Phase topology of one nonclassical integrable problem of dynamics

TL;DR

This paper analyzes the phase topology of a nonclassical integrable system with three degrees of freedom, providing explicit formulas for integrals, identifying invariant submanifolds, and describing bifurcation diagrams using critical subsystems.

Contribution

It introduces explicit formulas for integrals K and G, identifies invariant submanifolds, and applies the method of critical subsystems to analyze phase topology.

Findings

Explicit formulas for integrals K and G are derived.

Two invariant four-dimensional submanifolds are identified.

Bifurcation diagrams of Liouville tori are constructed.

Abstract

We consider the integrable system with three degrees of freedom for which Sokolov and Tsiganov specified Lax representation. Lax representation generalizes L-A pair of the Kowalevski gyrostat in two constant fields, found by A.G.Reyman and M.A.Semenov-Tian-Shansky. In the paper, we give the explicit formulas for the (independent almost everywhere) additional first integrals K and G. These integrals are functionally connected with factors of a spectral curve of L-A pair by Sokolov and Tsiganov. Due to this form of additional integrals K and G, without constant gyrostatic moment, we managed to find analytically two invariant four-dimensional submanifolds on which the induced dynamic system is almost everywhere Hamiltonian system with two degrees of freedom. System of equations that describes one of these invariant submanifolds is a generalization of invariant relations of an integrable Bogoyavlensky case in dynamics of a rigid body. To describe phase topology of the system as a whole we use the method of critical subsystems. For each subsystem, we construct the bifurcation diagrams and specify the bifurcations of Liouville tori both in subsystems, and in the system as a whole.