The Energy-Momentum Method for the Stability of Nonholonomic Systems
Zenkov, D. V., A. M. Bloch and J. E. Marsden
Dyn. Stab. of Systems, 13, 123-166
Abstract:
In this paper we analyze the stability of relative equilibria
of nonholonomic systems (that is, mechanical systems with
nonintegrable 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 the rolling
disk, the roller racer, and the rattleback top.