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Hamilton-Jacobi Equation

A first-order partial differential equation that describes the evolution of a system's phase space using the Hamiltonian as a function of position and time.

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The statement of the theorem

Let

H(q, p, t)

be the Hamiltonian of a system, and let

S(q, \boldsymbol{\beta}, t)

be the principal function, where

\boldsymbol{\beta}

are the constants of integration (or initial momenta). The function

S

must satisfy the first-order partial differential equation:

H(q, \frac{\partial S}{\partial q}, t) + \frac{\partial S}{\partial t} = 0

This equation determines the time evolution of the system's action in the phase space.

Source: Wikipedia