TITLE: Theory and numerics of stability transitions for falling
spinning underwater vehicles

ABSTRACT: Because of the presence of noncompact symmetry, the
stability of the motion of a bottom heavy falling, spinning underwater
vehicle, in the Kirchhoff approximation, is extremely delicate and
complicated. If the vehicle falls too quickly, then this motion is
spectrally unstable. As the speed is reduced, there is a (spin
dependent) transition to spectral stability, and full nonlinear
stability can be demonstrated by Kolmogorov-Arnold-Moser
confinement. As the speed is further reduced, there is a (spin
independent) transition to Energy-momentum stability. In the gap
between the transitions, it is expected on general grounds that the
addition of arbitrarily small dissipation will induce spectral
instability. Since the EM transition is spin independent, and the
stability analysis is general for Euclidean symmetry, the implication
for gyroscopically stabilized devices is startling: in the presence of
noncompact symmetry, robust stability may not be achievable by the use
of spin. The stability analysis depends on the language, results, and
insights, of modern mechanics set in the context of symplectic
geometry. The numerical validation of the results ought therefore to
preserve this context. The Kirchhoff approximation will be reviewed,
the stability analysis will be summarized, the numerical validation of
the results will be presented.

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.