- #1
itler
- 8
- 0
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?
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?