轴对称充液腔体旋转的定态解及其稳定性

本文根据势能的极值条件,讨论了轴对称充液腔体绕定轴旋转的各种可能的平衡态,证明了充满粘性液体的腔体在倒立情形和下悬情形中都只存在绕铅垂的对称轴整体旋转形式的稳定终态解,并且应用关于连续系统的Лядунов直接方法研究了这种旋转状态在大扰动下的稳定性.本文还描述了充液腔体倒立旋转运动和旋转的球碗中小球的运动之间有趣的类比关系.有关结果为长期稳定性的真实性提供了一种理论依据.

Stationary-State Solutions to the Rotation of Solid Bodies with Liquid-Filled Cavities and Their Stability

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.