Computation of action integrals for nonseparable classical dynamical systems

Abstract

It is shown that classical action integrals J can be expressed as local functions in phase space, computed from a classical trajectory if the frequency constants ω are known. The mean square deviation of components of J from constant values given by this formalism defines a variational objective function whose absolute minimum determines ω and J. Illustrative calculations are presented for a two-dimensional double-well model problem, equivalent to imposing specular reflection at a barrier cutting across a single-well potential. Solutions obtained over a range of values of initial trajectory parameters determine regular solutions of the classical dynamical problem. The proposed formalism essentially constructs a solution of the Hamilton–Jacobi equation from the phase integral computed along a trajectory.