`a`
Discrete and Continuous Dynamical Systems - Series S (DCDS-S)
 

Two rolling disks or spheres

Pages: 129 - 140, Volume 3, Issue 1, March 2010

doi:10.3934/dcdss.2010.3.129       Abstract        Full Text (209.8K)       Related Articles

George W. Patrick - Applied Mathematics and Mathematical Physics, Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada (email)
Tyler Helmuth - Applied Mathematics and Mathematical Physics, Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada (email)

Abstract: The mechanical system of two disks, moving freely in the plane, while in contact and rolling against each other without slipping, may be written as a Lagrangian system with three degrees of freedom and one holonomic rolling constraint. We derive simple geometric criteria for the rotational relative equilibria and their stability. Extending to three dimensions, we derive the kinematics of the analogous system where two spheres replace two disks, and we verify that the rolling disk system occurs as a holonomic subsystem of the rolling sphere system.

Keywords:  Relative equilibria, rolling rigid bodies, nonholonomic mechanics.
Mathematics Subject Classification:  Primary: 70F25; Secondary: 37J60, 70E, 70K42.

Received: July 2008;      Revised: July 2009;      Published: December 2009.