Let $y_0 \in C^2([a,b])$ be an extremal of $J[y] = \int_a^b L(x,y,y')\,dx$. Suppose that: 1. $y_0$ can be embedded in a field of extremals over some open region $D$ containing the graph of $y_0$. 2. The Weierstrass excess function satisfies $\mathcal{E}(x,y,q,p) \ge 0$ for all $(x,y) \in D$ and all $q \in \mathbb{R}$. Then $y_0$ is a strong local minimum. If condition 2 holds with strict inequality for all $q \ne p(x,y)$, then $y_0$ is a strict strong local minimum.