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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 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
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