Dynamics of the Kirchhoff equations I: Coincident centers of gravity and buoyancy

Abstract

We study the Kirchhoff equations for a rigid body immersed in an incompressible, irrotational, inviscid fluid in the case that the centers of buoyancy and gravity coincide. The resulting dynamical equations form a non-canonical Hamiltonian system with a six-dimensional phase space, which may be reduced to a four-dimensional (two-degree-of-freedom, canonical) system using the two Casimir invariants of motion. Restricting ourselves to ellipsoidal bodies, we identify several completely integrable subcases. In the general case, we analyze existence, linear and nonlinear stability, and bifurcations of equilibria corresponding to steady translations and rotations, including mixed modes involving simultaneous motion along two body axes, some of which we show can be stable. By perturbing from the axisymmetric, integrable case, we show that slightly asymmetric ellipsoids are typically non-integrable, and we investigate their dynamics with a view to using motions along homo- and -heteroclinic orbits to execute specific maneuvers in autonomous underwater vehicles.