As discussed here
The typical form of a functional for the Euler-Lagrange equation is $$ J[\tau] = \int_a^b L(e,\tau(e),\tau'(e)) de. $$
In the link above they discuss a situation where the bounds of the integral depend on the inverse function. I am facing a situation where the functional itself depends on the inverse function. I'm not sure if the solution from the link applies to mine, and I'm having trouble applying that solution anyways.
Functional dependent on inverse
In my setting $$ J[\tau] = \int_{a}^{b}L(e,\tau(e),\tau^{-1}(e)) de $$ Notably, my functional does not depend on $\tau'$ yet. And $\tau^{-1}(\tau(e)) = e$
Specific problem and inspiration
The inspiration for the problem actually comes from a game theoretic problem I am working on. One of the specific functionals I need to find a maximum of is $$ 2 u^2\,\mu^{(\tau)}[\overline{e}] \int\limits_{\underline{e}}^{\overline{e}} \left( \frac{1}{2}\left( \overline{v}^2 - \tau[e]^2 \right) + e\left( \overline{v} - \tau[e]\right)\right) \: de \\ \quad + 2 u^{2} \left( \int\limits_{\underline{e}}^{\overline{e}} \int\limits_{\tau[e]}^{\overline{v}} \left( v +e \right) \left[v\overline{e} - v\tau^{-1}[v] - \mu^{(\tau)}[\overline{e}] + \mu^{(\tau)}[\tau^{-1}[v]]\right] \: dv\: de \right) \\\\$$ where $$\mu^{(\tau)}[e] = \int_{\underline{e}}^{e} \tau[e]\, de$$
apologies for the bad over and under line formatting. Those are just meant to be max and mins of a range of e and v.
Is there a possible, or best way to solve this? My best idea is below.
Chain rule based derivative of inverse with respect to function
I have asked separate focused questions about partial derivatives for this approach in two Mathematics exchange questions and received some helpful answers.
Since the Euler-Lagrange equation in this setting would be $$ \frac{d}{de}\frac{\partial L}{\partial \tau'} - \frac{\partial L}{\partial \tau} = 0$$ I think we could pull $\frac{\partial L}{\partial \tau}$ through the inner integral using Leibniz integral rule, right? But then would still need to solve for $\frac{\partial \tau^{-1}}{\partial \tau}$.
Alternatively, a more general statement would be to say $$\frac{\partial L}{\partial \tau} = \frac{\partial L}{\partial \tau^{-1}} \frac{\partial \tau^{-1}}{\partial \tau}$$ If you can apply that rule for these partial derivativesw/ respect to a function? This would still require solving for $\frac{\partial \tau^{-1}}{\partial \tau}$.
I have two attempts at applying the chain rule for $\frac{\partial \tau^{-1}}{\partial \tau}$ and I get different answers: First is $$v = \tau(e) \\ e = \tau^{-1}(v) $$ $$\frac{\partial}{\partial\tau} \tau^{-1}(v) = \frac{\partial}{\partial\tau}e \\ \frac{\partial}{\partial\tau} \tau^{-1}(v) = 0 \:(?) $$
Second is $$\frac{\partial\tau^{-1}}{\partial\tau} = \frac{\partial\tau^{-1}}{\partial v}\frac{\partial v}{\partial\tau}$$ and $\frac{\partial v}{\partial\tau} = \frac{\partial}{\partial\tau} \tau(e) = 1$...? And also applying the inverse function rule, we get: $$\frac{\partial\tau^{-1}}{\partial\tau} = \frac{1}{\tau(\tau^{-1}(v))} \cdot 1$$ evaluating at $v=a$ becomes $$\frac{\partial}{\partial\tau} \tau^{-1}(a) = \frac{1}{\tau(\tau^{-1}(a))}$$
Am I using the chain rule wrong in some way? While the inverse function rule tells how to solve the derivative of the inverse with respect to the parameter, what is the functional-derivative, with respect to the function, of the inverse-function?
The second method seems more reasonable, but still results in something depending on $\tau^{-1}$. Hopefully I'd still be able to algebra my way from that to a solution for an extremal $\tau$?
A couple approaches have been offered.
See the addendum to Abezhiko's response as posted to a related subquestion on mathematics. https://math.stackexchange.com/a/5025023/1541119 First it seems he derives the total derivative with $\tau^{-1}$ from the start (basically re-deriving a different Euler-Lagrange equation? Sorry if this is a wrong interpretation) At the end he offers a second substitution approach very similar but slightly different to the substitution suggested here. https://mathoverflow.net/a/239476/551527
In the comments to the Original Post here we also have Majer's suggestion of using Frechet derivative rules. Perhaps this is equivalent to the first approach Abezhiko gave?
