# Precession and Recession of the Rock'n'roller - Nonlinear Sciences > Exactly Solvable and Integrable Systems

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Abstract: We study the dynamics of a spherical rigid body that rocks and rolls on aplane under the effect of gravity. The distribution of mass is non-uniform andthe centre of mass does not coincide with the geometric centre.The symmetric case, with moments of inertia I 1=I 2, is integrable and themotion is completely regular. Three known conservation laws are the totalenergy 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 ofthe solutions is profoundly changed and new types of motion become possible. Wederive the equations governing the general motion and present analytical andnumerical evidence of the recession, or reversal of precession, that has beenobserved in physical experiments.We present an analysis of recession in terms of critical lines dividing theQ R,Q J plane into four dynamically disjoint zones. We prove that recessionimplies the lack of conservation of Jellett-s and Routh-s quantities, byidentifying individual reversals as crossings of the orbit Q Rt,Q Jtthrough the critical lines. Consequently, a method is found to produce a largenumber of initial conditions so that the system will exhibit recession.

Author: ** Peter Lynch, Miguel D. Bustamante**

Source: https://arxiv.org/