Euler-Lagrange differential equation (advanced)

Let $M$ and $N$ be $C^{2}$ manifolds. Let $L\colon M\times TN\to\mathbb{R}$ be twice differentiable. Define a functional $F\colon D\subset C^{2}(M,N)\to\mathbb{R}$ as

F(q)=\int_{M}L\left(x,q(x),{\bf D}q(x)\right)\,d^{m}x

where $D$ is the subset of http://planetmath.org/node/5555 $C^{2}(M,N)$ for which this integral converges.

Note that if $f\in D$ and $g\in C^{2}(M,N)$ and the set $\{x\in M\mid f(x)\neq g(x)\}$ is compact, then $g\in D$ . We may impose a topology on $D$ as follows: Suppose that $f\in D$ , that $K\subset M$ is compact, and that $U_{0}\subset C^{2}(K,N)$ is open. Then we define an open set $U\subset D$ as the set of all functions $g\in D$ such that $f(x)=g(x)$ when $x\notin K$ and such that the restriction of $g$ to $K$ lies in $U_{0}$ .

It is not hard to show that the functional $F$ is continuous in this topology, and hence it makes sense to speak of local extrema of $F$ . Suppose that $q_{0}\in C^{2}(M,N)$ is a local extremum. Furthermore, suppose that $f\colon M\times[-1,+1]\to N$ is twice differentiable and $f(x,0)=q_{0}(x)$ for all $x\in q_{0}$ and $f(x,y)=q_{0}(x)$ for all $y\in[-1,+1]$ when $x$ does not lie in a certain compact subset $K\subset M$ . Then, viewed as a map from $[-1,+1]$ to $D$ , $f$ will be continuous. Therefore, since $q_{0}$ is a local extremum of $F$ , $0$ wil be a local extremum of the function $y\mapsto F(f(\cdot,y))$ . Because the function $y\mapsto F(f(\cdot,y))$ is differentiable, it will be the case that

{d\over d\lambda}F(f(\cdot,\lambda))\big{|}_{\lambda=0}=0

It can be shown (see the addendum to this entry) that this condition will be satisfied if and only if $q_{0}$ is a solution of the following differential equation:

dL-d\,\left({\partial L\over\partial(dq)}\right)=0.

(1)

This differential equation is known as the Euler-Lagrange differential equation (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 identity.

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