EulerLagrangeDifferentialEquationAdvanced 2004-10-21 22:00:43 2009-03-27 06:10:29 Definition Euler-Lagrange differential equation CalculusOfVariations % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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 \PMlinkid{$C^2 (M, N)$}{5555} 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: \begin{equation}\label{el} dL - d \, \left({\partial L \over \partial(dq)}\right) = 0. \end{equation} This differential equation is known as the \emph{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 $d L = 0$, the Euler-Lagrange equation can be replaced by the Beltrami identity.