VARIATIONALPRINCIPLES IN CLASSICALMECHANICS
DouglasCline
UniversityofRochester
9August2017
c °2017DouglasCline
ISBN:978-0-9988372-4-6e-book(AdobePDFcolor)
ISBN:978-0-9988372-5-3print(Paperbackgrayscale)
VariationalPrinciplesinClassicalMechanics
Contributors
Author:DouglasCline
Illustrator:MeghanSarkis
PublishedbyUniversityofRochesterRiverCampusLibraries UniversityofRochester Rochester,NY14627
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Version1.0
1Abriefhistoryofclassicalmechanics1
1.1Introduction..............................................1
1.2Prehistoricastronomy........................................1
1.3Greekantiquity............................................1
1.4MiddleAges..............................................2
1.5AgeofEnlightenment........................................3
1.6 19 century..............................................5
1.7The 20 centuryrevolutioninphysics...............................7
2ReviewofNewtonianmechanics9
2.1Introduction..............................................9
2.2Newton’sLawsofmotion......................................9
2.3Inertialframesofreference......................................10
2.4First-orderintegralsinNewtonianmechanics...........................11
2.4.1LinearMomentum......................................11
2.4.2Angularmomentum.....................................11
2.4.3Kineticenergy........................................12
2.5Conservationlawsinclassicalmechanics..............................12
2.6Motionof finite-sizedandmany-bodysystems...........................12
2.7Centerofmassofamany-bodysystem...............................13
2.8Totallinearmomentumofamany-bodysystem..........................14
2.8.1Center-of-massdecomposition................................14
2.8.2Equationsofmotion.....................................14
2.9Angularmomentumofamany-bodysystem............................16
2.9.1Center-of-massdecomposition................................16
2.9.2Equationsofmotion.....................................16
2.10Workandkineticenergyforamany-bodysystem.........................18
2.10.1Center-of-masskineticenergy................................18
2.10.2Conservativeforcesandpotentialenergy..........................18
2.10.3Totalmechanicalenergy...................................19
2.10.4Totalmechanicalenergyforconservativesystems.....................20
2.11VirialTheorem............................................22
2.12ApplicationsofNewton’sequationsofmotion...........................24
2.12.1Constantforceproblems...................................24
2.12.2LinearRestoringForce....................................25
2.12.3Position-dependentconservativeforces...........................25
2.12.4Constrainedmotion.....................................27
2.12.5VelocityDependentForces..................................28
2.12.6SystemswithVariableMass.................................29
2.12.7Rigid-bodyrotationaboutabody-fixedrotationaxis...................31
2.12.8Timedependentforces....................................34
2.13Solutionofmany-bodyequationsofmotion............................37
2.13.1Analyticsolution.......................................37
2.13.2Successiveapproximation..................................37
2.13.3Perturbationmethod.....................................37
2.14Newton’sLawofGravitation....................................38
2.14.1Gravitationalandinertialmass...............................38
2.14.2Gravitationalpotentialenergy
2.14.3Gravitationalpotential
2.14.4Potentialtheory.......................................41
2.14.5Curlofthegravitational
2.14.6Gauss’sLawforGravitation.................................43
2.14.7CondensedformsofNewton’sLawofGravitation.....................44 2.15Summary...............................................46
3.4.4Planependulum.......................................57
3.5Linearly-dampedfreelinearoscillator
3.5.1Generalsolution.......................................58 3.5.2Energydissipation......................................61
3.6Sinusoidally-drive,linearly-damped,linearoscillator...
3.6.1Transientresponseofadrivenoscillator
3.6.2Steadystateresponseofadrivenoscillator ........................63
3.6.3Completesolutionofthedrivenoscillator.........................64
3.6.4Resonance...........................................65
3.6.5Energyabsorption......................................65
3.7Waveequation............................................68
3.8Travellingandstandingwavesolutionsofthewaveequation...................69
3.9Waveformanalysis..........................................70
3.9.1Harmonicdecomposition...................................70
3.9.2Thefreelinearly-dampedlinearoscillator .........................70
3.9.3Dampedlinearoscillatorsubjecttoanarbitraryperiodicforce.............71
3.10Signalprocessing...........................................72
3.11Wavepropagation ..........................................73
3.11.1Phase,group,andsignalvelocitiesofwavepackets....................74
3.11.2Fouriertransformofwavepackets.............................79
3.11.3Wave-packetUncertaintyPrinciple.............................80
3.12Summary...............................................82
4Nonlinearsystemsandchaos89
4.1Introduction..............................................89
4.2Weaknonlinearity..........................................90
4.3Bifurcation,andpointattractors..................................92
4.4Limitcycles..............................................93
4.4.1Poincaré-Bendixsontheorem................................93
4.4.2vanderPoldampedharmonicoscillator:..........................94
4.5Harmonically-driven,linearly-damped,planependulum......................97
4.5.1Closetolinearity.......................................97
4.5.2Weaknonlinearity......................................99
4.5.3Onsetofcomplication....................................100
4.5.4Perioddoublingandbifurcation...............................100
4.5.5Rollingmotion ........................................100
4.5.6Onsetofchaos........................................101
4.6Differentiationbetweenorderedandchaoticmotion........................102
4.6.1Lyapunovexponent.....................................102
4.6.2Bifurcationdiagram.....................................103
4.6.3PoincaréSection.......................................104
4.7Wavepropagationfornon-linearsystems... ...........................105
4.7.1Phase,group,andsignalvelocities.............................105
4.7.2Solitonwavepropagation ..................................107
4.8Summary...............................................108 Workshopexercises.............................................110 Problems..................................................110
5Calculusofvariations
5.1Introduction..............................................111
5.2Euler’sdifferentialequation.....................................112
5.3ApplicationsofEuler’sequation...................................114
5.4Selectionoftheindependentvariable................................117
5.5Functionswithseveralindependentvariables () ........................119
5.6Euler’sintegralequation.......................................121
