The Remarkable Rock'n'roller

The rock'n'roller is a rigid body, spherical in form but having an asymmetric distribution of mass. It rolls, without slipping, on a horizontal surface. The moments of inertia are I1 < I2 < I3 and the geometric centre lies on the principal axis corresponding to I3.

The rock'n'roller has a facinating pattern of behaviour: When released from a tilting position, it rocks back and forth and precesses in the azimuthal direction. But this precession reverses from time to time, a phenomenon we call recession. Recession represents a dramatic change in the character of the motion arising from a breaking of the inertial symmetry I1=I2.

Recession can be seen in the animation below, and is fully discussed in a paper in J. Phys. A (see link to PDF below).

[picture of RnR]

The Rock'n'roller

Animation of the Rock'n'roller

[RnR Animation]
[Movie produced by Miguel Bustamante]

Precession and Recession of the Rock'n'roller

Peter Lynch and Miguel D Bustamante


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 I1=I2, is integrable and the motion is completely regular. Three known conservation laws are the total energy E Jellett's quantity QJ and Routh's quantity QR.

When the inertial symmetry I1=I2 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 (QR,QJ) 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 (QR(t),QJ(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.

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Updated June 2012.