Precession and recession of the rock'n'roller

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 < I3, is integrable and themotion 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.