I would like to minimize a functional of the type: $$L[\gamma]=\int_a^b F(T(\gamma(t))dt$$ on the space of paths $\gamma$, where $T=T(\gamma,t)$. Now, usually I would simply apply Euler-Lagrange's equations, but in this case $T$ is defined implicitly by an equation of type: $$f(t,\gamma,\dot\gamma,\ddot\gamma,T)=0$$ So, how can I do to find a path $\gamma$ minimizing $L$?
A possible way could maybe be the following (I will skip all formality à la physicist):
Assuming $f$ to be well behaved (i.e. a small variation in the path $\gamma$ leads to a small variation of the solution $T$), take a variation $\delta\gamma$ of the path. It will lead to a variation of $T$ in the following way: $$0=f(t,\gamma+\delta\gamma,\dot\gamma+\delta\dot\gamma,\ddot\gamma+\delta\ddot\gamma,T+\delta T)$$ And this (in first order in the variation) gives: $$0=f+f_\gamma\delta\gamma+f_{\dot\gamma}\delta\dot\gamma+f_{\ddot\gamma}\delta\ddot\gamma+f_T\delta T$$ where $f_x=\frac{\partial f}{\partial x}$ and $f=f(t,\gamma,\dot\gamma,\ddot\gamma,T)$. Assuming $f_T$ to be invertible (in particular this gives us unicity of the solution, by the implicit function theorem), then we find: $$\delta T = -f_T^{-1}(f+f_\gamma\delta\gamma+f_{\dot\gamma}\delta\dot\gamma+f_{\ddot\gamma}\delta\ddot\gamma)$$ And thus we obtain: $$\begin{array}{ll}\delta L=L[\gamma+\delta\gamma]-L[\gamma]&=\int F(T+\delta T)-F(T)dt\\&=\int F_T(T)\delta Tdt\\&=\int -F_T(T)f_T^{-1}(f+f_\gamma\delta\gamma+f_{\dot\gamma}\delta\dot\gamma+f_{\ddot\gamma}\delta\ddot\gamma)\end{array}$$ Integrating by parts (and omitting boundary terms) the term in the integral becomes: $$\left(-F_T(T)f_T^{-1}f+\frac{d}{dt}(F_T(T)f_T^{-1}f_\gamma)-\frac{d^2}{dt^2}(F_T(T)f_T^{-1}f_{\dot\gamma})+\frac{d^3}{dt^3}(F_T(T)f_T^{-1}f_{\ddot\gamma})\right)\delta\gamma$$ By variational principle, $\delta L=0$ if, and only if the term in the big brackets is $0$.
Is this correct? And does this give us an effective way to find $\gamma$ (at least numerically)?