Precession and Recession of the Rock'n'roller

TL;DR

This paper analyzes the dynamics of a non-uniform spherical body rolling on a plane, revealing how slight asymmetries cause complex motions like precession recession, supported by analytical and numerical evidence.

Contribution

It provides a detailed analysis of how breaking inertial symmetry leads to recession phenomena, including the derivation of governing equations and identification of critical conditions.

Findings

Recession occurs when the system crosses critical lines in the (Q_R,Q_J) plane.

Recession is associated with the loss of conservation of Jellett's and Routh's quantities.

The study offers a method to generate initial conditions that induce recession.

Abstract

We study the dynamics of a spherical rigid body that rocks and rolls on a plane under the effect of gravity. The distribution of mass is non-uniform and the centre of mass does not coincide with the geometric centre. The symmetric case, with moments of inertia I_1=I_2, is integrable and the motion is completely regular. Three known conservation laws are the total energy E, Jellett's quantity Q_J and Routh's quantity Q_R. When the inertial symmetry I_1=I_2 is broken, even slightly, the character of the solutions is profoundly changed and new types of motion become possible. We derive the equations governing the general motion and present analytical and numerical evidence of the recession, or reversal of precession, that has been observed in physical experiments. We present an analysis of recession in terms of critical lines dividing the (Q_R,Q_J) plane into four dynamically disjoint zones. We prove that recession implies the lack of conservation of Jellett's and Routh's quantities, by identifying individual reversals as crossings of the orbit (Q_R(t),Q_J(t)) through the critical lines. Consequently, a method is found to produce a large number of initial conditions so that the system will exhibit recession.