Abstract
In this paper, we discuss all the possible equilibrium states of axi-symmetrical solid bodies with liquid-filled cavities rotating around fixed axes according to the extremum conditions on the potential energy, and conclude that there exists a unique stable final-state solution, for which the system uniformly rotates around its vertical symmetrical axis, for both the inverted and suspended ones. And then applying the Lyapunov direct approach for a continuous system, we investigate the stability of the rotating systems subject to large disturbances. In addition, we describe an interesting analogue between the rotation of a solid body with a liquid-filled cavity in the inverted case and the motion of a small ball in a spinning spherical bowl. The results obtained herein theoretically provide an evidence of the reality of the secular stability.
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