Let $L : [a,b] \times \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ be a $C^2$ Lagrangian satisfying the strengthened Legendre condition $\partial_{y'y'} L > 0$ throughout a region $D \subset [a,b] \times \mathbb{R}$. Let $y_0 : [a,b] \to \mathbb{R}$ be an extremal whose graph lies in $D$. The following conditions are equivalent: 1. There exists a field of extremals over $D$ in which $y_0$ is embedded. 2. There exists a $C^2$ solution $S : D \to \mathbb{R}$ of the Hamilton–Jacobi equation $S_x + H(x, y, S_y) = 0$ such that $S_y(x, y_0(x)) = \partial_{y'} L(x, y_0, y_0')$ along $y_0$. When either (and hence both) conditions hold, the extremal $y_0$ satisfies \begin{align*} J[y] - J[y_0] = \int_a^b \mathcal{E}(x,\, y(x),\, y'(x),\, p(x, y(x)))\, dx \ge 0 \end{align*} for every competing curve $y$ with the same endpoints whose graph lies in $D$, where $\mathcal{E}(x,y,q,p) = L(x,y,q) - L(x,y,p) - (q-p)\partial_{y'} L(x,y,p)$ is the Weierstrass excess function. Under the strengthened Legendre condition, $\mathcal{E} \ge 0$ with equality iff $q = p$, so $y_0$ is the unique minimum among curves lying in $D$.