Hamilton–Jacobi equation

A first-order partial differential equation for an action function whose gradients encode momentum and whose characteristics trace classical mechanical trajectories.

Hamilton principal function: The classical action evaluated along a motion between specified endpoints and times. Its derivatives supply the momentum and energy that connect solutions to motion.
Hamiltonian mechanics: A formulation of classical mechanics using generalized coordinates, momenta, and a Hamiltonian function. The equation is the Hamiltonian formalism's partial-differential counterpart.
William Rowan Hamilton: An Irish mathematician and physicist who developed influential formulations of mechanics and optics. He introduced the action-based mechanics that led to the equation.
Central-force problem: A mechanical problem in which a body's force is directed toward or away from a fixed center and depends only on distance. Separation of the equation yields radial and angular actions for central-force motion.
Hamilton–Jacobi–Bellman equation: A nonlinear partial differential equation characterizing optimal value functions in dynamic programming and control. It adapts Hamilton–Jacobi structure to optimization rather than classical mechanical action.
Hamilton–Jacobi theory: A formulation of classical mechanics based on solving for an action function. The equation is the central partial differential equation of this formulation.
Hamiltonian: A function that represents a system's total energy or generates its time evolution in Hamiltonian mechanics. It appears directly in the equation as a function of position and action gradients.
Carl Gustav Jacob Jacobi: A German mathematician whose work developed the theory and applications of Hamilton–Jacobi methods. His contributions shaped the equation's mature form and its use in mechanics.
Kepler problem: The motion of bodies interacting through an inverse-square central force, including idealized planetary orbits. Its separable Hamilton–Jacobi solution produces the orbital action variables.
Schrödinger equation: A linear partial differential equation governing the evolution of quantum state amplitudes. Its semiclassical limit yields an equation closely related to the classical Hamilton–Jacobi equation.

Linked from 13 pages