Let $a,b \in \mathbb{R}$ with $a < b$, let $q_a,q_b \in \mathbb{R}^n$, and let $F \in C^2(\mathbb{R}^n \times \mathbb{R}^n;\mathbb{R})$ be autonomous. Let $q \in C^2([a,b];\mathbb{R}^n)$ satisfy $q(a)=q_a$ and $q(b)=q_b$. Suppose that $b$ lies in the interior of the admissible terminal-time interval and that $(q,b)$ is a local minimizer of