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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?

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# Rigurous treatment of variational calculus

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