27.6: Stability of Top Spinning about Vertical Axis

(from Landau) For $\theta=\dot{\theta}=0, \quad L_{3}=L_{Z}, E^{\prime}=0 . \text { Near } \theta=0$,

$$\begin{equation} \begin{aligned} V_{\text {effective }}(\theta) &=\dfrac{\left(L_{Z}-L_{3} \cos \theta\right)^{2}}{2 I_{1}^{\prime} \sin ^{2} \theta}-M g \ell(1-\cos \theta) \\ & \cong \dfrac{L_{3}^{2}\left(\dfrac{1}{2} \theta^{2}\right)^{2}}{2 I_{1}^{\prime} \theta^{2}}-\dfrac{1}{2} M g \ell \theta^{2} \\ &=\left(L_{3}^{2} / 8 I_{1}^{\prime}-\dfrac{1}{2} M g \ell\right) \theta^{2} \end{aligned} \end{equation}$$

The vertical position is stable against small oscillations provided $L_{3}^{2}>4 I_{1}^{\prime} M g \ell, \text { or }$, or $\Omega_{3}^{2}>4 I_{1}^{\prime} M g \ell / I_{3}^{2}$

Exercise $\PageIndex{1}$

Suppose you set the top vertical, but spinning at less than $\Omega_{3 \text { crit }}$, the value at which it is just stable. It will fall away, but bounce back, and so on. Show the maximum angle it reaches is given by $\cos (\theta / 2)=\Omega_{3} / \Omega_{3 \mathrm{crit}}$.