| Version current |
Version 12 |
| 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 |
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 |
| $$F(q) = \int_M L \left( x, q(x), {\bf D} q(x) \right) \, d^m x$$ |
$$F(q) = \int_M L \left( x, q(x), {\bf D} q(x) \right) \, d^m x$$ |
|
where $D$ is the subset of \PMlinkid{$C^2 (M, N)$}{5555} for which this integral converges.
|
where $D$ is the subset of $C^2 (M, N)$ 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 $U_0$. |
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 |
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 |
| $${d \over d\lambda} F (f(\cdot,\lambda)) \big|_{\lambda = 0} = 0$$ |
$${d \over d\lambda} F (f(\cdot,\lambda)) \big|_{\lambda = 0} = 0$$ |
|
|
| 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: |
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} |
\begin{equation}\label{el} |
| dL - d \, \left({\partial L \over \partial(dq)}\right) = 0. |
dL - d \, \left({\partial L \over \partial(dq)}\right) = 0. |
| \end{equation} |
\end{equation} |
| This differential equation is known as the \emph{Euler-Lagrange differential equation} (or Euler-Lagrange condition). |
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. |
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. |
In the special cases $d L = 0$, the Euler-Lagrange equation can be replaced by the Beltrami identity. |
