# A question about the tennis racket theorem with degenerate eigenvalues $I_1, I_2 , I_3$

If a rigid body has a symmetry such that two of the principal moments of inertia are equals, i.e. $$I_1=I_2> I_3 \qquad{\rm or}\qquad I_1>I_2=I_3.$$ Are the rotations around the principal axes stable?

Repeated application of Euler's equations

$$\dot{L}_i~\equiv~ I_i \dot{\Omega}_i~=~\Omega_{i+1}(I_{i+1}-I_{i-1}) \Omega_{i-1}, \qquad \forall i ~\in~\mathbb{Z}_3, \tag{1}$$

$$I_1I_2 I_3 \ddot{\Omega}_i~\stackrel{(1)}{=}~(I_{i+1}-I_{i-1}) \Omega_i\left\{(I_i -I_{i+1}) I_{i+1}\Omega_{i+1}^2- (I_i -I_{i-1}) I_{i-1}\Omega_{i-1}^2\right\} , \qquad\forall i ~\in~\mathbb{Z}_3. \tag{2}$$

Note that

$$I_{i+1}=I_{i-1}\qquad \stackrel{(1)}{\Rightarrow} \qquad \Omega_{i}, L_{i}\text{ are constants}, \qquad \forall i ~\in~\mathbb{Z}_3.\tag{3}$$

Assume $$I_1~\geq~ I_2~ \geq~ I_3 .\tag{4}$$

There are several cases:

• Case $$I_1>I_2>I_3$$: Euler Eqs. (1) have only the three principal axes as equilibrium points $$\dot{\vec{\Omega}}=0$$.

1. The major and the minor principal axes are stable, cf. a standard geometric argument where the intersection of an angular momentum sphere and an energy ellipsoid is a small loop, see e.g. the Phys.SE answers by Emilio Pisanty, Michael Seifert and ZeroTheHero.

2. The intermediate axis $$\vec{\Omega}\approx (0,\Omega_2,0)$$ is $$\color{red}{\text{unstable}}$$, cf. a standard analytic argument $$\ddot{\Omega}_i~\stackrel{(2)}{\approx}~\color{red}{+}\omega^2_2 \Omega_i, \qquad i\in{1,3}, \tag{5}$$ where $$\omega_2~:=~\Omega_2\sqrt{\frac{(I_1-I_2)(I_2-I_3)}{I_1I_3}}, \tag{6}$$ see e.g. the Phys.SE answer by David Bar Moshe.

• Case $$I_1=I_2>I_3$$: Then $$\Omega_3$$ and $$L_3$$ are constants, cf. eq. (3). Then $$\ddot{\Omega}_i~\stackrel{(2)}{=}~-\omega^2_3 \Omega_i, \qquad i\in{1,2}, \tag{7}$$ where $$\omega_3~:=~\Omega_3\sqrt{\frac{(I_1-I_3)(I_2-I_3)}{I_1I_2}}~=~{\rm const}. \tag{8}$$ Conclusion: There is a (slow) precession of $$\vec{\Omega}$$ and $$\vec{L}$$ around the third axis with angular frequency $$\omega_3$$. In other words: If $$\vec{\Omega}$$ is close to the third axis, it will stay close; while if $$\vec{\Omega}$$ is close to the principal plane, it will not stay put, but precess in the principal plane.

• Case $$I_1>I_2=I_3$$: Then $$\Omega_1$$ and $$L_1$$ are constants, cf. eq. (3). Then $$\ddot{\Omega}_i~\stackrel{(2)}{=}~-\omega^2_1 \Omega_i, \qquad i\in{2,3}, \tag{9}$$ where $$\omega_1~:=~\Omega_1\sqrt{\frac{(I_1-I_2)(I_1-I_3)}{I_2I_3}}~=~{\rm const}. \tag{10}$$ Conclusion: There is a (slow) precession of $$\vec{\Omega}$$ and $$\vec{L}$$ around the first axis with angular frequency $$\omega_1$$. In other words: If $$\vec{\Omega}$$ is close to the first axis, it will stay close; while if $$\vec{\Omega}$$ is close to the principal plane, it will not stay put, but precess in the principal plane.

• Case $$I_1=I_2=I_3$$: $$\vec{\Omega}$$ and $$\vec{L}$$ are constants, cf. eq. (3).

Interestingly, the degenerate cases can be solved exactly with closed formulas.

[Above we have implicitly assumed that $$\omega_i$$ in eqs. (6), (8), and (10) are never exact zero, but strictly positive. In practice, this is true.]