Let $a,b \in \mathbb{R}$ satisfy $a<b$, let $n\in\mathbb{N}$, let $M\subset\mathbb{R}^n$ be a smooth embedded submanifold without boundary, and let $q_a\in\mathbb{R}^n$. Let
and that $q\in\mathcal{A}$ is a local minimizer of $I$ on $\mathcal{A}$ with respect to the $C^1([a,b];\mathbb{R}^n)$ norm. Assume also that $q$ satisfies the Euler-Lagrange equation