Nonholonomic Euler-Poincaré equations and stability in Chaplygin's sphere

Abstract

A method of reducing several classes of nonholonomic mechanical systems that are defined on semidirect products of Lie groups is developed. The method reduces the Lagrange-d'Alembert principle to obtain a reduced constrained principle that determines Euler-Poincare equations on the reduced space. The theory is demonstrated by deriving the reduced equations of motion for several examples of rigid bodies that roll without slipping. This method of reduction is a generalization to the nonholonomic setting of the method developed in [CHMR] which reduces Hamilton's principle and derives Euler-Poincare equations for a class of unconstrained systems on Lie Groups.Our method of reduction is then used as a framework for obtaining several results related to a particular nonholonomic system: Chaplygin's sphere. Chaplygin's sphere is a ball that rolls without slipping on a horizontal plane and has a nonhomogeneous mass distribution. The principal moments of inertia of the ball need not be equal, however the ball's center of mass coincides with its geometric center. The first result determines the stability of the relative equilibria of Chaplygin's sphere: the ball spins stably about its long and short axis and unstably about the middle axis. The next results use two different approaches to stabilize the rotation of the ball about its middle axis by making use of the idea of controlled Lagrangians introduced in [BLMa]. The first approach controls an internal rotor that has been added to the ball and generalizes the solution presented in [BLMa] to the analogous problem of using an internal rotor to stabilize a free rigid body. The second approach controls the plane the ball rolls upon by forces of horizontal translations. For the system in which the plane is allowed to react to the motion of the ball, we derive the equations of motion, identify the relative equilibria and consider the problem of stabilizing the ball by controlling the plane with forces of horizontal translations.