  In the presence of noncompact symmetry, the stability of relative equilibria
under momentum-preserving perturbations does not generally imply robust
stability under momentum-changing perturbations. For axisymmetric relative
equilibria of Hamiltonian systems with Euclidean symmetry, we investigate
different mechanisms of stability: stability by energy-momentum confinement,
KAM, and Nekhoroshev stability, and we explain the transitions between these.
We apply our results to the Kirchhoff model for the motion of an axisymmetric
underwater vehicle, and we numerically study dissipation induced instability of
KAM stable relative equilibria for this system.
