## 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
*I*_{1} < I_{2} < I_{3} and the
geometric centre lies on the principal
axis corresponding to *I*_{3}.
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
*I*_{1}=I_{2}.

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

# The Rock'n'roller

### Animation of the Rock'n'roller

##### [Movie produced by Miguel Bustamante]

- Peter Lynch & Miguel D Bustamante, 2009:
**Precession and Recession of the Rock'n'roller.**

*J. Phys. A: Math. Theor.* **42 **(2009) 425203 (25pp).
PDF. DOI: 10.1088/1751-8113/42/42

Paper chosen for inclusion in
IOP Select
- Peter Lynch & Miguel D Bustamante, 2012:

**Quaternion Solution for the Rock'n'roller:
Box Orbits, Loop Orbits and Recession.
**

Submitted to *Reg. & Chaotic Dyn.*
PDF.

###
Precession and Recession of the Rock'n'roller

#### Peter Lynch and Miguel D Bustamante

### 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.

**
Link to Miguel Bustamante's Home Page **

**
Back to Peter Lynch's Home Page **

##### **Updated June 2012. **