Explicit determination of certain periodic motions of a generalized two-field gyrostat

TL;DR

This paper explicitly determines special periodic motions in a generalized two-field gyrostat system, revealing their algebraic structure and relation to the bifurcation diagram and Lax curve discriminant surface.

Contribution

It identifies points of rank 1 in the momentum map and expresses the phase variables of these motions using algebraic functions satisfying elliptic differential equations.

Findings

Periodic motions correspond to singular points of the bifurcation diagram.

Phase variables are algebraic functions of a single variable.

Points belong to the intersection of two sheets of the discriminant surface.

Abstract

The case of motion of a generalized two-field gyrostat found by V.V.Sokolov and A.V.Tsiganov is known as a Liouville integrable Hamiltonian system with three degrees of freedom. We find a set of points at which the momentum map has rank 1. This set consists of special periodic motions which correspond to the singular points of a bifurcation diagram on an iso-energetic surface. For such motions the phase variables can be expressed in terms of algebraic functions of a single auxiliary variable. These algebraic functions satisfy a differential equation integrable in elliptic functions of time. It is shown that the corresponding points in the three-dimensional space of the constants of the integrals belong to the intersection of two sheets of the discriminant surface of the Lax curve.