Euler-Lagrange differential equation (advanced)

Let M and N be C2 manifoldsMathworldPlanetmath. Let L:M×TN be twice differentiableMathworldPlanetmathPlanetmath. Define a functional F:DC2(M,N) as


where D is the subset of,N) for which this integral converges.

Note that if fD and gC2(M,N) and the set {xMf(x)g(x)} is compact, then gD. We may impose a topology on D as follows: Suppose that fD, that KM is compact, and that U0C2(K,N) is open. Then we define an open set UD as the set of all functions gD such that f(x)=g(x) when xK and such that the restriction of g to K lies in U0.

It is not hard to show that the functional F is continuousMathworldPlanetmath in this topology, and hence it makes sense to speak of local extrema of F. Suppose that q0C2(M,N) is a local extremum. Furthermore, suppose that f:M×[-1,+1]N is twice differentiable and f(x,0)=q0(x) for all xq0 and f(x,y)=q0(x) for all y[-1,+1] when x does not lie in a certain compact subset KM. Then, viewed as a map from [-1,+1] to D, f will be continuous. Therefore, since q0 is a local extremum of F, 0 wil be a local extremum of the function yF(f(,y)). Because the function yF(f(,y)) is differentiable, it will be the case that


It can be shown (see the addendum to this entry) that this condition will be satisfied if and only if q0 is a solution of the following differential equationMathworldPlanetmath:

dL-d(L(dq))=0. (1)

This differential equation is known as the Euler-Lagrange differential equationMathworldPlanetmathPlanetmath (or Euler-Lagrange condition).

The Euler-Lagrange equation can only be used to investigate local extrema which are smooth functions. To a certain extent, this limitation can be ameliorated — one can study piecewise smooth functions by supplementing the Euler-Lagrange equation with auxiliary conditions at discontinuities and, in some cases, one can consider non-smooth solutions as weak solutions of the Euler-Lagrange equation.

In the special cases dL=0, the Euler-Lagrange equation can be replaced by the Beltrami identityMathworldPlanetmath.

Title Euler-Lagrange differential equation (advanced)
Canonical name EulerLagrangeDifferentialEquationadvanced
Date of creation 2013-03-22 14:45:32
Last modified on 2013-03-22 14:45:32
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 16
Author rspuzio (6075)
Entry type Definition
Classification msc 47A60
Synonym Euler-Lagrange condition
Defines Euler-Lagrange differential equation