5.7Constrainedvariationalsystems...................................122
5.7.1Holonomicconstraints....................................122
5.7.2Geometric(algebraic)equationsofconstraint.......................122
5.7.3Kinematic(differential)equationsofconstraint......................122
5.7.4Isoperimetric(integral)equationsofconstraint......................123
5.7.5Propertiesoftheconstraintequations...........................123
5.7.6Treatmentofconstraintforcesinvariationalcalculus...................124
5.8Generalizedcoordinatesinvariationalcalculus..........................125
5.9Lagrangemultipliersforholonomicconstraints..........................126
5.9.1Algebraicequationsofconstraint..............................126
5.9.2Integralequationsofconstraint...............................128
5.10Geodesic................................................130
5.11Variationalapproachtoclassicalmechanics............................131
5.12Summary...............................................132 Workshopexercises.............................................133
6Lagrangiandynamics 135
6.1Introduction..............................................135
6.2NewtonianplausibilityargumentforLagrangianmechanics ...................136
6.3Lagrangeequationsfromd’Alembert’sPrinciple..........................138
6.3.1d’Alembert’sPrincipleofvirtualwork...........................138
6.3.2Transformationtogeneralizedcoordinates.........................139
6.3.3Lagrangian..........................................140
6.4LagrangeequationsfromHamilton’sPrinciple...........................141
6.5Constrainedsystems.........................................142
6.5.1Choiceofgeneralizedcoordinates..............................142
6.5.2Minimalsetofgeneralizedcoordinates...........................142
6.5.3Lagrangemultipliersapproach...............................142
6.5.4Generalizedforcesapproach.................................144
6.6ApplyingtheEuler-Lagrangeequationstoclassicalmechanics..................144
6.7Applicationstounconstrainedsystems...............................146
6.8Applicationstosystemsinvolvingholonomicconstraints.....................148
6.9Applicationsinvolvingnon-holonomicconstraints.........................161
6.10Velocity-dependentLorentzforce..................................168
6.11Time-dependentforces........................................169
6.12Impulsiveforces............................................170
6.13TheLagrangianversustheNewtonianapproachtoclassicalmechanics.............172
6.14Summary...............................................173 Workshopexercises.............................................176 Problems..................................................178
7Symmetries,InvarianceandtheHamiltonian179
7.1Introduction..............................................179
7.2Generalizedmomentum.......................................179
7.3InvarianttransformationsandNoether’sTheorem.........................181
7.4Rotationalinvarianceandconservationofangularmomentum..................183
7.5Cycliccoordinates..........................................184
7.6Kineticenergyingeneralizedcoordinates.............................185
7.7GeneralizedenergyandtheHamiltonianfunction.........................186
7.8Generalizedenergytheorem.....................................187
7.9Generalizedenergyandtotalenergy................................187
7.10Hamiltonianinvariance........................................188
7.11Hamiltonianforcycliccoordinates.................................193
7.12Symmetriesandinvariance.....................................193
7.13Hamiltonianinclassicalmechanics.................................193
7.14Summary...............................................194
Workshopexercises.............................................196 Problems..................................................197
8Hamiltonianmechanics199
8.1Introduction..............................................199
8.2LegendreTransformationbetweenLagrangianandHamiltonianmechanics...........200
8.3Hamilton’sequationsofmotion...................................201
8.3.1Canonicalequationsofmotion...............................202
8.4Hamiltonianindifferentcoordinatesystems............................203
8.4.1Cylindricalcoordinates ................................203
8.4.2Sphericalcoordinates, .................................204
8.5ApplicationsofHamiltonianDynamics...............................205
8.6Routhianreduction..........................................210
8.6.1R -RouthianisaHamiltonianforthecyclicvariables................211
8.6.2R -RouthianisaHamiltonianforthenon-cyclicvariables...........212
8.7Dissipativedynamicalsystems....................................216
8.7.1Generalizeddragforce....................................216
8.7.2Rayleigh’sdissipationfunction...............................217
8.8Summary...............................................221
Workshopexercises.............................................223 Problems..................................................224
9Conservativetwo-bodycentralforces227
9.1Introduction..............................................227
9.2Equivalentone-bodyrepresentationfortwo-bodymotion.....................228
9.3Angularmomentum L ........................................230
9.4Equationsofmotion.........................................231
9.5Differentialorbitequation:......................................232
9.6Hamiltonian..............................................233
9.7Generalfeaturesoftheorbitsolutions...............................234
9.8Inverse-square,two-body,centralforce...............................235
9.8.1Boundorbits.........................................236
9.8.2Kepler’slawsforboundplanetarymotion.........................237
9.8.3Unboundorbits........................................238
9.8.4Eccentricityvector... ...................................239
9.9Isotropic,linear,two-body,centralforce..............................241
9.9.1Polarcoordinates.......................................242
9.9.2Cartesiancoordinates....................................243
9.9.3Symmetrytensor A0 .....................................244
9.10Closed-orbitstability.........................................245
9.11Thethree-bodyproblem.......................................250
9.12Two-bodyscattering.........................................251
9.12.1Totaltwo-bodyscatteringcrosssection..........................251
9.12.2Differentialtwo-bodyscatteringcrosssection.......................252
9.12.3Impactparameterdependenceonscatteringangle....................252
9.12.4Rutherfordscattering....................................254
9.13Two-bodykinematics.........................................256 9.14Summary...............................................262
10Non-inertialreferenceframes267
10.1Introduction..............................................267
10.2Translationalaccelerationofareferenceframe ...........................267
10.3Rotatingreferenceframe.......................................268
10.3.1Spatialtimederivativesinarotating,non-translating,referenceframe.........268
10.3.2Generalvectorinarotating,non-translating,referenceframe..............269
10.4Referenceframeundergoingrotationplustranslation... ....................270
