DYNAMICS OF THE KIRCHHOFF EQUATIONS I - COINCIDENT CENTERS OF GRAVITYAND BUOYANCY

We study the Kirchhoff equations for a rigid body immersed in an incom pressible, irrotational, inviscid fluid in the case that the centers o f buoyancy and gravity coincide. The resulting dynamical equations for m a non-canonical Hamiltonian system with a six-dimensional phase spac e, which may be reduced to a four-dimensional (two-degree-of-freedom, canonical) system using the two Casimir invariants of motion. Restrict ing ourselves to ellipsoidal bodies, we identify several completely in tegrable subcases. In the general case, we analyze existence, linear a nd nonlinear stability, and bifurcations of equilibria corresponding t o steady translations and rotations, including mixed modes involving s imultaneous motion along two body axes, some of which we show can be s table. By perturbing from the axisymmetric, integrable case, we show t hat slightly asymmetric ellipsoids are typically non-integrable, and w e investigate their dynamics with a view to using motions along homo- and -heteroclinic orbits to execute specific maneuvers in autonomous u nderwater vehicles. (C) 1998 Elsevier Science B.V.