The Principle of Least Action: History and Physics

270 Pages·2018·5.03 MB·English

by  Alberto Rojo

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
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9780521869027
DOI:10.1017/9781139021029
(cid:2)c AlbertoRojoandAnthonyBloch2018
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Firstpublished2018
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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
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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