An alternative to the Hamilton-Jacobi approach in classical mechanics

Abstract

A time-dependent function on the phase space, the phase action, is introduced, which is related to the Hamilton principal function by means of the Legendre transformation. The phase action is the (unique) solution of the Cauchy problem for a first-order partial differential equation, analogous to the Hamilton-Jacobi equation. The result is a manifestly invariant phase-space formalism. General properties of the phase action are analysed; in particular, the symmetry under time inversion and the continuous group property. Some examples are considered: the anisotropic oscillator, free motion, the multi-dimensional rotator, and the Kepler problem in Fock variables. As application of the formalism, an invariant perturbation theory is developed. The relation to semiclassical methods in the quantum theory is briefly discussed. A generalised dynamics on a manifold with non-flat symplectic structure is considered in the appendix.