On the generalized integrable Chaplygin system

TL;DR

This paper explores two polynomial bi-Hamiltonian structures for a generalized integrable system on the sphere, providing explicit methods for variable separation and algebraic curve transformation.

Contribution

It introduces new polynomial bi-Hamiltonian structures and detailed procedures for variable separation in the generalized integrable Chaplygin system.

Findings

Two polynomial bi-Hamiltonian structures identified

Explicit separation variables and relations derived

Transformation of genus two algebraic curves explained

Abstract

We discuss two polynomial bi-Hamiltonian structures for the generalized integrable Chaplygin system on the sphere S^2 with an additional integral of fourth order in momenta. An explicit procedure to find the variables of separation, the separation relations and the transformation of the corresponding algebraic curves of genus two is considered in detail.