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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: $\delta q(t_f)=\delta q_f-{\dot{q}}_f\delta t_f$. ConsequentlyHence $p_i=\partial S/\partial q_i$ and $\partial S/\partial t=-H$. Eliminating the momenta yields the Hamilton-Jacobi equationA 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.

The general action identity isTreating $q,Q,t$ as independent and comparing coefficients givesChoose a complete integral of the Hamilton-Jacobi equation,Then $\mathcal{H}=0$, and Hamilton's equations become ${\dot{Q}}_i={\dot{P}}_i=0$. Completeness and local invertibility are needed: a solution with too few independent parameters cannot generate an arbitrary trajectory.