= Solution
Fix an initial event $(q_i,t_i)$. For each reachable final event $(q_f,t_f)$, evaluate the <action> on a classical path satisfying those endpoints. On a chosen local branch this produces Hamilton's principal function $S(q_f,t_f)$, rather than a functional of every possible path. Multiple classical paths may make this function multivalued, so the derivation is local to a branch.
Allow the final event to vary. The integrated interior variation vanishes by the <Euler-Lagrange equations>. The endpoint variation must distinguish the total coordinate change from the change at fixed time: $\delta q(t_f)=\delta q_f-\dot q_f\delta t_f$. Consequently
$$
\delta S=\sum_ip_{i,f}(\delta q_{i,f}-\dot q_{i,f}\delta t_f)+L_f\delta t_f
=\sum_ip_{i,f}\delta q_{i,f}-H_f\delta t_f.
$$
Hence $p_i=\partial S/\partial q_i$ and $\partial S/\partial t=-H$. Eliminating the <momenta> yields the <Hamilton-Jacobi equation>
$$
\boxed{\frac{\partial S}{\partial t}+H\left(q,\frac{\partial S}{\partial q},t\right)=0.}
$$
A <complete integral of the Hamilton-Jacobi equation> depends on $n$ suitable independent constants for $n$ coordinates; differentiating it with respect to those constants supplies the conjugate integration constants needed to reconstruct <trajectories>.
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