  We develop a general stability theory for equilibrium points of Poisson
dynamical systems and relative equilibria of Hamiltonian systems with
symmetries, including several generalisations of the Energy-Casimir and
Energy-Momentum methods. Using a topological generalisation of Lyapunov's
result that an extremal critical point of a conserved quantity is stable, we
show that a Poisson equilibrium is stable if it is an isolated point in the
intersection of a level set of a conserved function with a subset of the phase
space that is related to the non-Hausdorff nature of the symplectic leaf space
at that point. This criterion is applied to generalise the Energy-Momentum
method to Hamiltonian systems which are invariant under non-compact symmetry
groups for which the coadjoint orbit space is not Hausdorff. We also show that
a $G$-stable relative equilibrium satisfies the stronger condition of being
$A$-stable, where $A$ is a specific group-theoretically defined subset of $G$
which contains the momentum isotropy subgroup of the relative equilibrium.
