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1、FoundationsofComputationalGeometricMechanicsThesisbyMelvinLeokInPartialFulfillmentoftheRequirementsfortheDegreeofDoctorofPhilosophyCaliforniaInstituteofTechnologyPasadena,California2004(DefendedMay6,2004)iiiiic2004MelvinLeokAllRightsReservedivv“Thoughtisonlyaflashbetwee
2、ntwolongnights,butthisflashiseverything”HenriPoincar´e,1854–1912.viviiPrefaceThisthesiswassubmittedattheCaliforniaInstituteofTechnologyonApril19,2004,inpartialfulfillmentoftherequirementsforthedegreeofDoctorofPhilosophyinControlandDynamicalSystemswithaminorinAppliedandC
3、omputationalMathematics.ThethesiswasdefendedonMay6,2003,inPasadena,CA,andwasapprovedbythefollowingthesiscommittee:•ThomasY.Hou,AppliedandComputationalMathematics,CaliforniaInstituteofTechnology,•JerroldE.Marsden(Chair),ControlandDynamicalSystems,CaliforniaInstituteofT
4、echnology,•RichardM.Murray,ControlandDynamicalSystems,andMechanicalEngineering,CaliforniaInstituteofTechnology,•MichaelOrtiz,Aeronautics,andMechanicalEngineering,CaliforniaInstituteofTechnology,•AlanD.Weinstein,Mathematics,UniversityofCalifornia,Berkeley.Thisthesisdra
5、wsuponthefollowingpapersthathaveorwillbesubmittedforpublication,butanefforthasbeenmadetoclarifythemannerinwhichtheresearchrelatestoeachotherbyincorporatingextendedintroductionsandconclusionswithineachchapter,andthroughtheuseofexamplesthatdrawuponthevariouschapters.•S.M
6、.Jalnapurkar,M.Leok,J.E.Marsden,andM.West,DiscreteRouthReduction,J.FoCM,submitted.•M.Desbrun,A.N.Hirani,M.Leok,andJ.E.Marsden,DiscreteExteriorCalculus,inprepara-tion.•M.Desbrun,A.N.Hirani,M.Leok,andJ.E.Marsden,DiscretePoincar´eLemma,Appl.Nu-mer.Math.,submitted.viii•M.
7、Leok,J.E.Marsden,A.D.Weinstein,ADiscreteTheoryofConnectionsonPrincipalBun-dles,inpreparation.•M.Leok,GeneralizedGalerkinVariationalIntegrators,inpreparation.Areviewarticlebasedonpreliminaryversionsoftheworkondiscreteexteriorcalculus,discretePoincar´elemma,anddiscretec
8、onnectionsonprincipalbundles,receivedtheSIAMStudentPaperPrize,andtheLeslieFoxPrizeinNumericalAnalysis(secondprize),bothin200
