Bifurcation diagrams of the Kowalevski top in two constant fields

TL;DR

This paper analyzes the bifurcation diagrams of the Kowalevski top in two constant fields, revealing its critical sets, bifurcation structures, and admissible regions, thus advancing the topological understanding of this integrable Hamiltonian system.

Contribution

It provides the first topological analysis of the Kowalevski top in two fields, including critical set determination and bifurcation diagram equations.

Findings

Critical set of the integral map identified

Bifurcation diagram equations derived in R^3

Admissible regions characterized by inequalities

Abstract

The Kowalevski top in two constant fields is known as the unique profound example of an integrable Hamiltonian system with three degrees of freedom not reducible to a family of systems in fewer dimensions. As the first approach to topological analysis of this system we find the critical set of the integral map; this set consists of the trajectories with number of frequencies less than three. We obtain the equations of the bifurcation diagram in R^3. A correspondence to the Appelrot classes in the classical Kowalevski problem is established. The admissible regions for the values of the first integrals are found in the form of some inequalities of general character and boundary conditions for the induced diagrams on energy levels.