stationary point
StationaryPoint
2005-10-29 11:34:43
2006-02-04 08:44:11
Definition
calculus of variations
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Suppose $V$ is a vector space and $L\colon V\to \R$ is a map.
Then $v\in V$ is a \emph{stationar{y} point} of $L$
provided that
$$
\frac{d}{dt}L(v+t u)\Big|_{t=0} =0
$$
for all $u\in V$. In this case $u$ is called a variation of $v$.
% This needs to be rewritten. For curved the functional depends on
% the tangent.
%
%One can also talk about stationary points for curves. Then
%the natural definition reads as follows.
%
%Suppose $C$ is the set of all curves in a space $M$, and $L$ is a mapping
%$L\colon C\to \R$.
%Suppose $c\colon (0,1)\to M$ is a curve in $M$ and let
%$s\mapsto c_s$ for $s\in(-\varepsilon,\varepsilon)$ be a family
%of curves such that $c_0=c$. Then $c$ is stationary
%if
%$$
% \frac{d}{ds}L(c_s)\Big|_{s=0} =0.
%$$
%for all variations $c_s$ of $c$.
%Typically, however, in the case of curves, the mapping $L$ does not
%only depend on $c(t)$, but also on the derivative of $c$.