Lagrange equation and Hamiltonian

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1、LagrangianandHamiltonianMechanicsD.G.Simpson,Ph.D.DepartmentofPhysicalSciencesandEngineeringPrinceGeorge’sCommunityCollegeDecember5,2007IntroductionInthiscoursewehavebeenstudyingclassicalmechanicsasformulatedbySirIsaacNewton;thisiscalledNewtonianmechanics.Newtonianmechanicsismathematicallyfairly

2、straightforward,andcanbeappliedtoawidevarietyofproblems.Itisnotauniqueformulationofmechanics,however;otherformulationsarepossible.Herewewilllookattwocommonalternativeformulationsofclassicalmechanics:LagrangianmechanicsandHamiltonianmechanics.Itisimportanttounderstandthatalloftheseformulationsofm

3、echanicsequivalent.Inprinciple,anyofthemcouldbeusedtosolveanyprobleminclassicalmechanics.Thereasonthey’reimportantisthatinsomeproblemsoneofthealternativeformulationsofmechanicsmayleadtoequationsthataremucheasiertosolvethantheequationsthatarisefromNewtonianmechanics.UnlikeNewtonianmechanics,neith

4、erLagrangiannorHamiltonianmechanicsrequirestheconceptofforce;instead,thesesystemsareexpressedintermsofenergy.Althoughwewillbelookingattheequationsofmechanicsinonedimension,alltheseformulationsofmechanicsmaybegeneralizedtotwoorthreedimensions.NewtonianMechanicsWebeginbyreviewingNewtonianmechanics

5、inonedimension.Inthisformulation,webeginbywritingNewton’ssecondlaw,whichgivestheforceFrequiredtogiveanaccelerationatoamassm:FDma:(1)Generallytheforceisafunctionofx.SincetheaccelerationaDd2x=dt2,Eq.(1)maybewrittend2xF.x/Dm:(2)dt2Thisisasecond-orderordinarydifferentialequation,whichwesolveforx.t/t

6、ofindthepositionxatanytimet.SolvingaprobleminNewtonianmechanicsthenconsistsofthesesteps:1.WritedownNewton’ssecondlaw(Eq.2);2.SubstituteforF.x/thespecificforcepresentintheproblem;3.Solvetheresultingdifferentialequationforx.t/.1PartialDerivativesTheequationsofLagrangianandHamiltonianmechanicsareexpr

7、essedinthelanguageofpartialdifferentialequations.Wewillleavethemethodsforsolvingsuchequationstoamoreadvancedcourse,butwecanstillwritedowntheequationsandexploresomeoftheirconsequences.First,inordertounderstandtheseequations,w