the Hamilton-Jacobi equation
/ HAM-il-tun yah-KOH-bee /
The Hamilton-Jacobi equation is the ultimate expression of the idea 'find coordinates in which the problem solves itself'. It asks for the one generating function so powerful that, after the canonical transformation it produces, the new Hamiltonian is zero -- meaning all the new coordinates and momenta are constant, and the motion has been reduced to standing still. Solving mechanics becomes solving a single (nonlinear, first-order) partial differential equation for that function.
Seeking a type-2 generating function S(q, alpha, t) -- called Hamilton's principal function -- that makes K = 0 gives, via K = H + partial S / partial t and p = partial S / partial q, the Hamilton-Jacobi equation: H(q_1, ..., q_n, partial S / partial q_1, ..., partial S / partial q_n, t) + partial S / partial t = 0. This is one first-order PDE for S in the n coordinates and time. A complete solution depends on n constants alpha_i (the new, conserved momenta); the new coordinates beta_i = partial S / partial alpha_i are then also constant, and inverting gives q(t) directly. When H does not depend on time explicitly, one separates S = W(q, alpha) - E t, and W (Hamilton's characteristic function) satisfies H(q, partial W / partial q) = E.
The Hamilton-Jacobi equation is the deepest classical route to a solution and the closest classical structure to quantum mechanics. S is precisely the classical action as a function of the endpoint, and the surfaces of constant S propagate like wavefronts with trajectories as their perpendicular rays -- the optical-mechanical analogy that guided de Broglie and Schrodinger. Indeed the time-dependent Schrodinger equation reduces to the Hamilton-Jacobi equation in the limit hbar -> 0 (with psi ~ exp(i S / hbar)), which is the basis of the WKB approximation.
For a free particle in one dimension H = p^2/(2m), the equation is (1/2m)(partial W / partial x)^2 = E, so partial W / partial x = sqrt(2 m E) and W = sqrt(2 m E) x. Then S = sqrt(2 m E) x - E t, and beta = partial S / partial E = sqrt(m/(2E)) x - t = const reproduces x = sqrt(2E/m)(t + const): uniform motion, extracted by solving a PDE.
For the free particle the Hamilton-Jacobi equation reproduces uniform motion; the characteristic function W = sqrt(2mE) x is the spatial part of the action.
The Hamilton-Jacobi equation is powerful mainly when it separates -- when S splits into a sum of one-variable pieces. Separability is a special property tied to symmetry and to integrability; a generic (non-integrable) Hamiltonian admits no complete separable solution, which is precisely why most systems cannot be solved this way.