On stability of uniformly-accelerated motions of an axially-symmetric heavy rigid body in an ideal fluid

We consider the problem of heavy rigid body dynamics in an infinite volume of an ideal incompressible fluid performing a potential motion. If the body is axially-symmetric, then the system admits partial solutions, when the axis of symmetry is vertical, and the body sinks and rotates around its symmetry axis. These solutions were found by V.A.Steklov already at the end of the 19th century, and he also pointed out that in general these motions are unstable (as they are uniformly accelerated).Here we consider the more delicate question, namely we derive the conditions for stability of the rotation axis, direction. The method of constructing the Lyapunov function may be generalized for deriving stability conditions for mechanical systems with nonstationary force-fields.