TOPOLOGY AND PERIODIC ORBITS OF RING-SHAPED POTENTIALS AS A GENERALIZED 4-D ISOTROPIC OSCILLATOR

Abstract

In this work we study a generalized integrable biparametric family of 4-D isotropic oscillators. This family allows to treat, in a unified way, oscillators defined by the potentials given by Hartmann and Quesne and other ring-shaped systems. Using the Liouville–Arnold theorem and the analysis of the momentum map in its critical points, we obtain a complete topological classification of the different invariant sets of the phase flow of this problem. By this topological study and the calculation of the action-angle variables we obtain the full classification of periodic and quasiperiodic orbits for this system.