Euler-Lagrange differential equation (advanced)
where is the subset of http://planetmath.org/node/5555 for which this integral converges.
Note that if and and the set is compact, then . We may impose a topology on as follows: Suppose that , that is compact, and that is open. Then we define an open set as the set of all functions such that when and such that the restriction of to lies in .
It is not hard to show that the functional is continuous in this topology, and hence it makes sense to speak of local extrema of . Suppose that is a local extremum. Furthermore, suppose that is twice differentiable and for all and for all when does not lie in a certain compact subset . Then, viewed as a map from to , will be continuous. Therefore, since is a local extremum of , wil be a local extremum of the function . Because the function is differentiable, it will be the case that
It can be shown (see the addendum to this entry) that this condition will be satisfied if and only if is a solution of the following differential equation:
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 , the Euler-Lagrange equation can be replaced by the Beltrami identity.
|Title||Euler-Lagrange differential equation (advanced)|
|Date of creation||2013-03-22 14:45:32|
|Last modified on||2013-03-22 14:45:32|
|Last modified by||rspuzio (6075)|
|Defines||Euler-Lagrange differential equation|