I'm stuck on this question, let $B = \{x\in \mathbb{R}^n:|x|\leq 1\}$ be the unit ball in $\mathbb{R}^n$, consider the following minimizing problem $$ \inf_{x(\cdot) \in \mathcal{A}} \int_0^\infty e^{-s} \Big(|x'(s)|^2 - V(x(s))\Big)\;ds$$ where $V:\mathbb{R}^n\longrightarrow \mathbb{R}$ us of class $C^1$ and is bounded $|V(x)| \leq C$, subjected to a somewhat unsual constraint $$ \mathcal{A} = \Big\{x(\cdot):[0,\infty)\longrightarrow B: x'(\cdot)\in L^1_{\mathrm{loc}}([0,\infty)), x(0) = x_0\in B \Big\}.$$
How can I find the correct Euler-Lagrange equation for this problem? The problems appear when I need to find a good test function space $\gamma(\cdot)$ such that $\eta+\gamma \in \mathcal{A}$ for all $\eta$ and $\gamma$, which is not clear how to make $\eta(s)+\gamma(s) \in B$ for all $s$, and also the boundary for the Euler-Lagrange equation is unclear.