Euler–Lagrange-equation

Euler–LagrangeequationFromWikipedia,thefreeencyclopediaJumpto:navigation,searchIncalculusofvariations,theEuler–Lagrangeequation,orLagrange'sequation,isadifferentialequationwhosesolutionsarethefunctionsforwhichagivenfunctionalisstationary.ItwasdevelopedbySwissmathematicianLeonhardEulerandItalianmathematicianJosephLouisLagrangeinthe1750s.Becauseadifferentiablefunctionalisstationaryatitslocalmaximaandminima,theEuler–Lagrangeequationisusefulforsolvingoptimizationproblemsinwhich,givensomefunctional,oneseeksthefunctionminimizing(ormaximizing)it.ThisisanalogoustoFermat'stheoremincalculus,statingthatwhereadifferentiablefunctionattainsitslocalextrema,itsderivativeiszero.InLagrangianmechanics,becauseofHamilton'sprincipleofstationaryaction,theevolutionofaphysicalsystemisdescribedbythesolutionstotheEuler–Lagrangeequationfortheactionofthesystem.Inclassicalmechanics,itisequivalenttoNewton'slawsofmotion,butithastheadvantagethatittakesthesameforminanysystemofgeneralizedcoordinates,anditisbettersuitedtogeneralizations(see,forexample,theFieldtheorysectionbelow).Contents1History2Statement3Exampleso3.1Classicalmechanics3.1.1Basicmethod3.1.2Particleinaconservativeforcefieldo3.2Fieldtheory4Variationsforseveralfunctions,severalvariables,andhigherderivativeso4.1Singlefunctionofsinglevariablewithhigherderivativeso4.2Severalfunctionsofonevariableo4.3Singlefunctionofseveralvariableso4.4Severalfunctionsofseveralvariableso4.5Singlefunctionoftwovariableswithhigherderivatives5Notes6References7SeealsoHistoryTheEuler–Lagrangeequationwasdevelopedinthe1750sbyEulerandLagrangeinconnectionwiththeirstudiesofthetautochroneproblem.Thisistheproblemofdeterminingacurveonwhichaweightedparticlewillfalltoafixedpointinafixedamountoftime,independentofthestartingpoint.Lagrangesolvedthisproblemin1755andsentthesolutiontoEuler.ThetwofurtherdevelopedLagrange'smethodandappliedittomechanics,whichledtotheformulationofLagrangianmechanics.Theircorrespondenceultimatelyledtothecalculusofvariations,atermcoinedbyEulerhimselfin1766.[1]StatementTheEuler–Lagrangeequationisanequationsatisfiedbyafunctionqofarealargumenttwhichisastationarypointofthefunctionalwhere:qisthefunctiontobefound:suchthatqisdifferentiable,q(a)=xa,andq(b)=xb;q′isthederivativeofq:TXbeingthetangentbundleofX(thespaceofpossiblevaluesofderivativesoffunctionswithvaluesinX);Lisareal-valuedfunctionwithcontinuousfirstpartialderivatives:TheEuler–Lagrangeequation,then,istheordinarydifferentialequationwhereLxandLvdenotethepartialderivativesofLwithrespecttothesecondandthirdarguments,respectively.IfthedimensionofthespaceXisgreaterthan1,thisisasystemofdifferentialequations,oneforeachcomponent:Derivationofone-dimensionalEuler-LagrangeequationAlternatederivationofone-dimensionalEuler-LagrangeequationExamplesAstandardexampleisfindingthereal-valuedfunctionontheinterval[a,b],suchthatf(a)=candf(b)=d,thelengthofwhosegraphisasshortaspossible.Thelengthofthegraphoffis:theintegrandfunctionbeing2'1)',,(yyyxLevaluatedat(x,y,y′)=(x,f(x),f′(x)).ThepartialderivativesofLare:BysubstitutingtheseintotheEuler–Lagrangeequation,weobtainthatis,thefunctionmusthaveconstantfirstderivative,andthusitsgraphisastraightline.ClassicalmechanicsBasicmethodTofindtheequationsofmotionsforagivensystem,oneonlyhastofollowthesesteps:FromthekineticenergyT,andthepotentialenergyV,computetheLagrangianL=T−V.Compute.Computeandfromit,.Itisimportantthatbetreatedasacompletevariableinitsownright,andnotasaderivative.Equate.Thisis,ofcourse,theEuler–Lagrangeequation.Solvethedifferentialequationobtainedintheprecedingstep.Atthispoint,istreatednormally.Notethattheabovemightbeasystemofequationsandnotsimplyoneequation.ParticleinaconservativeforcefieldThemotionofasingleparticleinaconservativeforcefield(forexample,thegravitationalforce)canbedeterminedbyrequiringtheactiontobestationary,byHamilton'sprinciple.Theactionforthissystemiswherex(t)isthepositionoftheparticleattimet.ThedotaboveisNewton'snotationforthetimederivative:thusẋ(t)istheparticlevelocity,v(t).Intheequationabove,ListheLagrangian(thekineticenergyminusthepotentialenergy):where:misthemassoftheparticle(assumedtobeconstantinclassicalphysics);viisthei-thcomponentofthevectorvinaCartesiancoordinatesystem(thesamenotationwillbeusedforothervectors);Uisthepotentialoftheconservativeforce.Inthiscase,theLagrangiandoesnotvarywithitsfirstargumentt.(ByNoether'stheorem,suchsymmetriesofthesystemcorrespondtoconservationlaws.Inparticular,theinvarianceoftheLagrangianwithrespecttotimeimpliestheconservationofenergy.)BypartialdifferentiationoftheaboveLagrangian,wefind:wheretheforceisF=−∇U(thenegativegradientofthepotential,bydefinitionofconservativeforce),andpisthemomentum.BysubstitutingtheseintotheEuler–Lagrangeequation,weobtainasystemofsecond-orderdifferentialequationsforthecoordinatesontheparticle'strajectory,whichcanbesolvedontheinterval[t0,t1],giventheboundaryvaluesxi(t0)andxi(t1).Invectornotation,thissystemreadsor,usingthemomentum,whichisNewton'ssecondlaw.FieldtheoryThissectioncontainstoomuchjargonandmayneedsimplif