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Birkhäuser Advanced Texts Basler Lehrbücher Gerardo F. Torres del Castillo An Introduction to Hamiltonian Mechanics BirkhäuserAdvancedTextsBaslerLehrbücher Serieseditors StevenG.Krantz,WashingtonUniversity,St.Louis,USA ShrawanKumar,UniversityofNorthCarolinaatChapelHill,ChapelHill,USA JanNekováˇr,UniversitéPierreetMarieCurie,Paris,France Moreinformationaboutthisseriesathttp://www.springer.com/series/4842 Gerardo F. Torres del Castillo An Introduction to Hamiltonian Mechanics GerardoF.TorresdelCastillo InstitutodeCiencias,BUAP Puebla,Puebla,Mexico ISSN1019-6242 ISSN2296-4894 (electronic) BirkhäuserAdvancedTextsBaslerLehrbücher ISBN978-3-319-95224-6 ISBN978-3-319-95225-3 (eBook) https://doi.org/10.1007/978-3-319-95225-3 LibraryofCongressControlNumber:2018948684 Mathematics Subject Classification (2010): 34A34, 70-01, 70E17, 70F20, 70H03, 70H05, 70H15, 70H20,70H25,70H33,78A05 ©SpringerNatureSwitzerlandAG2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface The aim of this book is to present in an elementary manner the fundamentals of the Hamiltonian formulation of classical mechanics, making use of a basic knowledgeoflinearalgebra(matrices,propertiesofthedeterminantandthetrace), differential calculus in several variables (the differential of a function of several variables,thechainrule,andtheinversefunctiontheorem),analyticgeometry,and ordinary differential equations. Even though the main purpose of this book is the expositionof the Hamiltonian formalismof classical mechanics, the first chapters aredevotedtotheLagrangianformalismand,forareadernotfamiliarizedwiththe Lagrange equations, these introductory chapters should suffice to understand the basicelementsofanalyticalmechanics. Thisbookisintendedforadvancedundergraduateorgraduatestudentsinphysics orappliedmathematicsandforresearchersworkinginrelatedsubjects.Itisassumed that the reader has some familiarity with some elementarynotionsaboutclassical mechanics, such as the inertial reference frames and Newton’s second law. This book has been written having in mind readers trying to learn the subject by themselves,includingdetailedexamplesandexerciseswithcompletesolutions,but itcanalsobeusedasaclasstext. Thisbookdoesnotattempttobeanexhaustivetreatmentofanalyticalmechanics orevenoftheHamiltonianformulationofclassicalmechanics.Deliberately,some subjectsarenottreatedinthisbook;thesubjectsnotincludedherehavebeenomitted for at least one of the following reasons: they are not essential to understand the basicformalism,orthereexistbooksorarticlescontainingagooddiscussionofthe subjectthatwouldbedifficulttoimprove. Some of the subjects not treated here are dissipative systems, nonholonomic constraints,adiabaticinvariants,action-anglevariablesforsystemswithmorethan onedegreeoffreedom,perturbationtheory,continuoussystems, normalmodesof vibration, integral invariants, relativistic mechanics, singular Lagrangians, KAM theory,andchaos. v vi Preface Throughout the book I have avoided the use of terms like “small parameter,” “small variation,” “infinitesimal parameter,” “infinitesimal transformation,” and a ˜ ˜˜ diversityofvariationsdenotedbysymbolslikeΔ,δ,d,δ,δ,andsoon.Apartfrom this, the notation employed throughoutthe book coincides with that found in the traditionalbooksonanalyticalmechanics. WhilethereseemstoexistaconsensusabouttheimportanceoftheHamiltonian formalism, in the standard textbooks on analytical mechanics one finds a variety of opinionsaboutthe Hamilton–Jacobiequation,rangingfromthe convictionthat the Hamilton–Jacobiformalism is the most powerfultool of analyticalmechanics totheclaimthattheHamilton–Jacobiequationhasnopracticalvalueandthatitis onlyinterestingbecauseofitsrelationwiththeSchrödingerequation.Thepointof viewadoptedhereisthat,apartfromitsdeepandusefulconnectionswithquantum mechanics,the Hamilton–Jacobiformalism is interesting and usefulby itself, as I havetriedtoillustrateinChapter6. Among the differences between this book and the existing textbooks on the subjectarethepresenceof: (cid:129) Application of the various formulations to equations not related to classical mechanics. (cid:129) Systematic use of equivalent Hamiltonians, which allows us to relate different problems and to find constants of motion without integrating the equations of motion. (cid:129) Detailed derivationsof the canonicaltransformations, emphasizing the distinc- tionbetweenwhatisageneratingfunctionandwhatisnot. (cid:129) Studyofthecontinuousgroupsofcanonicaltransformations,avoidingtheuseof “infinitesimals.” (cid:129) Precise definition and examplesof the symmetries of a Hamiltonian, including thecaseoftransformationsthatinvolvethetimeexplicitly. (cid:129) Emphasisonthefactthat,intheHamiltonianformalism,thereareinfinitelymany generating functionsof translations and rotations(and, therefore, that, e.g., the linearmomentumcannotbedefinedasthegeneratingfunctionoftranslations). (cid:129) Studyofthecanonoidtransformationsandtheassociatedconstantsofmotion. (cid:129) DefinitionandexamplesofR-separablesolutionsoftheHamilton–Jacobiequa- tion. (cid:129) General statement, simplified proof, and detailed examples of the Liouville theoremonsolutionsoftheHamilton–Jacobiequation. (cid:129) Discussion and examples of the mapping of solutions of the Hamilton–Jacobi