Hamilton–Jacobi theory of the calculus of variations

Summary

Hamilton–Jacobi theory links a problem of the calculus of variations with a single first-order partial differential equation, the Hamilton–Jacobi equation. A solution $S$ of this equation, such as the action along extremals, generates a whole family of extremals, a field of extremals. Conversely, a suitable family of solutions, a complete integral, gives all solutions of Hamilton's canonical equations by Jacobi's theorem. The theory gives sufficient conditions for minima and is the classical ancestor of dynamic programming.