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1、Euler–LagrangeequationFromWikipedia,thefreeencyclopediaJumpto:navigation,searchIncalculusofvariations,theEuler–Lagrangeequation,orLagrange'sequation,isadifferentialequationwhosesolutionsarethefunctionsforwhichagivenfunctionalisstationary.Itwasdevelop
2、edbySwissmathematicianLeonhardEulerandItalianmathematicianJosephLouisLagrangeinthe1750s.Becauseadifferentiablefunctionalisstationaryatitslocalmaximaandminima,theEuler–Lagrangeequationisusefulforsolvingoptimizationproblemsinwhich,givensomefunctional,o
3、neseeksthefunctionminimizing(ormaximizing)it.ThisisanalogoustoFermat'stheoremincalculus,statingthatwhereadifferentiablefunctionattainsitslocalextrema,itsderivativeiszero.InLagrangianmechanics,becauseofHamilton'sprincipleofstationaryaction,theevolutio
4、nofaphysicalsystemisdescribedbythesolutionstotheEuler–Lagrangeequationfortheactionofthesystem.Inclassicalmechanics,itisequivalenttoNewton'slawsofmotion,butithastheadvantagethatittakesthesameforminanysystemofgeneralizedcoordinates,anditisbettersuitedt
5、ogeneralizations(see,forexample,the"Fieldtheory"sectionbelow).Contents1History·2Statement·3Exampleso3.1Classicalmechanics§3.1.1Basicmethod§3.1.2Particleinaconservativeforcefieldo3.2Fieldtheory·4Variationsforseveralfunctions,severalvariables,andhigher
6、derivativeso4.1Singlefunctionofsinglevariablewithhigherderivativeso4.2Severalfunctionsofonevariableo4.3Singlefunctionofseveralvariableso4.4Severalfunctionsofseveralvariableso4.5Singlefunctionoftwovariableswithhigherderivatives·5Notes·6References·7See
7、alsoHistoryTheEuler–Lagrangeequationwasdevelopedinthe1750sbyEulerandLagrangeinconnectionwiththeirstudiesofthetautochroneproblem.Thisistheproblemofdeterminingacurveonwhichaweightedparticlewillfalltoafixedpointinafixedamountoftime,independentofthestart
8、ingpoint.Lagrangesolvedthisproblemin1755andsentthesolutiontoEuler.ThetwofurtherdevelopedLagrange'smethodandappliedittomechanics,whichledtotheformulationofLagrangianmechanics.Theircorrespondenceultimatelyledtothecalculusofvariations,atermcoinedbyEuler
