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1、Euler–LagrangeequationJumpto:navigation,searchIncalculusofvariations,theEuler–Lagrangeequation,orLagrange'sequation,isadifferentialequationwhosesolutionsarethefunctionsforwhichagivenfunctionalisstationary.ItwasdevelopedbySwissmathematicianLeonhardEulerandItalianmathema
2、ticianJosephLouisLagrangeinthe1750s.Becauseadifferentiablefunctionalisstationaryatitslocalmaximaandminima,theEuler–Lagrangeequationisusefulforsolvingoptimizationproblemsinwhich,givensomefunctional,oneseeksthefunctionminimizing(ormaximizing)it.ThisisanalogoustoFermat'st
3、heoremincalculus,statingthatwhereadifferentiablefunctionattainsitslocalextrema,itsderivativeiszero.InLagrangianmechanics,becauseofHamilton'sprincipleofstationaryaction,theevolutionofaphysicalsystemisdescribedbythesolutionstotheEuler–Lagrangeequationfortheactionofthesys
4、tem.Inclassicalmechanics,itisequivalenttoNewton'slawsofmotion,butithastheadvantagethatittakesthesameforminanysystemofgeneralizedcoordinates,anditisbettersuitedtogeneralizations(see,forexample,the"Fieldtheory"sectionbelow).HistoryTheEuler–Lagrangeequationwasdevelopedint
5、he1750sbyEulerandLagrangeinconnectionwiththeirstudiesofthetautochroneproblem.Thisistheproblemofdeterminingacurveonwhichaweightedparticlewillfalltoafixedpointinafixedamountoftime,independentofthestartingpoint.Lagrangesolvedthisproblemin1755andsentthesolutiontoEuler.Thet
6、wofurtherdevelopedLagrange'smethodandappliedittomechanics,whichledtotheformulationofLagrangianmechanics.Theircorrespondenceultimatelyledtothecalculusofvariations,atermcoinedbyEulerhimselfin1766.[1]StatementTheEuler–Lagrangeequationisanequationsatisfiedbyafunctionqofare
7、alargumenttwhichisastationarypointofthefunctionalwhere:·qisthefunctiontobefound:suchthatqisdifferentiable,q(a)=xa,andq(b)=xb;·q′isthederivativeofq:TXbeingthetangentbundleofX(thespaceofpossiblevaluesofderivativesoffunctionswithvaluesinX);·Lisareal-valuedfunctionwithcont
8、inuousfirstpartialderivatives:TheEuler–Lagrangeequation,then,istheordinarydifferentialequationwhereLxandLvdenotethepa
