Tennis racket theorem

The tennis racket theorem is a result in classical mechanics describing movement of a rigid body with three distinct angular momenta. It also dubbed Dzhanibekov effect named after Russian astronaut Vladimir Dzhanibekov who discovered the theorem's consequences while in space in 1985.[1]

Qualitative Proof

The tennis racket theorem can be qualitatively analysed with the help of Euler's equations.

Under torque free conditions, they take the following form:

\begin{align} I_1\dot{\omega}_{1}&=(I_2-I_3)\omega_2\omega_3~~~~~~~~~~~~~~~~~~~~\text{(1)}\\ I_2\dot{\omega}_{2}&=(I_3-I_1)\omega_3\omega_1~~~~~~~~~~~~~~~~~~~~\text{(2)}\\ I_3\dot{\omega}_{3}&=(I_1-I_2)\omega_1\omega_2~~~~~~~~~~~~~~~~~~~~\text{(3)} \end{align}

Let $I_1 > I_2 > I_3$

Consider the situation when the object is rotating about axis with moment of inertia $I_1$. To determine the nature of equilibrium, assume small initial angular velocities along the other two axes. As a result, according to equation (1), $~\dot{\omega}_{1}$ is very small. Therefore the time dependence of $~\omega_1$ may be neglected.

Now, differentiating equation (2) and substituting $\dot{\omega}_3$ from equation (3),

\begin{align} I_2 I_3 \ddot{\omega}_{2}&= (I_3-I_1) (I_1-I_2) \omega_1\omega_{2}\\ \text{i.e.}~~~~ \ddot{\omega}_2 &= \text{(negative quantity)} \times \omega_2 \end{align}

Note that $\omega_2$ is being opposed and so rotation around this axis is stable for the object.

Similar reasoning also gives that rotation around axis with moment of inertia $I_3$ is also stable.

Now apply the same thing to axis with moment of inertia $I_2$. This time $\dot{\omega}_{2}$ is very small. Therefore the time dependence of $~\omega_2$ may be neglected.

Now, differentiating equation (1) and substituting $\dot{\omega}_3$ from equation (3),

\begin{align} I_1 I_3 \ddot{\omega}_{1}&= (I_2-I_3) (I_1-I_2) \omega_1\omega_{2}\\ \text{i.e.}~~~~ \ddot{\omega}_1 &= \text{(positive quantity)} \times \omega_1 \end{align}

Note that $\omega_1$ is not opposed and so rotation around this axis is unstable. Therefore even a small disturbance along other axes causes the object to 'flip'.