In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics.

The Hamilton–Jacobi equation is a formulation of mechanics in which the motion of a particle can be represented as a wave. In this sense, it fulfilled a long-held goal of theoretical physics (dating at least to Johann Bernoulli in the eighteenth century) of finding an analogy between the propagation of light and the motion of a particle. The wave equation followed by mechanical systems is similar to, but not identical with, the Schrödinger equation, as described below; for this reason, the Hamilton–Jacobi equation is considered the "closest approach" of classical mechanics to quantum mechanics. The qualitative form of this connection is called Hamilton's optico-mechanical analogy.

In mathematics, the Hamilton–Jacobi equation is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations. It can be understood as a special case of the Hamilton–Jacobi–Bellman equation from dynamic programming.

The Hamilton–Jacobi equation is a first-order, non-linear partial differential equation

${\displaystyle -{\frac {\partial S}{\partial t}}=H{\left(\mathbf {q} ,{\frac {\partial S}{\partial \mathbf {q} }},t\right)}.}$

for a system of particles at coordinates ⁠${\displaystyle \mathbf {q} }$⁠. The function ${\displaystyle H}$ is the system's Hamiltonian giving the system's energy. The solution of this equation is the action, ⁠${\displaystyle S}$⁠, called Hamilton's principal function. The solution can be related to the system Lagrangian ${\displaystyle \ {\mathcal {L}}\ }$ by an indefinite integral of the form used in the principle of least action: $${\displaystyle \ S=\int {\mathcal {L}}\ \mathrm {d} t+~{\mathsf {some\ constant}}~}$$ Geometrical surfaces of constant action are perpendicular to system trajectories, creating a wavefront-like view of the system dynamics. This property of the Hamilton–Jacobi equation connects classical mechanics to quantum mechanics.

Boldface variables such as ${\displaystyle \mathbf {q} }$ represent a list of ${\displaystyle N}$ generalized coordinates, $${\displaystyle \mathbf {q} =(q_{1},q_{2},\ldots ,q_{N-1},q_{N})}$$

A dot over a variable or list signifies the time derivative (see Newton's notation). For example, $${\displaystyle {\dot {\mathbf {q} }}={\frac {d\mathbf {q} }{dt}}.}$$