充液型腔陀螺的非线性稳定性分析
Nonlinear Stability Analysis of a Spinning Top with an Interior Liquid-Filled Cavity
摘要：
考虑耦合系统$\mathscr S$的运动，该耦合系统由（不一定对称的）顶部$\mathscr B$和内腔$\mathscr C$构成，内腔中完全充满了纳维尔-斯托克斯液体$\mathscr L$。$\mathscr S$的特定稳态运动$\bar{\sf s}$（比方说）是当$\mathscr L$相对于$\mathscr B$静止时，并且作为整个刚性体的$\mathscr S$在其最高位置（{ \em直立旋转顶部}）围绕穿过其质心$G$的垂直轴以恒定角速度$\bar{\Vω}$旋转。然后，我们提供了一个完整的表征的非线性稳定性的$\bar{\sf s}$显示，粗略地说，$\bar{\sf s}$是稳定的，当且仅当$|\bar{\Vω}|$是足够大的，所有其他的物理参数是固定的。此外，我们表明，与$\mathscr C$是空的情况下，在上述稳定性条件下，顶部最终将返回到未扰动直立配置。
英文摘要：
Consider the motion of the the coupled system, $\mathscr S$, constituted by a (non-necessarily symmetric) top, $\mathscr B$, with an interior cavity, $\mathscr C$, completely filled up with a Navier-Stokes liquid, $\mathscr L$. A particular steady-state motion $\bar{\sf s}$ (say) of $\mathscr S$, is when $\mathscr L$ is at rest with respect to $\mathscr B$, and $\mathscr S$, as a whole rigid body, spins with a constant angular velocity $\bar{\Vω}$ around a vertical axis passing through its center of mass $G$ in its highest position ({\em upright spinning top}). We then provide a completely characterization of the nonlinear stability of $\bar{\sf s}$ by showing, roughly speaking, that $\bar{\sf s}$ is stable if and only if $|\bar{\Vω}|$ is sufficiently large, all other physical parameters being fixed. Moreover we show that, unlike the case when $\mathscr C$ is empty, under the above stability conditions, the top will eventually return to the unperturbed upright configuration.
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