Richard Talman Geometric Mechanics 1807–2007 Knowledge for Generations Each generation has its unique needs and aspirations. When Charles Wiley first openedhis small printing shop in lower Manhattan in 1807, it was a generation of boundless potential searching for an identity. And we were there, helping to define a new American literary tradition. Over half a century later, in the midst of the Second Industrial Revolution, it was a generation focused on building the future. Onceagain, we werethere, supplying the criticalscientific,technical, and engineering knowledge that helped frame the world. Throughout the 20th Century, and into the new millennium, nations began to reach out beyond their own bordersand a new international community was born.Wiley was there, ex- panding its operations around the world to enable a global exchange of ideas, opinions,and know-how. 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Nevertheless, authors, editors, and publisher do not warrant the information Prof. Richard Talman contained in these books, including this book, to Cornell University be free of errors. Readers are advised to keep in Laboratory of Elementary Physics mind that statements, data, illustrations, procedural details or other items may Ithaca, NY 14853 inadvertently be inaccurate. USA [email protected] Library of Congress Card No.: applied for British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at <http://dnb.d-nb.de>. (cid:164) 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim All rights reserved (including those of translation into other languages). 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Pacifico Printed in the Federal Republic of Germany 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 GeometricMechanics:TowardaUnificationofClassicalPhysics.2ndEdition.RichardTalman Copyright(cid:1)c 2007WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim ISBN:978-3-527-40683-8 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 X Contents 10.3 ConservationofAngularMomentum: LagrangianApproach 342 10.4 ConservationofEnergy 343 10.5 CyclicCoordinatesandRouthianReduction 344 10.5.1 Integrability;GeneralizationofCyclicVariables 347 10.6 Noether’sTheorem 348 10.7 ConservationLawsinFieldTheory 352 10.7.1 IgnorableCoordinatesandtheEnergyMomentumTensor 352 10.8 TransitionFromDiscretetoContinuousRepresentation 356 10.8.1 The4-CurrentDensityandChargeConservation 356 10.8.2 EnergyandMomentumDensities 360 10.9 AngularMomentumofaSystemofParticles 362 10.10 AngularMomentumofaField 363 Bibliography 364 11 ElectromagneticTheory 365 11.1 TheElectromagneticFieldTensor 367 11.1.1 TheLorentzForceEquationinTensorNotation 367 11.1.2 LorentzTransformationandInvariantsoftheFields 369 11.2 TheElectromagneticFieldEquations 370 11.2.1 TheHomogeneousPairofMaxwellEquations 370 11.2.2 TheActionfortheField,ParticleSystem 370 11.2.3 TheElectromagneticWaveEquation 372 11.2.4 TheInhomogeneousPairofMaxwellEquations 373 11.2.5 EnergyDensity,EnergyFlux,andtheMaxwellStressEnergy Tensor 374 Bibliography 377 12 RelativisticStrings 379 12.1 Introduction 379 12.1.1 IsStringTheoryAppropriate? 379 12.1.2 ParameterizationInvariance 381 12.1.3 PostulatingaStringLagrangian 381 12.2 AreaRepresentationinTermsoftheMetric 383 12.3 TheLagrangianDensityandActionforStrings 384 12.3.1 ARevisedMetric 384 12.3.2 ParameterizationofStringWorldSurfacebyσandτ 385 12.3.3 TheNambu–GotoAction 385 12.3.4 StringTensionandMassDensity 387 12.4 EquationsofMotion,BoundaryConditions,andUnexcited Strings 389 12.5 TheActioninTermsofTransverseVelocity 391 12.6 OrthogonalParameterizationbyEnergyContent 394