Separation of variables in the generalized 4th Appelrot class

TL;DR

This paper extends the analysis of the Kowalevski top's 4th Appelrot class to two constant force fields, deriving explicit separation of variables and describing the motion's existence regions.

Contribution

It introduces a separation of variables method for the generalized 4th Appelrot class in two force fields, providing explicit parametric equations and analyzing the Hamiltonian structure.

Findings

Explicit parametric equations for the bifurcation sheet

Identification of the Hamiltonian structure with two degrees of freedom

Determination of the motion existence region

Abstract

We consider the analogue of the 4th Appelrot class of motions of the Kowalevski top for the case of two constant force fields. The trajectories of this family fill the four-dimensional surface O^4 in the six-dimensional phase space. The constants of three first integrals in involution restricted to this surface fill one of the sheets of the bifurcation diagram in R^3. We point out the pair of partial integrals to obtain the explicit parametric equations of this sheet. The induced system on O^4 is shown to be Hamiltonian with two degrees of freedom having the thin set of points where the induced symplectic structure degenerates. The region of existence of motions in terms of the integral constants is found. We provide the separation of variables on O^4 and the algebraic formulae for the initial phase variables.