Topological analysis of classical integrable systems in the dynamics of the rigid body

TL;DR

This paper extends the topological analysis of classical integrable systems in rigid body dynamics, specifically addressing complex cases with non-linear integrals by modifying Smale's scheme to classify bifurcations and transformations of integral tori.

Contribution

It introduces a modified approach to Smale's program for analyzing the phase topology of the Kovalevskaya and Goryachev-Chaplygin cases with non-linear integrals.

Findings

Identified bifurcation sets for complex integrable cases.

Classified transformations of integral tori in these cases.

Discovered new non-trivial bifurcations of tori.

Abstract

The general integrability cases in the rigid-body dynamics are the solutions of Lagrange, Euler, Kovalevskaya, and Goryachev-Chaplygin. The first two can be included in Smale's scheme for studying the phase topology of natural systems with symmetries. We modify Smale's program to suit the most complicated last two cases with non-linear first integrals. The bifurcation sets are found and all transformations of the integral tori are described and classified. New non-trivial bifurcation of a torus is established in the Kovalevskaya and Goraychev-Chaplygin cases.