10.5Newton’slawofmotioninanon-inertialframe..........................270
10.6Lagrangianmechanicsinanon-inertialframe...........................271
10.7Centrifugalforce...........................................272
10.8Coriolisforce.............................................273
10.9Routhianreductionforrotatingsystems..............................277
10.10EffectivegravitationalforcenearthesurfaceoftheEarth....................280
10.11Freemotionontheearth.......................................282
10.12Weathersystems...........................................284
10.12.1Low-pressuresystems:....................................284
10.12.2High-pressuresystems:....................................286
10.13Foucaultpendulum..........................................286
10.14Summary...............................................288
Workshopexercises.............................................289 Problems..................................................290
11Rigid-bodyrotation 291
11.1Introduction..............................................291
11.2Rigid-bodycoordinates........................................292
11.3Rigid-bodyrotationaboutabody-fixedpoint...........................292
11.4Inertiatensor.............................................294
11.5Matrixandtensorformulationsofrigid-bodyrotation......................295
11.6Principalaxissystem.........................................295
11.7Diagonalizetheinertiatensor. ...................................296
11.8Parallel-axistheorem.........................................297 11.9Perpendicular-axistheoremforplanelaminae...........................300
11.10Generalpropertiesoftheinertiatensor...............................301 11.10.1Inertialequivalence......................................301
11.10.2Orthogonalityofprincipalaxes ...............................302
11.11Angularmomentum L andangularvelocity ω vectors......................303
11.12Kineticenergyofrotatingrigidbody................................305
11.13Eulerangles..............................................307
11.14Angularvelocity ω ..........................................309
11.15KineticenergyintermsofEulerangularvelocities........................310 11.16Rotationalinvariants.........................................311
11.17Euler’sequationsofmotionforrigid-bodyrotation........................312
11.18Lagrangeequationsofmotionforrigid-bodyrotation.......................313
11.19Hamiltonianequationsofmotionforrigid-bodyrotation.....................315
11.20Torque-freerotationofaninertially-symmetricrigidrotor....................315 11.20.1Euler’sequationsofmotion:.................................315 11.20.2Lagrangeequationsofmotion:...............................319
11.21Torque-freerotationofanasymmetricrigidrotor.........................321
11.22Stabilityoftorque-freerotationofanasymmetricbody.. ....................322
12.7Two-bodycoupledoscillatorsystems
12.11Dampedcoupledlinearoscillators
13Hamilton’sprincipleofleastaction381 13.1Introduction..............................................381
13.2PrincipleofLeastAction......................................382
13.2.1Hamilton’sPrinciple.....................................382 13.2.2Least-actionprincipleinHamiltonianmechanics.....................383 13.2.3Abbreviatedaction......................................384
13.3StandardLagrangian.........................................385
13.4GaugeinvarianceoftheLagrangian.................................385
13.5Non-standardLagrangians......................................387
13.6Inversevariationalcalculus.....................................387
13.7DissipativeLagrangians.......................................388
13.8Linearvelocity-dependentdissipation................................389 13.9Summary...............................................392
14AdvancedHamiltonianmechanics393 14.1Introduction..............................................393
14.2PoissonbracketrepresentationofHamiltonianmechanics....................395 14.2.1PoissonBrackets.......................................395 14.2.2FundamentalPoissonbrackets:...............................395 14.2.3Poissonbracketinvariancetocanonicaltransformations.................396 14.2.4CorrespondenceofthecommutatorandthePoissonBracket...............397 14.2.5ObservablesinHamiltonianmechanics...........................398 14.2.6Hamilton’sequationsofmotion...............................401
14.2.7Liouville’sTheorem.. ...................................405 14.3CanonicaltransformationsinHamiltonianmechanics.......................407 14.3.1Generatingfunctions.....................................408 14.3.2Applicationsofcanonicaltransformations.........................410 14.4Hamilton-Jacobitheory.......................................412 14.4.1Time-dependentHamiltonian................................412 14.4.2Time-independentHamiltonian...............................414 14.4.3Separationofvariables....................................415 14.4.4Visualrepresentationoftheactionfunction .......................422 14.4.5AdvantagesofHamilton-Jacobitheory...........................422 14.5Action-anglevariables........................................423 14.5.1Canonicaltransformation..................................423
14.5.2Adiabaticinvarianceoftheactionvariables........................426
14.6Canonicalperturbationtheory...................................428 14.7Symplecticrepresentation......................................430 14.8ComparisonoftheLagrangianandHamiltonianformulations..................430 14.9Summary...............................................432 Workshopexercises.............................................435 Problems..................................................436
15Analyticalformulationsforcontinuoussystems437 15.1Introduction..............................................437
15.2Thecontinuousuniformlinearchain................................437
15.3TheLagrangiandensityformulationforcontinuoussystems...................438 15.3.1Onespatialdimension....................................438 15.3.2Threespatialdimensions..................................439
15.4TheHamiltoniandensityformulationforcontinuoussystems..................440
15.5Linearelasticsolids..........................................441
15.5.1Stresstensor.........................................442
15.5.2Straintensor.........................................442
15.5.3Moduliofelasticity......................................443
15.5.4Equationsofmotioninauniformelasticmedia......................444
15.6Electromagnetic fieldtheory.....................................445
15.6.1Maxwellstresstensor....................................445
15.6.2Momentumintheelectromagnetic field..........................446
15.7Ideal fluiddynamics.........................................447
15.7.1Continuityequation.....................................447
15.7.2Euler’shydrodynamicequation...............................447
15.7.3Irrotational flowandBernoulli’sequation
15.7.4Gas flow............................................448
15.8Viscous fluiddynamics........................................450
15.8.1Navier-Stokesequation....................................450
