Let $I\subset\mathbb{R}$, $V\subset\mathbb{R}^n$, and $A\subset\mathbb{R}^n$ be open sets. Let $H:I\times V\times\mathbb{R}^n\to\mathbb{R}$ be $C^2$, and let $S:I\times V\times A\to\mathbb{R}$ be $C^2$. Assume that $S$ is a complete integral of the Hamilton-Jacobi equation with the convention
for every $(x,y,\alpha)\in I\times V\times A$. Fix $\alpha_0\in A$ and $\beta\in\mathbb{R}^n$. Suppose that $(x_0,y_0,\alpha_0)\in I\times V\times A$ satisfies