The Rolling Body Motion Of a Rigid Body on a Plane and a Sphere. Hierarchy of Dynamics

TL;DR

This paper investigates the dynamics of a rigid body rolling on a plane and a sphere, identifying conditions for invariant measures, integrals, and Poisson structures, and presenting a hierarchy of these properties across different cases.

Contribution

It provides a comprehensive analysis of the existence of invariant measures, integrals, and Poisson structures in rolling rigid body problems, including new cases with algebraic expressions.

Findings

Identification of conditions for invariant measures and integrals.

Hierarchy of tensor invariants across different rolling scenarios.

Tables summarizing the existence of geometric structures.

Abstract

In this paper we consider cases of existence of invariant measure, additional first integrals, and Poisson structure in a problem of rigid body's rolling without sliding on plane and sphere. The problem of rigid body's motion on plane was studied by S.A. Chaplygin, P. Appel, D. Korteweg. They showed that the equations of motion are reduced to a second-order linear differential equation in the case when the surface of dynamically symmetric body is a surface of revolution. These results were partially generalized by P. Woronetz, who studied the motion of body of revolution and the motion of round disk with sharp edge on the surface of sphere. In both cases the systems are Euler-Jacobi integrable and have additional integrals and invariant measure. It turns out that after some change of time defined by reducing multiplier, the reduced system is a Hamiltonian system. Here we consider different cases when the integrals and invariant measure can be presented as finite algebraic expressions. We also consider the generalized problem of rolling of dynamically nonsymmetric Chaplygin ball. The results of studies are presented as tables that describe the hierarchy of existence of various tensor invariants: invariant measure, integrals, and Poisson structure in the considered problems.