Hamilton's principal function

Summary

Fix a starting point and a starting time. Hamilton's principal function $S(q, t)$ is the action of the true motion from there to the position $q$ at the time $t$. As a function of the endpoint, it solves the Hamilton–Jacobi equation $\partial_t S + H(q, \nabla_q S, t) = 0$. Its gradient in $q$ is the final momentum, so the single function $S$ encodes a whole family of motions. In many texts, the name also covers a complete solution of the Hamilton–Jacobi equation used as a generating function. See the definition.