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 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. PACS numbers: 45.20.dc, 45.20.Jj, 45.40.Cc (Some figures in this article are in colour only in the electronic version) 1