15.8.2Reynoldsnumber.......................................451
15.8.3Laminarandturbulent fluid flow..............................451
15.9Summaryandimplications.....................................453
16Relativisticmechanics
16.3.3TimeDilation:........................................458
16.3.4LengthContraction.....................................459
16.3.5Simultaneity.........................................459
16.4Relativistickinematics........................................462
16.4.1Velocitytransformations...................................462
16.4.2Momentum..........................................462
16.4.3Centerofmomentumcoordinatesystem..........................463 16.4.4Force.............................................463
16.4.5Energy.............................................463
16.5Geometryofspace-time.......................................465
16.5.1Four-dimensionalspace-time................................465
16.5.2Four-vectorscalarproducts.................................466
16.5.3Minkowskispace-time....................................467
16.5.4Momentum-energyfourvector...............................468
16.6Lorentz-invariantformulationofLagrangianmechanics......................469
16.6.1Parametricformulation...................................469
16.6.2ExtendedLagrangian....................................469
16.6.3Extendedgeneralizedmomenta...............................471
16.6.4ExtendedLagrangeequationsofmotion..........................471
16.7Lorentz-invariantformulationsofHamiltonianmechanics.....................474
16.7.1Extendedcanonicalformalism................................474
16.7.2ExtendedPoissonBracketrepresentation.........................476
16.7.3ExtendedcanonicaltransformationandHamilton-Jacobitheory.............476
16.7.4ValidityoftheextendedHamilton-Lagrangeformalism..................476
16.8TheGeneralTheoryofRelativity..................................478
16.8.1Fundamentalconcepts....................................478
16.8.2Einstein’spostulatesoftheGeneralTheoryofRelativity.................479
16.8.3Experimentalevidence....................................479
16.9Implicationsofrelativistictheorytoclassicalmechanics.....................480
16.10Summary...............................................481
Workshopexercises.............................................482 Problems..................................................482
17Thetransitiontoquantumphysics483
17.1Introduction..............................................483
17.2Briefsummaryoftheoriginsofquantumtheory.........................483
17.2.1Bohrmodeloftheatom...................................485
17.2.2Quantization.........................................485
17.2.3Wave-particleduality....................................486
17.3Hamiltonianinquantumtheory...................................487
17.3.1Heisenberg’smatrix-mechanicsrepresentation.......................487
17.3.2Schrödinger’swave-mechanicsrepresentation.......................489
17.4Lagrangianrepresentationinquantumtheory...........................490
17.5CorrespondencePrinciple......................................491
17.6Summary...............................................492
18Epilogue 493
Appendices
AMatrixalgebra 495
A.1Mathematicalmethodsformechanics................................495
A.2Matrices................................................495
A.3Determinants.............................................499
A.4Reductionofamatrixtodiagonalform... ...........................501
BVectoralgebra 505
B.1Linearoperations...........................................505 B.2Scalarproduct............................................505
B.3Vectorproduct............................................506
B.4Tripleproducts............................................507
COrthogonalcoordinatesystems509
C.1Cartesiancoordinates ( ) ....................................509
C.2Curvilinearcoordinatesystems ...................................509
C.2.1Two-dimensionalpolarcoordinates ( ) ..........................510
C.2.2CylindricalCoordinates ( ) ..............................512
C.2.3SphericalCoordinates () ................................512
C.3Frenet-Serretcoordinates......................................513
DCoordinatetransformations515
D.1Translationaltransformations....................................515
D.2Rotationaltransformations.....................................515
D.2.1Rotationmatrix.......................................515
D.2.2Finiterotations........................................518
D.2.3Infinitessimalrotations....................................519
D.2.4Properandimproperrotations...............................519
D.3Spatialinversiontransformation...................................520
D.4Timereversaltransformation....................................521
ETensoralgebra 523
E.1Tensors................................................523
E.2Tensorproducts............................................524
E.2.1Tensorouterproduct.....................................524
E.2.2Tensorinnerproduct.....................................524
E.3Tensorproperties...........................................525
E.4Contravariantandcovarianttensors................................526
E.5Generalizedinnerproduct......................................527
E.6Transformationpropertiesofobservables..............................528
FAspectsofmultivariatecalculus529
F.1Partialdifferentiation........................................529
F.2Linearoperators...........................................529
F.3TransformationJacobian.......................................531
F.3.1Transformationofintegrals:.................................531
F.3.2Transformationofdifferentialequations:..........................531
F.3.3PropertiesoftheJacobian:.................................531
F.4Legendretransformation.......................................532
GVectordifferentialcalculus533
G.1Scalardifferentialoperators.....................................533
G.1.1Scalar field..........................................533
G.1.2Vector field..........................................533
G.2Vectordifferentialoperatorsincartesiancoordinates.......................533
G.2.1Scalar field..........................................533
G.2.2Vector field..........................................534
G.3Vectordifferentialoperatorsincurvilinearcoordinates.. ....................535
G.3.1Gradient:...........................................535
G.3.2Divergence:..........................................536
G.3.3Curl:..............................................536
G.3.4Laplacian:...........................................536
HVectorintegralcalculus537
H.1Lineintegralofthegradientofascalar field............................537
H.2Divergencetheorem.........................................537
H.2.1Fluxofavector fieldforGaussiansurface.........................537
H.2.2Divergenceincartesiancoordinates.............................538
H.3StokesTheorem............................................540
H.3.1Thecurl............................................540
H.3.2Curlincartesiancoordinates................................541
H.4Potentialformulationsofcurl-freeanddivergence-free fields...................543
