The existence conditions for periodic motions of the Kowalevski gyrostat in double force field

TL;DR

This paper analyzes the conditions for the existence of special periodic motions in a Kowalevski gyrostat within a double force field, identifying bifurcation diagrams and singularities using analytical and computational methods.

Contribution

It provides the analytical equations of the separating set and its singularities, and studies the parameter-dependent transformations of special periodic motions.

Findings

Identified regions with different sets of motions.

Derived the bifurcation diagrams on iso-energetic levels.

Mapped the singularities and the separating set in parameter space.

Abstract

Consider the integrable problem of motion of a gyrostat with the Kowalevski type inertia tensor in a double force field. We study the special periodic motions (the rank 1 critical points of the integral mapping) found by M.P. Kharlamov (Mekh. Tverd. Tela, No 37, 2007). Possible transformations inside the set of such motions depending on three essential parameters are studied. We obtain the analytical equations of the separating set and its singularities, point out the regions with different sets of motions. We find the image of the separating set in the space of parameters defining the bifurcation diagrams on iso-energetic levels. The most complicated calculations are fulfilled with the help of the computer system Mathematica 7.