# Integrable discretization and deformation of the nonholonomic Chaplygin ball

## TL;DR

This paper explores integrable discretizations and deformations of the nonholonomic Chaplygin ball system, resulting in new geodesic flows on the sphere with higher-order polynomial integrals, advancing understanding of nonholonomic integrable systems.

## Contribution

It introduces novel integrable discretizations and deformations of the Chaplygin ball using Abel quadratures, leading to new geodesic flows with polynomial integrals.

## Key findings

- New integrable discretizations of the Chaplygin ball system
- Development of deformations with preserved integrability
- Discovery of geodesic flows with fourth-order polynomial integrals

## Abstract

The rolling of a dynamically balanced ball on a horizontal rough table without slipping was described by Chaplygin using Abel quadratures. We discuss integrable discretizations and deformations of this nonholonomic system using the same Abel quadratures. As a by-product one gets new geodesic flow on the unit two-dimensional sphere whose additional integrals of motion are polynomials in the momenta of fourth order.