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

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).