The Rolling Motion of a Ball on a Surface. New Integrals and Hierarchy of Dynamics

TL;DR

This paper investigates the dynamics of a rolling homogeneous ball on various surfaces, introducing new integrals and hierarchies, and extends classical results with novel solutions and properties of the motion.

Contribution

It presents new integrals and solutions for the rolling problem on arbitrary surfaces and generalizes classical problems like the Jacobi problem to nonholonomic systems.

Findings

Solutions by quadratures for new cases of rolling motion

Existence of an additional integral and invariant measure in a special case

Ball's motion on a cylinder in gravity is bounded and non-descending on average

Abstract

The paper is concerned with the problem on rolling of a homogeneous ball on an arbitrary surface. New cases when the problem is solved by quadratures are presented. The paper also indicates a special case when an additional integral and invariant measure exist. Using this case, we obtain a nonholonomic generalization of the Jacobi problem for the inertial motion of a point on an ellipsoid. For a ball rolling, it is also shown that on an arbitrary cylinder in the gravity field the ball's motion is bounded and, on the average, it does not move downwards. All the results of the paper considerably expand the results obtained by E. Routh in XIX century.