Geometrical approach to separation of variables in mechanical systems

TL;DR

This paper reviews geometrical methods for separating variables in the integrable motion of a top in two constant fields, highlighting new coordinate systems that facilitate solving subsystems of the Kowalevski top.

Contribution

It introduces two new local planar coordinate systems that enable separation of variables in subsystems of the generalized Kowalevski top.

Findings

Established geometrical foundations for variable separation.

Presented coordinate systems leading to solvable subsystems.

Connected separation techniques with Liouville integrability.

Abstract

The article presents a compact review of the analytical results (2002-2009) in the study of the system describing the motion of a top in two constant fields. The Liouville integrability of this system under certain condition of the Kowalevski type was established by A.G.Reyman and M.A.Semenov-Tian-Shansky. We present some geometrical foundations of finding separations of variables. Two systems of local planar coordinates are introduced leading to separation of variables for two subsystems with two degrees of freedom in the dynamics of the generalized Kowalevski top.