Are the solutions of the optimization of 2 general functionals with $S_1[f]=S_2[f^{-1}]$ the same?

Question

I have two real functionals $S_1,S_2$ with the property of $S_1[f]=S_2[f^{-1}]$. These functionals are integrals or sum of integrals. If I suppose that they have a maximum or minimum, and I use the same method for the derivation of Euler-Lagrange equation and give me differential equations with these two functionals, Do the solutions of these two equations give me the same intituive result? i.e. if the solution of $S_1$ equation is $g_1$ and the solution of $S_2$ equation is $g_2$, Is it true $g_1=g_2^{-1}$ for certain subsets of the domain of $g_1,g_2$? Else, When this is true?