The topology of Liouville foliation for the Kovalevskaya integrable case on the Lie algebra so(4)

TL;DR

This paper investigates the topological structure of the Liouville foliation for an integrable system on the Lie algebra so(4), revealing how classical Kovalevskaya properties relate to this generalized case through bifurcation analysis.

Contribution

It provides a detailed topological analysis of the Kovalevskaya integrable case on so(4), including bifurcation diagrams, critical points, and loop molecules, extending classical results.

Findings

Bifurcation diagrams of the momentum map are constructed for all parameter values.

Types of critical points of rank 0 are classified.

Loop molecules for all singular points are computed.

Abstract

In this paper we study topological properties of an integrable case for Euler's equations on the Lie algebra $\textrm{so}(4)$, which can be regarded as an analogue of the classical Kovalevskaya case in rigid body dynamics. In particular, for all values of the parameters of the system under consideration the bifurcation diagrams of the momentum mapping are constructed, the types of critical points of rank $0$ are determined, the bifurcations of Liouville tori are described and the loop molecules for all singular points of the bifurcation diagram are computed. It follows from the obtained results that some topological properties of the classical Kovalevskaya case can be obtained from the corresponding properties of the considered integrable case on the Lie algebra $\textrm{so}(4)$ by taking a natural limit.