On bi-hamiltonian geometry of some integrable systems on the sphere with cubic integral of motion

TL;DR

This paper develops a bi-Hamiltonian framework for certain integrable systems on the sphere with cubic integrals, providing explicit methods for separation of variables in classical problems like the Goryachev system.

Contribution

It introduces a bi-Hamiltonian structure for integrable systems on the sphere with cubic integrals and applies it to classical tops, deriving explicit separation procedures.

Findings

Established bi-Hamiltonian structures for specific integrable systems

Derived explicit separated coordinates for Goryachev system

Provided a method to find separated relations for these systems

Abstract

We obtain bi-Hamiltonian structure for a family of integrable systems on the sphere S with an additional integral of third order in momenta. These results are applied to the Goryachev system and Goryachev-Chaplygin top for which we give an explicit procedure to find the separated coordinates and the separated relations.