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The Euler-Lagrange equation - KAIST

Chapter2 TheEuler-LagrangeequationIn thischapter,we willgive necessaryconditionsforanextremumof a functionof thetypeI(x) =ZbaF(x(t); x0(t); t)dt;withvarioustypes of in theformof a di erentialequationthattheextremalcurve shouldsatisfy, andthisdi erentialequationis beginwiththesimplesttype of boundaryconditions,wherethecurves areallowed to varybetweentwo formulatedas follows:LetF( ; ; ) be a functionwithcontinuous rstandsecondpartialderivatives withrespectto( ; ; ). Then ndx2C1[a; b] such thatx(a) =yaandx(b) =yb, andwhich is anextremumforthefunctionI(x) =ZbaF(x(t); x0(t); t)dt:( )In otherwords,thesimplestoptimisationproble mconsistsof ndinganextremumof a functionof theform( ),wheretheclassof admissiblecurves comprisesallsmoothcurves joiningtwo xedpoints; willapplythenecessaryconditionforanextre mum(establishedin )to thesolve thesimplestoptimisationproblemdescribed [a; b]jx(a) =yaandx(b) =ybg, andletI:S!

Note that the Euler-Lagrange equation is only a necessary condition for the existence of an extremum (see the remark following Theorem 1.4.2). However, in many cases, the Euler-Lagrange equation by itself is enough to give a complete solution of the problem. In fact, the existence of an extremum is sometimes clear from the context of the problem.

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