Typically the Euler-Lagrange equations are defined for the functional

$$ J[u] = \int_a^b L(x,u,u') dx. $$

However, I was wondering if anyone knows if they can be solved when the expression involves the inverse of $u$? The way my problem is formulated, it is simplest to write as

$$ J[u] = \int_{u^{-1}(a)}^{u^{-1}(b)} L(x,u,u') dx, $$

but it could equivalently be written

$$ J[u] = \int_{u(a)}^{u(b)}L(x,u,u^{-1},u') dx $$

I had a bit of a play with it and I have a feeling it is not generally solvable, but I'm really just playing -- not an expert in this area.

Cheers!