Affine generalizations of the nonholonomic problem of a convex body rolling without slipping on the plane

TL;DR

This paper introduces affine generalizations of a nonholonomic rolling problem for convex bodies on a plane, analyzing their dynamics, integrability, and special cases like balanced spheres and bodies of revolution.

Contribution

It presents a novel affine generalization of the classical nonholonomic problem and studies its dynamical properties, including integrals and invariant measures.

Findings

Existence of first integrals in certain cases

Presence of smooth invariant measures

Conditions for integrability in special configurations

Abstract

We introduce a class of examples which provide an affine generalization of the nonholonomic problem of a convex body rolling without slipping on the plane. We investigate dynamical aspects of the system such as existence of first integrals, smooth invariant measure and integrability, giving special attention to the cases in which the convex body is a dynamically balanced sphere or a body of revolution.