IWaveformanalysis 545
I.1Harmonicwaveformdecomposition.................................545
I.1.1PeriodicsystemsandtheFourierseries...........................545
I.1.2AperiodicsystemsandtheFourierTransform.......................547
I.2Time-sampledwaveformanalysis..................................548
I.2.1Delta-functionimpulseresponse..............................549
I.2.2Green’sfunctionwaveformdecomposition.........................550
Example:Bolasthrownbygaucho
2.5 Example:Theidealgaslaw
2.6 Example:Themassofgalaxies
2.7 Example:Diatomicmolecule
2.8 Example:Rollercoaster
Example:Merry-go-round
2.14 Example:Centerofpercussionofabaseballbat
2.15 Example:Energytransferincharged-particlescattering
3.3 Example:Waterwavesbreakingonabeach
3.6 Example:FouriertransformofaGaussianwavepacket:
3.7 Example:Fouriertransformofarectangularwavepacket:
3.8 Example:Acousticwavepacket
3.9 Example:Gravitationalredshift ....................................82
3.10 Example:Quantumbaseball
4.1 Example:Non-linearoscillator
5.1 Example:Shortestdistancebetweentwopoints
5.2 Example:Brachistochroneproblem ...................................114
5.3 Example:Minimaltravelcost ......................................116
5.4 Example:Surfaceareaofacylindrically-symmetricsoapbubble
5.5 Example:Fermat’sPrinciple ......................................119
5.6 Example:Minimumof (∇)2 inavolume
5.7 Example:Twodependentvariablescoupledbyoneholonomicconstraint
5.8 Example:Catenary
5.9 Example:TheQueenDidoproblem
6.1 Example:Motionofafreeparticle,U=0 ...............................146
6.2 Example:Motioninauniformgravitational field
6.3 Example:Centralforces .........................................147
6.4 Example:Diskrollingonaninclinedplane
6.5 Example:Twoconnectedmassesonfrictionlessinclinedplanes
6.6 Example:Twoblocksconnectedbyafrictionlessbar
6.7 Example:Blockslidingonamovablefrictionlessinclinedplane
6.8 Example:Sphererollingwithoutslippingdownaninclinedplaneonafrictionless floor.
6.9 Example:Massslidingonarotatingstraightfrictionlessrod.
6.10
Example:Sphericalpendulum ......................................155
6.11 Example:Springplanependulum ....................................156
6.12 Example:Theyo-yo
6.13 Example:Massconstrainedtomoveontheinsideofafrictionlessparaboloid
6.14 Example:Massonafrictionlessplaneconnectedtoaplanependulum
6.15 Example:Twoconnectedmassesconstrainedtoslidealongamovingrod
6.16 Example:Massslidingonafrictionlesssphericalshell
6.17 Example:Rollingsolidsphereonasphericalshell
6.18 Example:Solidsphererollingplusslippingonasphericalshell
6.19 Example:Smallbodyheldbyfrictionontheperipheryofarollingwheel
6.20 Example:Planependulumhangingfromavertically-oscillatingsupport
6.21 Example:Series-coupleddoublependulumsubjecttoimpulsiveforce
7.1 Example:Feynman’sangular-momentumparadox
7.2 Example:Atwoodsmachine .......................................182
7.3 Example:Conservationofangularmomentumforrotationalinvariance: ..............183
7.4 Example:Diatomicmoleculesandaxially-symmetricnuclei .....................184
7.5 Example:Linearharmonicoscillatoronacartmovingatconstantvelocity
7.6 Example:Isotropiccentralforceinarotatingframe
7.7 Example:Theplanependulum .....................................191
7.8 Example:Oscillatingcylinderinacylindricalbowl
8.1 Example:Motioninauniformgravitational field ...........................205
8.2 Example:One-dimensionalharmonicoscillator ............................205
8.3 Example:Planependulum ........................................206
8.4 Example:Hooke’slawforceconstrainedtothesurfaceofacylinder
8.5 Example:Electronmotioninacylindricalmagnetron ........................208
8.6 Example:SphericalpendulumusingHamiltonianmechanics
8.7 Example:Sphericalpendulumusing
8.8 Example:Sphericalpendulumusing
8.9 Example:Singleparticlemovinginaverticalplaneundertheinfluenceofaninverse-square centralforce ...............................................216
8.10 Example:Driven,linearly-damped,coupledlinearoscillators
8.11 Example:Kirchhoff ’srulesforelectricalcircuits
9.1 Example:Centralforceleadingtoacircularorbit =2
9.2 Example:Orbitequationofmotionforafreebody
9.3 Example:Lineartwo-bodyrestoringforce
9.4 Example:Inversesquarelawattractiveforce
9.5 Example:Attractiveinversecubiccentralforce
9.6 Example:Spirallingmassattachedbyastringtoahangingmass
9.7 Example:Two-bodyscatteringbyaninversecubicforce
