This may seem like a weird question, but it's something which has been intriguing me for quite a while. In the Calculus of Variations we are told to find the extrema of a functional defined over a certain set, but has nobody tried to solve the inverse problem? That is: given a curve find the functional(s) for which it is the extrema.
Consider, for example, that the catenary solves the suspended chain problem but it is also the solution for the Minimal Surface of Revolution problem. Could there be other variational problems for which it's the solution?
On Scopus searching for "inverse problem" AND "calculus of variations" in Article Title, Abstract, Keywords gives me 113 results, almost all relating to the Inverse problem for Lagrangian mechanics
This is not so interesting. We can build a million functionals having optimum on the given curve. It is like if you have some function $f(x)$ to minimize and get say solution $x=1$. Now the question what function $g(x)$ will have $x=1$ as a minimum? Does it make sense? IMHO it does not.