\(G_2\) and the rolling distribution. (English) Zbl 1251.70008

Bor, Gil; Montgomery, Richard

Enseign. Math. (2) 55, No. 1-2, 157-196 (2009).

Two balls of different radii, \(r\) and \(R\), rolling along each other, without slipping or spinning, are considered. The configuration space for this system is a 5-dimensional manifold on which the no-slip/no-spin condition defines a rank 2 distribution, the rolling distribution which is related with the exceptional Lie group \(G_2\). A theorem concerning properties if this distribution was proved by Élie Cartan in 1898 which is valid only for the radii ratio \(3:1\) or \(1:3\). The authors propose another proof of this theorem avoiding the Cartan’s sophisticated approach.

Reviewer: Bojidar Cheshankov (Sofia)

MSC:

70E18	Motion of a rigid body in contact with a solid surface
37J60	Nonholonomic dynamical systems
70F25	Nonholonomic systems related to the dynamics of a system of particles

Keywords:

Lie groups