10.1 Example:Acceleratingspringplanependulum
10.2 Example:Surfaceofrotatingliquid
10.3 Example:Thepirouette
10.4 Example:Crankedplanependulum
10.5 Example:Nucleonorbitsindeformednuclei
10.6 Example:Freefallfromrest .......................................283
10.7 Example:Projectile firedverticallyupwards
10.8 Example:MotionparalleltoEarth’ssurface
11.1 Example:Inertiatensorofasolidcuberotatingaboutthecenterofmass.
11.2 Example:Inertiatensorofaboutacornerofasolidcube.
11.3 Example:Inertiatensorofahulahoop ................................301 11.4 Example:Inertiatensorofathinbook .................................301
11.5 Example:Rotationaboutthecenterofmassofasolidcube
11.6 Example:Rotationaboutthecornerofthecube
11.7 Example:Eulerangletransformation
11.8 Example:Rotationofadumbbell
11.9
Example:Precessionratefortorque-freerotatingsymmetricrigidrotor
11.10 Example:Tennisracquetdynamics ...................................323
11.11 Example:Rotationofasymmetrically-deformednuclei
11.12 Example:TheSpinning"Jack"
11.13 Example:TheTippeTop
11.14
Example:Tippingstabilityofarollingwheel
11.15 Example:Pivoting
11.16 Example:Rolling
11.17 Example:Forcesonthebearingsofarotatingcirculardisk
12.1 Example:TheGrandPiano
12.2 Example:Twocoupledlinearoscillators
12.3 Example:Twoequalmassesseries-coupledbytwoequalsprings
12.4 Example:Twoparallel-coupledplanependula
12.5 Example:Theseries-coupleddoubleplanependula
12.6 Example:Threeplanependula;mean-fieldlinearcoupling
12.7 Example:Threeplanependula;nearest-neighborcoupling
12.8 Example:Systemofthreebodiescoupledbysixsprings
12.9 Example:LineartriatomicmolecularCO 2
12.10 Example:Benzenering .........................................365
12.11 Example:Twolinearly-dampedcoupledlinearoscillators
12.12 Example:Collectivemotioninnuclei .................................375
13.1 Example:Gaugeinvarianceinelectromagnetism
13.2 Example:Thelinearly-damped,linearoscillator:
14.1 Example:Checkthatatransformationiscanonical
14.2 Example:Angularmomentum:
14.3 Example:Lorentzforceinelectromagnetism ..............................402
14.4 Example:Wavemotion: .........................................402
14.5 Example:Two-dimensional,anisotropic,linearoscillator
14.6 Example:Theeccentricityvector ....................................404 14.7 Example:Theidentitycanonicaltransformation
14.8 Example:Thepointcanonicaltransformation
14.9 Example:Theexchangecanonicaltransformation
14.10 Example:Infinitessimalpointcanonicaltransformation
14.11 Example:1-Dharmonicoscillatorviaacanonicaltransformation
14.12 Example:Freeparticle ..........................................415
14.13 Example:Pointparticleinauniformgravitational field
14.14 Example:One-dimensionalharmonicoscillator
14.15 Example:Thecentralforceproblem
14.16 Example:Linearly-damped,one-dimensional,harmonicoscillator
14.17 Example:Adiabaticinvarianceforthesimplependulum
14.18 Example:Harmonicoscillatorperturbation ..............................428
14.19 Example:Lindbladresonanceinplanetaryandgalacticmotion
15.1 Example:Acousticwavesinagas ...................................449 16.1 Example:Muonlifetime .........................................460
16.2 Example:RelativisticDopplerEffect ..................................461
16.3 Example:Twinparadox .........................................461
16.4 Example:Rocketpropulsion .......................................464
16.5 Example:Lagrangianforarelativisticfreeparticle ..........................472
16.6 Example:Relativisticparticleinanexternalelectromagnetic field
16.7 Example:TheBohr-Sommerfeldhydrogenatom ............................477
A.1 Example:Eigenvaluesandeigenvectorsofarealsymmetricmatrix
A.2 Example:Degenerateeigenvaluesofrealsymmetricmatrix
D.1 Example:Rotationmatrix: .......................................517
D.2 Example:Proofthatarotationmatrixisorthogonal
E.1 Example:Displacementgradienttensor
F.1 Example:Jacobianfortransformfromcartesiantosphericalcoordinates ..............531
H.1 Example:Maxwell’sFluxEquations ..................................539
H.2 Example:Buoyancyforcesin fluids ..................................540
H.3 Example:Maxwell’scirculationequations ...............................542
H.4 Example:Electromagnetic fields: ....................................543
I.1
Example:Fouriertransformofasingleisolatedsquarepulse: ....................548
