Two rolling disks or spheres

George W. Patrick; Tyler Helmuth

doi:10.3934/dcdss.2010.3.129

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2010, Volume 3, Issue 1: 129-140. Doi: 10.3934/dcdss.2010.3.129

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Two rolling disks or spheres

George W. Patrick^1, and
Tyler Helmuth^1,

1.
Applied Mathematics and Mathematical Physics, Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan

Received: June 30, 2008

Revised: June 30, 2009

Published: February 2010

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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.

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Mathematics Subject Classification: Primary: 70F25; Secondary: 37J60, 70E, 70K42.

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