Rigurous treatment of variational calculus

[itler]

Hi,

I have a general question about variational calculus (VC). I know the standard derivation of Euler-Lagrange equations and I´m able to use them. Nevertheless I think what I generally read cannot be the whole truth.

Generally if f:M->R (M: arbitrary topological space, R: real numbers), then m in M is a minimum of f iff there is an open set O in M such that f(x)>f(m) for all x in O.

Reasoning this way I´m wondering what are the open sets in standard VC?? I think the standard derivation of the Euler-Lagrange equations doesn´t cover this? Actually it is hard to understand what the variation of a functional means.

I can put my question differently:

1) Is there a general way to find extrema of real valued functions on general topological spaces?
2) Is there a way to make "the set of all paths between two points" into a topological space?

 

[cliowa]

Generally if f:M->R (M: arbitrary topological space, R: real numbers), then m in M is a minimum of f iff there is an open set O in M such that f(x)>f(m) for all x in O.

I think you should also require that m be in O, right?

Reasoning this way I´m wondering what are the open sets in standard VC?? I think the standard derivation of the Euler-Lagrange equations doesn´t cover this?

One important point about the standard derivation of Euler-Lagrange is that you use integrals and differentiation. However integration and differentiation is generally not defined for a function f:M->R, where M is an arbitrary topological space (In case this was not clear to you: Check the quotient you have to build for differentiating, and think about just how you would integrate).

The standard derivation of EL does cover the problems with the open sets: It simply assumes the standard topology (i.e. induced by the euclidean norm) on both real spaces. Can you figure out the relation between the requirement of finding an "open set around and including your point" (called an open neighborhood) and the derivative properties of a minimum?

I believe that the special properties of functions from the reals to the reals are quite important for the calculus of variations. By choosing silly topologies one can easily get quite uninteresting results.

1) Is there a general way to find extrema of real valued functions on general topological spaces?

None that I'd know of, for the reasons mentioned above.

2) Is there a way to make "the set of all paths between two points" into a topological space?

Yes, sure there is. I guess you're familiar with the requirements for a topology, aren't you? Simply check all three and see how you need to modify the set. Ever heard of the (in-) discrete topology?

 

[DeadWolfe]

An excellent rigorous resource for this material is Gelfand's calculus of variations book, which is a dover book and is thus very cheap. If you're interested in this stuff, it's a no brainer to by that book.

 

[itler]

I think you should also require that m be in O, right?

Of course, sorry, I forgot...

The standard derivation of EL does cover the problems with the open sets: It simply assumes the standard topology (i.e. induced by the euclidean norm) on both real spaces. Can you figure out the relation between the requirement of finding an "open set around and including your point" (called an open neighborhood) and the derivative properties of a minimum?

Hm, but the domain of f is not simply R^n, but the set of functions on R^n? So this is an infinitely dimensional vector space with all its complications like

- What´s meant by "eukidean norm" here (probably ||g|| = sqrt(integral(g^2)))
- Norms on infinite dimensional spaces are not equivalent (the topologies they generate are not the same, means the notion of an open set depends on your choice of a particular norm?), so choosing any particular one seems a little bit random? Whether a "point" (= function or path) is a minimum will depend on the (deliberate) choice of a norm on the domain?

I believe that the special properties of functions from the reals to the reals are quite important for the calculus of variations. By choosing silly topologies one can easily get quite uninteresting results.

Of course...

The main reason I was asking this is out of physics. I tried to understand discussions like "can all PDE´s be derived from some variational principle", "are PDE formulations of physics always equivalent to a VC formulation". It looks a bit dirty if all this depends on a human´s arbitrary choice of a norm.