Spherical and planar ball bearings -- nonholonomic systems with invariant measures

Abstract

We first construct nonholonomic systems of $n$ homogeneous balls ${\mathbf{B}}_1,\dots ,{\mathbf{B}}_n$ with centers $O_1,...,O_n$ and with the same radius $r$ that are rolling without slipping around a fixed sphere ${\mathbf{S}}_0$ with center $O$ and radius $R$. In addition, it is assumed that a dynamically nonsymmetric sphere $\mathbf{S}$ of radius $R+2r$ and the center that coincides with the center $O$ of the fixed sphere ${\mathbf{S}}_0$ rolls without slipping over the moving balls ${\mathbf{B}}_1,\dots ,{\mathbf{B}}_n$. We prove that these systems possess an invariant measure. As the second task, we consider the limit, when the radius $R$ tends to infinity. We obtain a corresponding planar problem consisting of $n$ homogeneous balls ${\mathbf{B}}_1,\dots ,{\mathbf{B}}_n$ with centers $O_1,...,O_n$ and the same radius $r$ that are rolling without slipping over a fixed plane ${\Sigma }_0$, and a moving plane $\Sigma$ that moves without slipping over the homogeneous balls. We prove that this system possesses an invariant measure and that it is integrable in quadratures according to the Euler-Jacobi theorem.

20 pages, 2 figures

References in corpus (2)

• Hamiltonisation, measure preservation and first integrals of the multi-dimensional rubber Routh sphere
• Demchenko's nonholonomic case of gyroscopic ball rolling without sliding over a sphere after his 1923 Belgrade doctoral thesis

Cited by in corpus (3)

• Gyroscopic Chaplygin systems and integrable magnetic flows on spheres
• Bifurcation analysis of the problem of a "rubber" ellipsoid of revolution rolling on a plane
• Spherical and Planar Ball Bearings -- a Study of Integrable Cases