THE PRINCIPLE OF LEAST ACTION HistoryandPhysics The principle of least action originates in the idea that Nature has a purpose and thusshouldfollowaminimumorcriticalpath.Thisbasicprinciple,withitsvariants andgeneralizations,appliestooptics,mechanics,electromagnetism,relativityand quantummechanics,andprovidesaguidetounderstandingthebeautyofphysics. This text provides an accessible introduction to the action principle across these variousfieldsofphysicsandexaminesitshistoryandfundamentalroleinscience. Itincludesexplanationsfromhistoricalsources,discussionsofclassicpapers,and originalworkedexamples. Differentsectionsrequiredifferentlevelsofmathematicalsophistication.How- ever,themainstorylineisaccessiblenotonlytoresearchersandstudentsinphysics and the history of physics, but also to those with a more modest mathematical background. ALBERTO ROJO is an Associate Professor at Oakland University. He is a Ful- bright Specialist in Physics Education and he was awarded the Jack Williams Endowed Chair in Science and Humanities from the University of Eastern New Mexico. His research focuses primarily on theoretical condensed matter, and he haspreviouslypublishedpopularsciencebooks. ANTHONY BLOCHistheAlexanderZiwetCollegiateProfessorofMathematics attheUniversityofMichiganwhereheistheDepartmentChair.Hehasservedon theeditorialboardsofvariousjournals,heisaFellowofseveralprofessionalsoci- eties,andhehasreceivedaPresidentialYoungInvestigatorAward,aGuggenheim FellowshipandaSimonsFellowship. THE PRINCIPLE OF LEAST ACTION History and Physics ALBERTO ROJO OaklandUniversity,Michigan ANTHONY BLOCH UniversityofMichigan UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre,NewDelhi–110025,India 79AnsonRoad,#06–04/06,Singapore079906 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9780521869027 DOI:10.1017/9781139021029 (cid:2)c AlbertoRojoandAnthonyBloch2018 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2018 PrintedintheUnitedKingdombyClays,StIvesplc AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. LibraryofCongressCataloging-in-PublicationData Names:Rojo,AlbertoG.,author.|Bloch,Anthony,author. Title:Theprincipleofleastaction:historyandphysics/AlbertoRojo (OaklandUniversity,Michigan),AnthonyBloch(UniversityofMichigan). Description:Cambridge,UnitedKingdom;NewYork,NY: CambridgeUniversityPress,2018. |Includesbibliographicalreferencesandindex. Identifiers:LCCN2017023575|ISBN9780521869027(hardback;alk.paper)| ISBN0521869021(hardback;alk.paper) Subjects:LCSH:Leastaction.|Variationalprinciples.|Mechanics.| Lagrangeequations.|Hamilton-Jacobiequations. Classification:LCCQA871.R652017|DDC530.1–dc23 LCrecordavailableathttps://lccn.loc.gov/2017023575 ISBN978-0-521-86902-7Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. Contents ListofIllustrations pageix Acknowledgments xii 1 Introduction 1 2 PrehistoryofVariationalPrinciples 6 2.1 QueenDidoandtheIsoperimetricProblem 6 2.1.1 Zenodorus’sSolution* 7 2.2 HeroofAlexandriaandtheLawofReflection 13 2.3 GalileoandtheCurveofSwiftestDescent 14 2.4 BendingofLightRaysandFermat’sMinimumPrinciple 18 2.4.1 Fermat’sMethodofMaximaandMinima 21 2.4.2 Huygens’SimplifiedDerivationofSnell’sLaw 24 2.5 NewtonandtheSolidofLeastResistance* 27 2.5.1 TheSphereandtheCylinder 28 2.5.2 AnApplication“intheBuildingofShips” 30 2.5.3 TheFirstGenuineVariationalCalculation 34 3 AnExcursiontoNewton’sPrincipia 38 3.1 Newton’sPropositionsontheLawsofMotion 38 3.2 GeometricalDerivationofKepler’sLawsofPlanetaryMotion 39 3.2.1 Proposition1:EqualAreasareSweptOutinEqualTimes 39 3.2.2 Proposition6:TheForceLawandtheGeometryoftheOrbit* 41 3.2.3 CircularOrbits 43 3.2.4 Proposition10:EllipticalOrbitwiththeCenterofForce attheCenteroftheEllipse 45 3.2.5 Proposition11:CenterofForceattheFocusoftheEllipse* 48 4 TheOptical-MechanicalAnalogy,PartI 51 4.1 Bernoulli’sChallengeandtheBrachistochrone 51 4.1.1 HuygensandtheHorologiumOscillatorium 52 v vi Contents 4.1.2 Leibniz’sSolutionoftheBrachistochrone 