HamiltonJacobiGleichung

The Hamilton–Jacobi equation is a formulation of classical mechanics that unites Hamiltonian dynamics with Jacobi's principle of least action. It is named for William Rowan Hamilton and Carl Gustav Jakob Jacobi and serves as a bridge between dynamical equations and the action principle. The central object is Hamilton's principal function S(q, t), which encodes the complete evolution of the system.

For a system with generalized coordinates q and momenta p, the Hamilton–Jacobi equation takes the form H(q, ∂S/∂q, t) + ∂S/∂t = 0, where H is the Hamiltonian. In time-independent cases, S can be separated as S(q, t) = W(q) − Et, yielding the reduced equation H(q, ∂W/∂q) = E. Solving this first-order partial differential equation by the method of characteristics gives trajectories as level surfaces of S and reduces the dynamics to quadrature.

The approach also provides a route to canonical transformations. A complete solution S can generate a transformation to new coordinates in which the Hamiltonian is constant or zero, leading to action-angle variables for integrable systems. The Hamilton–Jacobi framework underlies the semiclassical relation to quantum mechanics, where the phase S corresponds to the classical limit of the wavefunction in the WKB approximation.

Historically, Jacobi extended Hamilton’s least-action principle, and Hamilton formulated the corresponding PDE that bears his name. The equation has applications in celestial mechanics, optics, and general relativity, where it governs geodesic motion and other systems through corresponding Hamiltonians.