Historical and Modern Perspectives on Hamilton-Jacobi Equations

Abstract: We present the Hamilton-Jacobi equation as originally derived by Hamilton in 1834 and 1835 and its modern interpretation as determining a canonical transformation. We show that the method of characteristics for partial differential equations, applied to the Hamilton-Jacobi equation, yields Hamilton's canonical equations and the action as the solution. Canonical perturbation theory is applied to the Sun-Earth-Jupiter system to calculate perturbations of the orbital elements. We also study tidal locking and find a damped harmonic oscillator equation for a moon that is almost locked. Finally, we present a modern (1980s), weaker notion of solutions to the Hamilton-Jacobi equation, viscosity solutions. We reproduce proofs that this concept is consistent with the classical concept and that visocsity solutions are unique, for locally Lipschitzian Hamiltonians.