Nonholonomic Euler-Poincare Equations and Stability in Chaplygin's Sphere

A theory of reducing several classes of nonholonomic mechanical systems that are defined on semidirect products of Lie groups is discussed. The method reduces the Lagrange-d'Alembert principle to obtain a reduced constrained principle that determines Euler-Poincar\'e equations on the reduced space. We then use the theory as a framework to study a particular nonholonomic system: Chaplygin's sphere (a ball that rolls without slipping on the plane and whose moments of inertia may differ, but has center of mass coinciding with the center of the ball). Several results are discussed, the stability of relative equilibria and stabilizing the unstable equilibria with feedback control.