I.2 Example:FouriertransformoftheDiracdeltafunction: .......................548
Preface
Thegoalofthisbookistointroducethereadertotheintellectualbeauty,andphilosophicalimplications, ofthefactthatnatureobeysvariationalprinciplesthatunderlietheLagrangianandHamiltoniananalytical formulationsofclassicalmechanics.Thesevariationalmethods,whichweredevelopedforclassicalmechanics duringthe 18 19 century,havebecomethepreeminentformalismsforclassicaldynamics,aswellasfor manyotherbranchesofmodernscienceandengineering.Theambitiousgoalofthisbookistoleadthestudent fromtheintuitiveNewtonianvectorialformulation,tointroductionofthemoreabstractvariationalprinciples thatunderlietheLagrangianandHamiltoniananalyticalformulations.Thisculminatesindiscussionofthe contributionsofvariationalprinciplestothedevelopmentofrelativisticandquantummechanics.Thebroad scopeofthisbookattemptstounifytheundergraduatephysicscurriculumbybridgingthechasmthat dividestheNewtonianvector-differentialformulationandtheintegralvariationalformulationofclassical mechanics,andthecorrespondingchasmthatexistsbetweenclassicalandquantummechanics.Powerful variationaltechniquesinmathematics,thatunderliemuchofmodernphysics,areintroducedandproblem solvingskillsaredevelopedinordertochallengestudentsatthecrucialstagewhenthey firstencounterthis sophisticatedandchallengingmaterial.Theunderlyingfundamentalconceptsofclassicalmechanics,and theirapplicationstomodernphysics,areemphasizedthroughoutthecourse.
Afullunderstandingofthepowerandbeautyofvariationalprinciplesinclassicalmechanics,isbest acquiredby firstlearningtheconceptsofthevariational approach,andthenapplyingtheseconceptsto manyexamplesinclassicalmechanics.Classicalmechanicsistheidealtopicforlearningtheprinciplesand thepowerofusingthevariationalapproachpriortoapplyingthesetechniquestootherbranchesofscience andengineering.TheunderlyingphilosophicalapproachadoptedbythisbookwasespousedbyGalileo Galilei"Youcannotteachamananything;youcanonlyhelphim finditwithinhimself."
Thedevelopmentofthistextbookwasinfluencedbythreetextbooks:"TheVariationalPrinciplesof Mechanics "byCorneliusLanczos(1949)[La49],"ClassicalMechanics" (1950)byHerbertGoldstein[Go50], and"ClassicalDynamicsofParticlesandSystems "(1965)byJerryB.Marion[Ma65].Marion’sexcellent textbookwasunusualinpartiallybridgingthechasmbetweentheoutstandinggraduatetextsbyGoldstein andLanczos,andabevyofintroductorytextsbasedonNewtonianmechanicsthatwereavailableatthat time.Thepresenttextbookwasdevelopedtocoverthetechniquesandphilosophicalimplicationsofthe variationalapproachestoclassicalmechanics,withabreadthanddepthclosetothatprovidedbyGoldstein andLanczos,butinaformatthatbettermatchestheneedsoftheundergraduatestudent.Anadditional goalistobridgethegapbetweenclassicalandmodernphysicsintheundergraduatecurriculum.
Thisbookwaswritteninsupportofthephysicsjunior/seniorundergraduatecourseP235Wentitled "VariationalPrinciplesinClassicalMechanics"thattheauthortaughtattheUniversityofRochesterbetween 1993 2015.Theselecturenotesweredistributedtostudents toallowpre-lecturestudy,facilitatedaccurate transmissionofthecomplicatedformulae,andminimizednotetakingduringlectures.Theselecturenotes evolvedintothepresenttextbookthatwasusedforthiscourse.Thetargetaudienceofthecourse,upon whichthistextbookisbased,typicallycomprised ≈ 70% junior/seniorundergraduates, ≈ 25% sophomores, ≤ 5% graduatestudents,andtheoccasionalwell-prep aredfreshman.Thetargetaudiencewasphysics andastrophysicsmajors,butitattractedasignificantfractionofmajorsfromotherdisciplinessuchas mathematics,chemistry,optics,engineering,music,andthehumanities.Asaconsequence,thebookincludes appreciableintroductorylevelphysics,plusmathematicalreviewmaterial,toaccommodatethediverse rangeofpriorpreparationofthestudents.Thistextbookincludesmaterialthatextendsbeyondwhat reasonablycanbecoveredduringaone-termcourse.Thissupplementalmaterialispresentedtoshowthe importanceandbroadapplicabilityofvariationalconceptstoclassicalmechanics.Thebookincludes 162 workedexamplestoillustratetheconceptspresented. Advancedgroup-theoreticconceptsareminimizedto
betteraccommodatethemathematicalskillsofthetypicalundergraduatephysicsmajor.Forcompatibility withmodernliteratureinthis field,thisbookfollowsthewidely-adoptednomenclatureusedin"Classical Mechanics"byGoldstein[Go50],withrecentadditionsbyJohns[Jo05].