equationundercanonicaltransformations. (cid:129) Discussion of the Hamilton–Jacobi equation as an evolution equation for the principalfunction. (cid:129) PresentationofgeometricalopticsasanapplicationoftheHamiltonianformal- ism. (cid:129) Detailedsolutionofalltheexercises. Preface vii Many textbooks on analytical mechanics written in the last few decades make use of the language of modern differential geometry (manifolds, vector fields, differentialforms),which is particularlyusefulandelegantwhenthe Hamiltonian does not depend on the time. In fact, in classical mechanics one has one of the nicestapplicationsofthisformalism.Oneoftheaimsofthisbookistoshowthatit ispossibletoobtainmanyinterestingresultsmakinguseofelementarymathematics only. Throughout the book, there is a collection of examples worked out in detail, whichforman essentialpartof the book,anda setof exercisesis also given.Itis advisablethat the readerattempts to solve them all and to fill in the details of the computations presented in the book. The detailed solutions of all the exercises are collected at the end of the book, but the reader is encouraged to try to find the solutions before seeing the answers. Some sections go beyond the basic leveland can be skipped;these sectionsare Section 2.5(VariationalSymmetries), Section4.3.1(TheKeplerProblemRevisited),Section5.5(CanonoidTransforma- tions),Section6.3(MappingofSolutionsoftheHamilton–JacobiEquationUnder Canonical Transformations), Section 6.4 (Transformation of the Hamilton–Jacobi Equation Under Arbitrary Point Transformations), and Section 6.5 (Geometrical Optics). Throughout the book, references are given to some books or papers when the subjectisnotcommonlytreatedinthestandardtextbooks. Some words aboutthe notation:I have avoided the use of superscriptsto label coordinatesorcomponents,consideringthattheyarenotindispensableatthislevel and, sometimes, its use can complicate the expressions. The sign ≡ indicates a definition and in all cases it should be clear which side of the sign contains the objectbeingdefined. IwouldliketothankDr.IraísRubalcava-Garcíaforherhelpwiththefiguresand thereviewersfortheirhelpfulcomments.IwouldalsoliketothankSamuelDiBella atSpringerNatureforhisvaluablesupport. Puebla,Puebla,Mexico GerardoF.TorresdelCastillo May2018 Contents 1 TheLagrangianFormalism................................................. 1 1.1 IntroductoryExamples.TheD’AlembertPrinciple................... 1 1.2 TheLagrangeEquations................................................ 18 2 SomeApplicationsoftheLagrangianFormalism........................ 43 2.1 CentralForces........................................................... 43 2.2 FurtherExamples ....................................................... 55 2.3 TheLagrangiansCorrespondingtoaSecond-OrderOrdinary DifferentialEquation.................................................... 60 2.4 Hamilton’sPrinciple.................................................... 65 2.5 VariationalSymmetries................................................. 74 3 RigidBodies .................................................................. 81 3.1 TheConfigurationSpaceofaRigidBodywithaFixedPoint........ 81 3.2 TheInstantaneousAngularVelocityandtheInertiaTensor .......... 84 3.3 TheEulerAngles........................................................ 98 4 TheHamiltonianFormalism................................................ 103 4.1 TheHamiltonEquations................................................ 103 4.2 ThePoissonBracket .................................................... 114 4.2.1 Hamilton’sPrincipleinthePhaseSpace ...................... 122 4.3 EquivalentHamiltonians................................................ 123 4.3.1 TheKeplerProblemRevisited ................................. 135 5 CanonicalTransformations................................................. 143 5.1 SystemswithOneDegreeofFreedom................................. 144 5.2 SystemswithanArbitraryNumberofDegreesofFreedom.......... 167 5.3 One-ParameterGroupsofCanonicalTransformations................ 198 5.4 SymmetriesoftheHamiltonianandConstantsofMotion............ 205 5.5 CanonoidTransformations ............................................. 221 ix x Contents 6 TheHamilton–JacobiFormalism .......................................... 229 6.1 TheHamilton–JacobiEquation......................................... 230 6.1.1 RelationBetweenCompleteSolutionsoftheHJEquation... 242 6.1.2 AlternativeExpressionsoftheHJEquation................... 245 6.1.3 R-SeparableSolutionsoftheHJEquation.................... 247 6.2 TheLiouvilleTheoremonSolutionsoftheHJEquation ............ 252 6.3 MappingofSolutionsoftheHJEquationUnderCanonical Transformations......................................................... 259 6.3.1 TheHJEquationasanEvolutionEquation ................... 266 6.4 TransformationoftheHJEquationUnderArbitraryPoint Transformations......................................................... 269 6.5 GeometricalOptics...................................................... 272 Solutions........................................................................... 281 References......................................................................... 361 Index............................................................................... 363