Is there a package for computing and discretizing Euler-Lagrange equations?

Question

I have an energy functional that I want to optimize using the calculus of variations. Is there any software package that will do this automatically?

As far as I can tell, Maple and Mathematic have some relevant functionality, and they can generate C code as well. However, I could not find a working example. Are there any available working examples that use Mathematica?

Are there any other specialized packages that will solve these sorts of problems?

Answer 1

The question of course all depends on what kind of problem you have. If you have an energy functional that describes the position of a weight suspended from a spring, then the Euler-Lagrange equations are just an algebraic equation. On the other hand, if you have an energy equation that is described using an ODE, e.g. the position of a chain suspended from its end points as gravity acts on it, then you will get a two-point boundary value problem. Finally, if you have a multi-dimensional problem, then the Euler-Lagrange equations will be partial differential equations.

The point I want to make is that the type of problem you get in the Euler-Lagrange equations depends very much on the situation you are considering, and so the question of what discretization method to use can not be answered without seeing more context.

Answer 2

Mathematica has the package VariationalMethods which you may want to take a look at: http://reference.wolfram.com/mathematica/VariationalMethods/tutorial/VariationalMethods.html

For an example of how to set up the Euler-Lagrange equations for a problem and then use Mathematica to solve them, see the first example here: http://reference.wolfram.com/mathematica/VariationalMethods/ref/EulerEquations.html