The energy-momentum method for the stability of non-holonomic systems

In this paper, we analyze the stability of relative equilibria of non-holonomic systems (that is, mechanical systems with non-integrable constraints such as rolling constraints). In the absence of external dissipation, such systems conserve energy but nonetheless can exhibit both neutrally stable and asymptotically stable, as well as linearly unstable relative equilibria. To carry, out the stability analysis, we use a generalization of the energy-momentum method combined with the Lyapunov-Malkin theorem and the center manifold theorem. While this approach is consistent with the energy-momentum method for holonomic systems, it extends it in substantial ways. The theory is illustrated with several examples, including the rolling disk, the roller racer and the rattleback top.