Hamilton's principal function S=S(q,t;q_0,t_0) is obtained from the action functional \mathcal{S} by fixing the initial time t_0 and the initial endpoint q_0, while allowing the upper time limit t and the second endpoint q to vary. The Hamilton's principal function satisfies the Hamilton–Jacobi equation, a formulation of classical mechanics. Due to a similarity with the Schrödinger equation, the Hamilton–Jacobi equation provides, arguably, the most direct link with quantum mechanics.
