Can I derive an Euler-Lagrange equation for the following functional:$$F[u] = \phi(\int L(u,u^{'},x) dx)$$with constraint$$\int L(u,u^{'},x) dx = c \quad \forall c \in \mathbb{R} $$The function $\phi()$ can be any convex function, such as $(\cdot)^2$. When we set the function $\phi()$ to be a linear function, this functional will be degenerate to the classical form that I can derive the corresponding Euler–Lagrange equation. But how can I derive the Euler-Lagrange equation for more general form? I think that we can also consider $\phi(\int L(u,u^{'},x) dx)$ as a functinal of $u$ and derive the corresponding Gâteaux derivative. Do we have some reference (books or papers ) to support us to solve such optimization questions?
I'm new to the calculus of variation. If you need any clarification for this question, please let me know.