The Ideal Resonance Problem a Comparison of Two Formula Solutions I

Abstract

Garfinkel’s solution of the Ideal Resonance problem derived from a Bohlin-von Zeipel procedure, and Jupp’s solution, using Poincaré’s action and angle variables and an application of Lie series expansions, are compared. Two specific Hamiltonians are chosen for the comparison and both solutions are compared with the numerical solutions obtained from direct integrations of the equations of motion. It is found that in deep resonance the second-mentioned solution is generally more accurate, while in the classical limit the first solution gives excellent agreement with the numerical integrations.

This article represents a summary of a much more extensive programme of research, the complete results of which will be published in a future article.