Krishnaprasad, Perinkulam S.Manikonda, VikramIt has been known for some time that the classical work of Kirchhoff, Love,and Birkhoff on rigid bodies in incompressible, irrotational flows provideseffective models for treating control problems for underwater vehicles.This has also led to a better appreciation of the dynamics of suchsystems. In this paper, we develop results based on geometric mechanics andcenter manifold theory to solve controllability and stabilization questionsfor a class of under-actuated left invariant mechanical systems on Liegroups that include approximate models of underwater vehicles and surfacevehicles. We also provide numerical evidence to capture the globalproperties of certain interesting feedback laws.<p><I>(This work appears as an invited paper in the Proc. IFAC Sympo. on NonlinearControl Systems Design (NOLCOS'98), (1998), 1:139-144) </I>en-USgeometric controlkinematicslinear systemsnonlinear systemsroboticsstabilitysymmetryLie-Poisson reductionunderactuated systemscontrollabilitystabilizationhovercraftunderwater vehiclesIntelligent Control SystemsTechnical Report