33
1.7.2 TheContinuumLimit 34
1.8 FieldTheoreticalTreatmentandLagrangianDensity 36
1.9 HamiltonianDensityforTransverseStringMotion 39
1.10 StringMotionExpressedasPropagatingandReflectingWaves 40
1.11 Problems 42
Bibliography 44
2 GeometryofMechanics,I,Linear 45
2.1 PairsofPlanesasCovariantVectors 47
2.2 DifferentialForms 53
2.2.1 GeometricInterpretation 53
2.2.2 CalculusofDifferentialForms 57
2.2.3 FamiliarPhysicsEquationsExpressedUsingDifferentialForms 61
VI Contents
2.3 AlgebraicTensors 66
2.3.1 VectorsandTheirDuals 66
2.3.2 TransformationofCoordinates 68
2.3.3 TransformationofDistributions 72
2.3.4 Multi-indexTensorsandtheirContraction 73
2.3.5 RepresentationofaVectorasaDifferentialOperator 76
2.4 (PossiblyComplex)CartesianVectorsinMetricGeometry 79
2.4.1 EuclideanVectors 79
2.4.2 SkewCoordinateFrames 81
2.4.3 ReductionofaQuadraticFormtoaSumorDifferenceof
Squares 81
2.4.4 IntroductionofCovariantComponents 83
2.4.5 TheReciprocalBasis 84
Bibliography 86
3 GeometryofMechanics,II,Curvilinear 89
3.1 (Real)CurvilinearCoordinatesinn-Dimensions 90
3.1.1 TheMetricTensor 90
3.1.2 RelatingCoordinateSystemsatDifferentPointsinSpace 92
3.1.3 TheCovariant(orAbsolute)Differential 97
3.2 DerivationoftheLagrangeEquationsfromtheAbsolute
Differential 102
3.2.1 PracticalEvaluationoftheChristoffelSymbols 108
3.3 IntrinsicDerivativesandtheBilinearCovariant 109
3.4 TheLieDerivative–CoordinateApproach 111
3.4.1 Lie-DraggedCoordinateSystems 111
3.4.2 LieDerivativesofScalarsandVectors 115
3.5 TheLieDerivative–LieAlgebraicApproach 120
3.5.1 ExponentialRepresentationofParameterizedCurves 120
3.6 IdentificationofVectorFieldswithDifferentialOperators 121
3.6.1 LoopDefect 122
3.7 CoordinateCongruences 123
3.8 Lie-DraggedCongruencesandtheLieDerivative 125
3.9 CommutatorsofQuasi-Basis-Vectors 130
Bibliography 132
4 GeometryofMechanics,III,Multilinear 133
4.1 GeneralizedEuclideanRotationsandReflections 133
4.1.1 Reflections 134
4.1.2 ExpressingaRotationasaProductofReflections 135
4.1.3 TheLieGroupofRotations 136
4.2 Multivectors 138
Contents VII
4.2.1 VolumeDeterminedby3-andbyn-Vectors 138
4.2.2 Bivectors 140
4.2.3 MultivectorsandGeneralizationtoHigherDimensionality 141
4.2.4 LocalRadiusofCurvatureofaParticleOrbit 143
4.2.5 “Supplementary”Multivectors 144
4.2.6 Sumsof p-Vectors 145
4.2.7 BivectorsandInfinitesimalRotations 145
4.3 CurvilinearCoordinatesinEuclideanGeometry(Continued) 148
4.3.1 RepeatedExteriorDerivatives 148
4.3.2 TheGradientFormulaofVectorAnalysis 149
4.3.3 VectorCalculusExpressedbyDifferentialForms 151
4.3.4 DerivationofVectorIntegralFormulas 154
4.3.5 GeneralizedDivergenceandGauss’sTheorem 157
4.3.6 Metric-FreeDefinitionofthe“Divergence”ofaVector 159
4.4 SpinorsinThree-DimensionalSpace 161
4.4.1 DefinitionofSpinors 162
4.4.2 DemonstrationthataSpinorisaEuclideanTensor 162
×
4.4.3 Associatinga2 2Reflection(Rotation)MatrixwithaVector
(Bivector) 163
4.4.4 AssociatingaMatrixwithaTrivector(TripleProduct) 164
4.4.5 RepresentationsofReflections 164
4.4.6 RepresentationsofRotations 165
4.4.7 OperationsonSpinors 166
4.4.8 RealEuclideanSpace 167
4.4.9 RealPseudo-EuclideanSpace 167
Bibliography 167
5 Lagrange–PoincaréDescriptionofMechanics 169
5.1 ThePoincaréEquation 169
5.1.1 SomeFeaturesofthePoincaréEquations 179
5.1.2 InvarianceofthePoincaréEquation 180
5.1.3 TranslationintotheLanguageofFormsandVectorFields 182
5.1.4 Example: FreeMotionofaRigidBodywithOnePointFixed 183
5.2 VariationalDerivationofthePoincaréEquation 186
