Classification of singularities in the problem of motion of the Kovalevskaya top in a double force field

TL;DR

This paper analyzes the singularities of the Kovalevskaya top in a double force field, classifying all nondegenerate critical points and describing the bifurcation structure of the integrable Hamiltonian system.

Contribution

It provides a comprehensive classification of all nondegenerate singularities and critical points in the Kovalevskaya top problem with a double force field, including bifurcation diagrams.

Findings

Classification of all nondegenerate critical points

Description of bifurcation diagrams

Identification of equilibria and singular motions

Abstract

The problem of motion of the Kovalevskaya top in a double force field is investigated (the integrable case of A.G. Reyman and M.A. Semenov-Tian-Shansky without a gyrostatic momentum). It is a completely integrable Hamiltonian system with three degrees of freedom not reducible to a family of systems with two degrees of freedom. The critical set of the integral map is studied. The critical subsystems and bifurcation diagrams are described. The classification of all nondegenerate critical points is given. The set of these points consists of equilibria (nondegenerate singularities of rank 0), of singular periodic motions (nondegenerate singularities of rank 1), and also of critical two-frequency motions (nondegenerate singularities of rank 2).