Regular and Chaotic Dynamics

, Volume 18, Issue 1–2, pp 166–183 | Cite as

Quaternion solution for the rock’n’roller: Box orbits, loop orbits and recession

  • Peter LynchEmail author
  • Miguel D. Bustamante


We consider two types of trajectories found in a wide range of mechanical systems, viz. box orbits and loop orbits. We elucidate the dynamics of these orbits in the simple context of a perturbed harmonic oscillator in two dimensions. We then examine the small-amplitude motion of a rigid body, the rock’n’roller, a sphere with eccentric distribution of mass. The equations of motion are expressed in quaternionic form and a complete analytical solution is obtained. Both types of orbit, boxes and loops, are found, the particular form depending on the initial conditions. We interpret the motion in terms of epi-elliptic orbits. The phenomenon of recession, or reversal of precession, is associated with box orbits. The small-amplitude solutions for the symmetric case, or Routh sphere, are expressed explicitly in terms of epicycles; there is no recession in this case.


rolling body dynamics nonholonomic constraints Hamiltonian dynamics 

MSC2010 numbers

70E18 70E20 70H07 


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Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.School of Mathematical SciencesUCDBelfield, Dublin 4Ireland

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