Phase topology of the Kowalevski-Sokolov top

TL;DR

This paper analyzes the phase topology of an integrable Hamiltonian system generalizing the Kowalevski top, classifying equilibria, stability, and bifurcations, and providing detailed diagrams and topological descriptions.

Contribution

It offers a comprehensive topological analysis of the Kowalevski-Sokolov top, including classification of equilibria, stability, and bifurcation diagrams, extending previous work on integrable tops.

Findings

Classification of relative equilibria and their stability types

Explicit bifurcation diagrams and iso-energy manifold descriptions

Complete characterization of critical points and their types

Abstract

The phase topology of the integrable Hamiltonian system on $e(3)$ found by V.V.Sokolov (2001) and generalizing the Kowalevski case is investigated. The generalization contains, along with a homogeneous potential force field, gyroscopic forces depending on the configurational variables. Relative equilibria are classified, their type is calculated and the character of stability is defined. The Smale diagrams of the case are found and the classification of iso-energy manifolds of the reduced systems with two degrees of freedom is given. The set of critical points of the complete momentum map is represented as a union of critical subsystems; each critical subsystem is a one-parameter family of almost Hamiltonian systems with one degree of freedom. For all critical points we explicitly calculate the characteristic values defining their type. We obtain the equations of the surfaces bearing the bifurcation diagram of the momentum map. We give examples of the existing iso-energy diagrams with a complete description of the corresponding rough topology (of the regular Liouville tori and their bifurcations).