5.3 RestrictingthePoincaréEquationWithGroupTheory 189
5.3.1 ContinuousTransformationGroups 189
5.3.2 UseofInfinitesimalGroupParametersasQuasicoordinates 193
5.3.3 InfinitesimalGroupOperators 195
5.3.4 CommutationRelationsandStructureConstantsoftheGroup 199
5.3.5 QualitativeAspectsofInfinitesimalGenerators 201
5.3.6 ThePoincaréEquationinTermsofGroupGenerators 204
5.3.7 TheRigidBodySubjecttoForceandTorque 206
Bibliography 217
VIII Contents
6 Newtonian/GaugeInvariantMechanics 219
6.1 VectorMechanics 219
6.1.1 VectorDescriptioninCurvilinearCoordinates 219
6.1.2 TheFrenet–SerretFormulas 222
6.1.3 VectorDescriptioninanAcceleratingCoordinateFrame 226
6.1.4 ExploitingtheFictitiousForceDescription 232
6.2 SingleParticleEquationsinGaugeInvariantForm 238
6.2.1 Newton’sForceEquationinGaugeInvariantForm 239
6.2.2 ActiveInterpretationoftheTransformations 242
6.2.3 Newton’sTorqueEquation 246
6.2.4 ThePlumbBob 248
6.3 GaugeInvariantDescriptionofRigidBodyMotion 252
6.3.1 SpaceandBodyFramesofReference 253
×
6.3.2 ReviewoftheAssociationof2 2MatricestoVectors 256
×
6.3.3 “Association”of3 3MatricestoVectors 258
6.3.4 DerivationoftheRigidBodyEquations 259
6.3.5 TheEulerEquationsforaRigidBody 261
6.4 TheFoucaultPendulum 262
6.4.1 FictitiousForceSolution 263
6.4.2 GaugeInvariantSolution 265
6.4.3 “Parallel”TranslationofCoordinateAxes 270
6.5 TumblersandDivers 274
Bibliography 276
7 HamiltonianTreatmentofGeometricOptics 277
7.1 AnalogyBetweenMechanicsandGeometricOptics 278
7.1.1 ScalarWaveEquation 279
7.1.2 TheEikonalEquation 281
7.1.3 DeterminationofRaysfromWavefronts 282
7.1.4 TheRayEquationinGeometricOptics 283
7.2 VariationalPrinciples 285
7.2.1 TheLagrangeIntegralInvariantandSnell’sLaw 285
7.2.2 ThePrincipleofLeastTime 287
7.3 ParaxialOptics,GaussianOptics,MatrixOptics 288
7.4 Huygens’Principle 292
Bibliography 294
8 Hamilton–JacobiTheory 295
8.1 Hamilton–JacobiTheoryDerivedfromHamilton’sPrinciple 295
8.1.1 TheGeometricPicture 297
8.1.2 ConstantSWavefronts 298
8.2 TrajectoryDeterminationUsingtheHamilton–JacobiEquation 299
Contents IX
8.2.1 CompleteIntegral 299
8.2.2 FindingaCompleteIntegralbySeparationofVariables 300
8.2.3 Hamilton–JacobiAnalysisofProjectileMotion 301
8.2.4 TheJacobiMethodforExploitingaCompleteIntegral 302
8.2.5 CompletionofProjectileExample 304
8.2.6 TheTime-IndependentHamilton–JacobiEquation 305
8.2.7 Hamilton–JacobiTreatmentof1DSimpleHarmonicMotion 306
8.3 TheKeplerProblem 307
8.3.1 CoordinateFrames 308
8.3.2 OrbitElements 309
8.3.3 Hamilton–JacobiFormulation. 310
8.4 AnalogiesBetweenOpticsandQuantumMechanics 314
8.4.1 ClassicalLimitoftheSchrödingerEquation 314
Bibliography 316
9 RelativisticMechanics 317
9.1 RelativisticKinematics 317
9.1.1 FormInvariance 317
9.1.2 WorldPointsandIntervals 318
9.1.3 ProperTime 319
9.1.4 TheLorentzTransformation 321
9.1.5 TransformationofVelocities 322
9.1.6 4-VectorsandTensors 322
9.1.7 Three-IndexAntisymmetricTensor 325
9.1.8 Antisymmetric4-Tensors 325
9.1.9 The4-Gradient,4-Velocity,and4-Acceleration 326
9.2 RelativisticMechanics 327
9.2.1 TheRelativisticPrincipleofLeastAction 327
9.2.2 EnergyandMomentum 328
9.2.3 4-VectorNotation 329
9.2.4 ForcedMotion 329
9.2.5 Hamilton–JacobiFormulation 330
9.3 IntroductionofElectromagneticForcesintoRelativistic
Mechanics 332
9.3.1 GeneralizationoftheAction 332
9.3.2 DerivationoftheLorentzForceLaw 334
9.3.3 GaugeInvariance 335
Bibliography 338
10 ConservationLawsandSymmetry 339
10.1 ConservationofLinearMomentum 339
10.2 RateofChangeofAngularMomentum:PoincaréApproach 341