Let $U \subset \mathbb{R}^n$ be open, let $L: U \times \mathbb{R}^n \to \mathbb{R}$ be a $C^2$ autonomous Lagrangian, and let $q: [a,b] \to U$ be a $C^2$ curve satisfying the Euler-Lagrange equation
for every $t \in (a,b)$, where $\partial_x L$ and $\partial_v L$ denote the gradients of $L$ with respect to the first and second $\mathbb{R}^n$ variables. Define the energy