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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

user3083324
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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.

Alexander Vigodner
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  • That's because you didn't put enough constraints in the problem, consider the set of functions that have x=-1 and x=+1 as minimum, x=0 as maximum, are infinitely differentiable and whose area=pi, not even sure this set is non-empty – user3083324 Oct 02 '14 at 03:25
  • Back to calculus of variations, consider the minimal surface of revolution problem: does the catenary only solve the euclidean norm case or works for others norms? If the answer is yes, then is there a pattern? – user3083324 Oct 02 '14 at 03:48
  • I can set any constraints and after this build a million function/als having even a global optimum on the given curve which is by definition satisfies all your constraints. $\int (x(t)-yourcurve(t))^2 dt$ is the simplest example. – Alexander Vigodner Oct 02 '14 at 04:04
  • Yes, but for $\int(\left| x(t)-yourcurve(t)^{2} \right|)dt$ the trick of putting yourcurve(t) inside the functional doesn't work, so clearly there are rules of composition that must be followed
    • – user3083324 Oct 02 '14 at 14:46
  • Can u give an example that doesn't involve yourcurve(t) inside the functional?
    • – user3083324 Oct 02 '14 at 14:48
  • Example 2. Any monotonically increasing function from your original functional. Example 3. Assume you curve satisfies DE:$\dot x=g(x)$. Then $\int Z((\dot x - g(x))^2)dt$ where $Z$ is monotonically increasing function. Etc etc. – Alexander Vigodner Oct 02 '14 at 15:03