From order to chaos in a perturbed Kepler problem

The aim of this paper is twofold. First, we want to find angle-action variables suitable for the study of a generic perturbed Kepler problem: indeed, the unperturbed problem is degenerate, since its Hamiltonian depends on only one action variable (instead of three), and only a circle (instead of a three-dimensional torus) is intrinsically defined. Fortunately, the manifold of the orbits is compact, so the perturbed averaged system has always elliptic equilibrium points: nearby these points the reduced system behaves like a two-dimensional harmonic oscillator, which bears naturally the variables we seek. Second, we will apply the method of Numerical Frequencies Analysis in order to detect the transition from order to chaos. Four numerical examples are examined, by means of the free programs KEPLER and NAFF.