55 4.1.3 Bernoulli’sSolution:ParticlePathsasLightRays 57 4.2 Maupertuis,LeastAction,andMetaphysicalMechanics 59 4.3 EulerandtheMethodofMaximaandMinima* 62 4.3.1 Euler’sDerivationofOrbitsfromtheLeastActionPrinciple 65 4.4 ExamplesoftheOptical-MechanicalAnalogy 68 4.4.1 Conservationof“AngularMomentum”forLightRays 69 4.4.2 TheTerrestrialBrachistochrone 71 4.5 TheStringAnalogyandthePrincipleofLeastAction 72 4.5.1 TheLeastActionPrincipleandStretchableStrings 75 5 D’Alembert,Lagrange,andtheStatics-DynamicsAnalogy 79 5.1 ThePrincipleofVirtualWork 79 5.2 StaticsMeetsDynamics:Bernoulli’sCalculationoftheCenter ofOscillation 82 5.3 D’Alembert’sPrinciple 85 5.4 Lagrange’sDynamics 90 5.4.1 Lagrange’s“ScientificPoem” 93 5.4.2 Symmetries 96 5.5 Lagrange versus d’Alembert: Dissipative and Nonholonomic Systems 100 5.5.1 DissipationinaReversibleSystem:Lamb’sModel 101 5.5.2 NonholonomicSystems 103 5.6 Gauss’sPrincipleofLeastConstraint 105 5.7 Least Action with a Twist: the Elasticity of the Ether and Maxwell’sEquations* 107 6 TheOptical-MechanicalAnalogy,PartII:TheHamilton-Jacobi Equation 112 6.1 Hamilton’s“TheoryofSystemsofRays” 112 6.2 ConicalRefraction* 119 6.2.1 Fresnel’sEquationsforAnisotropicCrystals 119 6.2.2 AnalyticalDerivationoftheWaveSurface 122 6.2.3 Hamilton’sDerivationoftheConicalCusp 124 6.2.4 InternalConicalRefraction:“ThePlumLaidDownona Table” 125 6.3 Hamilton’sLawofVaryingAction* 128 6.4 AnExamplefromHamilton:TheCharacteristicFunction V foraParabolicOrbit* 131 6.5 Hamilton’s“SecondEssayonaGeneralMethodinDynamics”* 134 Contents vii 6.5.1 Example:ParticleinaUniformGravitationalField 140 6.6 Hamilton-JacobiandHuygens’Principle* 142 6.7 ApplicationsandExamples 143 6.7.1 TheEquationofaLightRay 143 6.7.2 HamiltonianoftheHarmonicOscillator 145 6.7.3 Hamilton-JacobiEquationforaParticleinaMagneticField 146 6.8 WhenthePrincipleofLeastActionLosesits“Least” 149 6.8.1 FocusandKineticFocus 149 6.8.2 KineticFocusforaFreeParticleonaSphere 151 6.8.3 SaddlePathsfortheHarmonicOscillator 152 6.8.4 KineticFocusofEllipticPlanetaryOrbits 154 6.8.5 Gouy’sPhaseandCriticalAction 155 6.8.6 Caustics 158 7 RelativityandLeastAction 162 7.1 SimultaneityandtheRelativityofTime 163 7.2 TheRelativistic“F = ma” 166 7.2.1 TheEnergy-MomentumFour-Vector 169 7.2.2 InvarianceoftheRelativisticAction 171 7.3 Hamilton-JacobiEquationforaRelativisticParticle 172 7.4 ThePrincipleofEquivalence 173 7.4.1 Bending of Light Rays According to the Equivalence Principle 176 7.4.2 BendingofLightRays,NewtonianCalculation 178 7.5 Space-TimeisCurved 180 7.6 WeakGravityaroundaStatic,SphericalStar 182 7.6.1 PrecessionofthePerihelionofMercury 182 7.6.2 BendingoftheLightRaysintheGeneralTheory 185 7.7 Hilbert’sLeastActionPrincipleforGeneralRelativity* 186 8 TheRoadtoQuantumMechanics 189 8.1 TheNeedforaNewMechanics 189 8.2 Bohr’s“Trilogy”of1913andSommerfeld’sGeneralization 192 8.2.1 SommerfeldandtheKeplerProblem 195 8.2.2 TheFineStructureoftheHydrogenSpectrum 198 8.3 AdiabaticInvariants 199 8.4 DeBroglie’sMatterWaves 203 8.5 Schrödinger’sWaveMechanics 205 8.5.1 TheEikonalEquation 206 8.5.2 Schrödinger’sDerivation 207 viii Contents 8.6 Dirac’sLagrangianViewofQuantumMechanics 209 8.7 Feynman’sThesisandPathIntegrals 211 8.8 Huygens’PrincipleinOpticsandQuantumMechanics* 214 8.8.1 First-Order(inTime)PropagatorfortheWaveEquation 215 8.8.2 Huygens’PrincipleandSphericalWavelets 217 8.8.3 CancellationoftheBackwardsWave 218 AppendixA Newton’sSolidofLeastResistance,UsingCalculus 221 AppendixB OriginalStatementofd’Alembert’sPrinciple 223 AppendixC EquationsofMotionofMcCullagh’sEther 224 AppendixD CharacteristicFunctionforaParabolicKeplerianOrbit 225 AppendixE SaddlePathsforReflectionsonaMirror 227 AppendixF KineticCausticsfromQuantumMotionin OneDimension 229 AppendixG Einstein’sProofoftheCovarianceof Maxwell’sEquations 233 AppendixH RelativisticFour-VectorPotential 235 AppendixI Ehrenfest’sProofoftheAdiabaticTheorem 238 References 241 Index 254