Thebookisbrokenintofourmajorsections.This firstreviewsectionsetsthestagebyincludinga briefhistoricalintroduction(chapter 1),reviewoftheNewtonianformulationofmechanicsplusgravitation (chapter 2),linearoscillatorsandwavemotion(chapter 3),andanintroductiontonon-lineardynamics andchaos(chapter 4).Extensivereadingassignmentsareassignedtominimizethetimespentonthis reviewofNewtonianvectorialmechanics.Buildingontheintroductorysection,thesecondsectionofthe bookintroducesthevariationalprinciplesofanalyticalmechanicsthatunderliethisbook.Itincludesan introductiontothecalculusofvariations(chapter 5),theLagrangianformulationofmechanicswithapplicationstoholonomicandnon-holonomicsystems(chapter 6),adiscussionofsymmetries,invariance,plus Noether’stheorem(chapter 7)andanintroductiontotheHamiltonianandtheHamiltonianformulation ofmechanicsplustheRouthianreductiontechnique(Chapter 8).Thethirdsectionofthebook,applies LagrangianandHamiltonianformulationsofclassicaldynamicstocentralforceproblems(chapter 9),motioninnon-inertialframes(chapter 10),rigid-bodyrotation(chapter 11),andcoupledoscillators(chapter 12).The finalsectionofthebookdiscussesHamilton’sPrincipleplusadvancedapplicationsofLagrangian mechanics(chapter 13),HamiltonianmechanicsincludingPoisson brackets,Liouville’stheorem,canonical transformations,Hamilton-Jacobitheory,theaction-angletechnique(chapter 14),andclassicalmechanics inthecontinua(chapter 15).ThisisfollowedbyabriefreviewoftherevolutioninclassicalmechanicsintroducedbyEinstein’stheoryofrelativisticmechanics.TheextendedtheoryofLagrangianandHamiltonian mechanicsisusedtoapplyvariationaltechniquesto theSpecialTheoryofRelativityfollowedbyasuperficial introductiontotheconceptsofGeneralTheoryofRelativity(chapter 16).Thebook finisheswithabrief reviewoftheroleofvariationalprinciplesinbridgingthegapbetweenclassicalmechanicsandquantum mechanics,(chapter 17).Theseadvancedtopicsextendbeyondthetypicalsyllabusforanundergraduate classicalmechanicscourse.Thereasonforintroducingtheseadvancedtopicsistostimulatestudentinterest inphysicsbygivingthemaglimpseofthephysicsatthesummitthattheyhavestruggledtoclimb.This glimpseillustratesthebreadthofclassicalmechanics, andtherolethatvariationalprincipleshaveplayed inthedevelopmentofclassical,relativistic, quantal,andstatisticalmechanics.These finalsupplemental lecturesillustratethebeautyandunityofclassicalmechanics,andthefoundationthatclassicalmechanics hasprovidedtothedevelopmentofmodernphysics.Theappendicessummarizeaspectsofthemathematical methodsthatareexploitedinclassicalmechanics.
Thepresenttextbookcontainsmorematerialthanrequiredforajunior/seniorundergraduateclassical mechanicscourse,andthus,itcouldserveasthetextforagraduatecoursebyfocussingthecourseonthe variationalprinciplescoveredbychapters 5 17.Thepartitioningandorderingofthetopicsinthebook aretheresultofmanypermutationstriedwhileteachingclassicalmechanicsformanyyears.Chapters 1 through 3 plusthemathematicalappendices,areusedasreadingassignmentsduringthe firstthreeweeks ofclasstominimizethetimespentreviewingNewtonianmechanics.Thismaximizestheclasstimeavailable tocoverthevariationalapproach,thatis,chapters 5 through 14.Thebriefreviewsofthemechanicsinthe continua,andthetransitiontoquantummechanics,providethestudentwithaglimpseoftheimplications ofanalyticalmechanicstothesemoreadvancedtopics.
InformationregardingtheassociatedP235undergraduatecourseattheUniversityofRochesterisavailableonthewebsiteathttp://www.pas.rochester.edu/~cline/P235/index.shtml.Informationaboutthe authorisavailableattheClinehomewebsite:http://www.pas.rochester.edu/~cline/index.html.
TheauthorthanksMeghanSarkiswhopreparedmanyof theillustrations,JoeEasterlywhodesignedthe bookcoverplusthewebpage,andMorianaGarciawhoorganizedpublication.AndrewSifaindevelopedthe diagnosticworkshopquestions.Theauthorappreciates thepermission,grantedbyProfessorStruckmeier,to quotehispublishedarticleontheextendedHamilton-Lagrangianformalism.Theauthoracknowledgesthe feedbackandsuggestionsmadebymanystudentswhohavetakenthiscourse,aswellashelpfulsuggestions byhiscolleagues;AndrewAbrams,AdamHayes,ConnieJones,AndrewMelchionna,DavidMunson,Alice Quillen,RichardSarkis,JamesSchneeloch,Steven Torrisi,DanWatson,andFrankWolfs.Theselecture notesweretypedinLATEXusingScientificWorkPlace(MacKichanSoftware,Inc.),whileAdobeIllustrator, Photoshop,Origin,Mathematica,andMUPAD,wereusedtopreparetheillustrations.
DouglasCline, UniversityofRochester,2017
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