Hamilton–Jacobi theory of the calculus of variations

Remarks

The basic fact is that each such curve is an extremal, with momentum $p(t) = \partial S / \partial q(t, q(t))$ solving Hamilton's canonical equations. These curves fill $G$ and form a field of extremals, and $S$ is the action along them. Example: $L = \frac12 \abs{v}^2$, so $H = \frac12 \abs{p}^2$, on $G = \set{(t, q) : t > t_0}$. The function $S(t, q) = \abs{q - a}^2 / (2(t - t_0))$ solves the equation, since $S_t = -\abs{q - a}^2 / (2 (t - t_0)^2)$ and $\frac12 \abs{\nabla_q S}^2 = \abs{q - a}^2 / (2 (t - t_0)^2)$. The curves solve $q' = (q - a)/(t - t_0)$, so they are the straight lines through $a$ at time $t_0$, which are the extremals of $L$. Non-example: for $L = \frac12 \abs{v}^2$ the function $S(t, q) = \abs{q}^2$ does not solve the equation, since $S_t + \frac12 \abs{\nabla S}^2 = 2 \abs{q}^2 \ne 0$ for $q \ne 0$. Jacobi's theorem completes the theory: from a family $S(t, q, a)$ of solutions with $\det(\partial^2 S / \partial q \, \partial a) \ne 0$, the equations $\partial S / \partial a = b$ define all solutions of Hamilton's equations near a point.