Tennis racket theorem

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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[edit]

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'.

See also[edit]

References[edit]

  1. ^ [1]