Euler-Lagrange differential equation (advanced)

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

$$F(q)={\int }_{M}L(x,q(x),𝐃q(x))d^{m}x$$

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

Note that if $f\in D$ and $g\in C^2(M,N)$ and the set $\{x\in M\mid f(x)\ne g(x)\}$ 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 $f:M\times [-1,+1]\to N$ 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

$${\frac{d}{d\lambda }F(f(\cdot ,\lambda ))|}_{\lambda =0}=0$$

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:

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