Integrable Euler top and nonholonomic Chaplygin ball

TL;DR

This paper explores the mathematical structures underlying the integrable Euler top and nonholonomic Chaplygin ball, focusing on Poisson structures, Lax matrices, and bi-Hamiltonian frameworks to deepen understanding of these dynamical systems.

Contribution

It provides a comprehensive analysis of the algebraic and geometric structures of these systems, highlighting new insights into their integrability and separation variables.

Findings

Identification of Poisson and bi-Hamiltonian structures

Construction of Lax matrices and r-matrices for both systems

Clarification of separation variables and integrability properties

Abstract

We discuss the Poisson structures, Lax matrices, $r$-matrices, bi-hamiltonian structures, the variables of separation and other attributes of the modern theory of dynamical systems in application to the integrable Euler top and to the nonholonomic Chaplygin ball.