Bifurcation diagrams and critical subsystems of the Kowalevski gyrostat in two constant fields

TL;DR

This paper analyzes the bifurcation diagrams and critical subsystems of the Kowalevski gyrostat in two constant fields, providing a topological classification of its phase space structure and critical sets.

Contribution

It offers the first topological analysis of the Kowalevski gyrostat in two fields by identifying its critical set and deriving bifurcation diagram equations.

Findings

Stratified critical set of the momentum map identified

Bifurcation diagram equations derived in three-dimensional space

Classification framework for bifurcation sets on iso-energetic levels

Abstract

The Kowalevski gyrostat in two constant fields is known as the unique example of an integrable Hamiltonian system with three degrees of freedom not reducible to a family of systems in fewer dimensions and still having the clear mechanical interpretation. The practical explicit integration of this system can hardly be obtained by the existing techniques. Then the challenging problem becomes to fulfil the qualitative investigation based on the study of the Liouville foliation of the phase space. As the first approach to topological analysis of this system we find the stratified critical set of the momentum map; this set consists of the trajectories with number of frequencies less than three. We obtain the equations of the bifurcation diagram in three-dimensional space. These equations have the form convenient for the classification of the bifurcation sets induced on 5-dimensional iso-energetic levels.