Conformally symplectic Chaplygin reduction in rubber rolling of surfaces of revolution over the plane

TL;DR

This paper investigates the conformally symplectic structure of Chaplygin reduction in rubber rolling of surfaces of revolution, highlighting special integrals of motion and connecting classical results with recent geometric frameworks.

Contribution

It establishes a link between rubber rolling on surfaces of revolution and conformally symplectic Chaplygin systems, extending understanding of integrals of motion in nonholonomic mechanics.

Findings

Special elementary integral for surfaces of revolution

Connection between rubber rolling and conformally symplectic structures

Extension of classical results with modern geometric insights

Abstract

Rubber rolling (no-slip and no-twist) of a convex body on the plane under the influence of gravity is a SE(2) Chaplygin system, that reduces to the sphere of Poisson vectors. I comment upon an observation by A.V Borisov and I.S. Mamaev (Regular and Chaotic Dynamics, 13(5):443-490, 2008) for the case of surfaces of revolution [also in A. V. Borisov, I. S. Mamaev and I. A. Bizyaev (Regular and Chaotic Dynamics, 18(3):277-328, 2013)]. They show that this case is quite special: the additional integral of motion is elementary, while for marble rolling it is not elementary. I connect this finding with recent work about Chaplygin reduced systems that are conformally symplectic (Luis Garcia Naranjo and Juan C. Marrero. The geometry of nonholonomic Chaplygin systems revisited. Nonlinearity, 33(3):1297, 2020).