Consider an interval $I = [t_0,t_1]$ and a finite dimensional Banach space $X$. Let $U$ be an open subset of $\mathbb{R} \times X \times X$ and let $V \subseteq \mathcal{C}^{1}(I,X)$ be the set of all curves $c:I \rightarrow X$, where $(t,c(t),c^\prime(t))$ is contained in $U$ for all $t$. Let further be
$$\lvert\lvert c \rvert\rvert_{\mathcal{C}^1(I)} := \sup_{t \in I} \lvert\lvert c(t) \rvert\rvert_X + \sup_{t \in I}\lvert\lvert c^\prime(t) \rvert\rvert_X.$$
We note that $V$ is open in $\mathcal{C}^1(I,X)$. For a $\mathcal{C}^r$ function $H: U \rightarrow \mathbb{R}$ ($r \ge 2$) consider the functional $$f: V \rightarrow \mathbb{R}, c \mapsto \int_I H(t,c(t),c^\prime(t)) dt $$ under side conditions $c(t_0) = x_0$ and $c(t_1) = x_1$. Find the Frechet derivative $Df_c$, where $C \in V$.
This setting strongly reminds me of Euler-Lagrange equation. The (first) proof given in Wikipedia computes the total derivative of a pertubation of $c$. Is it possible to transfer this here?