Asymptotic and Lyapunov stability of constrained and Poisson equilibria

This paper includes results centered around three topics, all of them related with the nonlinear stability of equilibria in constrained dynamical systems. First, we prove an energy-Casimir type sufficient condition for stability that uses functions that are not necessarily conserved by the flow and that takes into account the asymptotically stable behavior that may occur in certain constrained systems, such as Poisson and Leibniz dynamical systems. Second, this method is specifically adapted to Poisson systems obtained via a reduction procedure and we show in examples that the kind of stability that we propose is appropriate when dealing with the stability of the equilibria of some constrained mechanical systems. Finally, we discuss two situations in which the use of continuous Casimir functions in stability studies is equivalent to the topological stability methods introduced by Patrick et al. (Arch. Rational Mech. Anal., 2004, preprint arXiv:math.DS/0201239v1, to appear).