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1、LectureNotesinMathematics1844Editors:J.--M.Morel,CachanF.Takens,GroningenB.Teissier,Paris3BerlinHeidelbergNewYorkHongKongLondonMilanParisTokyoKarlFriedrichSiburgThePrincipleofLeastActioninGeometryandDynamics13AuthorKarlFriedrichSiburgFakultatf¨urMathematik¨Ruhr-UniversitatBochum¨4
2、4780Bochum,Germanye-mail:siburg@math.ruhr-uni-bochum.deLibraryofCongressControlNumber:2004104313MathematicsSubjectClassification(2000):37J,53D,58EISSN0075-8434ISBN3-540-21944-7Springer-VerlagBerlinHeidelbergNewYorkThisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeor
3、partofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,reproductiononmicrofilmorinanyotherway,andstorageindatabanks.DuplicationofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawof
4、September9,1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer-Verlag.ViolationsareliableforprosecutionundertheGermanCopyrightLaw.Springer-VerlagBerlinHeidelbergNewYorkamemberofBertelsmannSpringerScience+BusinessMediaGmbHhttp://www.springer.decSpringer-Ve
5、rlagBerlinHeidelberg2004PrintedinGermanyTheuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.Typesetting:Came
6、ra-readyTEXoutputbytheauthorSPIN:1100219241/3142/du-543210-Printedonacid-freepaperPrefaceThemotionofclassicalmechanicalsystemsisdeterminedbyHamilton’sdif-ferentialequations:x˙(t)=∂yH(x(t),y(t))y˙(t)=−∂xH(x(t),y(t))Forinstance,ifweconsiderthemotionofnparticlesinapotentialfield,theH
7、amiltonianfunctionn12H=yi−V(x1,...,xn)2i=1isthesumofkineticandpotentialenergy;thisisjustanotherformulationofNewton’sSecondLaw.AdistinguishedclassofHamiltoniansonacotangentbundleT∗Xcon-sistsofthosesatisfyingtheLegendrecondition.TheseHamiltoniansareob-tainedfromLagrangiansystemsont
8、heconfigurationspaceX,withcoordi-n
