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If a asymmetric body with three different principal moments of inertia, (\( I_1> I_2> I_3\)), is given initial spin around the first, or the third axis,its motion is stable. But if the body is given a nonzero initial angular velocity around the second axis, with slight disturbance the body acquires angular velocity around the other axes too and moves in an unstable manner, very different from the initial motion of the body. This is known as "Tennis Racket Theorem". We will give a simplified discussion of the Tennis Racket theorem. We start with the Euler's equations.
Initial spin along the first axis
We shall analyze the motion of the racket assuming that the racket is given an initial spin about the first axis so that \(\omega_1\ne 0\). Also that it acquires small nonzero angular velocities about the other two axis due to some disturbance, \(\omega_2\approx 0, \omega_3\approx 0\). % \begin{eqnarray}{EQ01} I_1 \dot{\omega}_1 + (I_3-I_1)\omega_2\omega_3 =0\\ I_2 \dot{\omega}_2 + (I_1-I_3)\omega_3\omega_1 =0\\{EQ02} I_3 \dot{\omega}_3 + (I_2-I_1)\omega_1\omega_2 =0.{EQ03} \end{eqnarray} Under the assumption that \(\omega_2, \omega_3\) are small, \EqRef{EQ01} implies that \(\omega_1\) remains constant. Differentiating \eqRef{EQ02} w.r.t. time, and neglecting time derivative of \(\omega_1\), we get \begin{equation} I_3 \DD[\omega_3]{t} + (I_2-I_1)\dd[\omega_2]{t} \omega_1 =0\\{EQ04}. \end{equation} Substituting \(\dot{\omega}_2= \frac{(I_1-I_3)\omega_3\omega_1}{I_2}\) from \eqRef{EQ02} leads to \begin{equation} I_3\DD[\omega_3]{t} + \frac{(I_2-I_1)(I_3-I_1)}{I_1} \omega_1^2 \omega_3 =0. \end{equation} The solution of this equation is of the form \begin{equation} \omega_3 = A \sin\nu t +B \cos\nu t \end{equation} where \(\nu=\sqrt{\frac{(I_1-I_3)(I_2-I_3)}{I_1I_3}}\omega_1\) and \(A,B\) are some constants. Thus \(\omega_3\) and, also \(\omega_2\), oscillate with frequency \(\nu\).
% Similar conclusions hold for the case of initial conditions \(\omega_3\ne0,\, \omega_1\approx0,\, \omega_2\approx0\). In the two cases of initial spin around the first, or the third axis, the angular velocity around other axes remains small. This can be demonstrated by doing a stability analysis.
Initial spin along the second axis
If the racket is spun around the second axis, \(\omega_2\ne0\), the situation is different. Proceeding in a manner similar to the previous case, but now assuming, \(\omega_1, \omega_3\) remain small, the equation for \(\omega_3\) takes the form \begin{equation} \DD[\omega_3]{t} + {\nu{'}}^2\omega_3 =0, \qquad \text{where } \nu{'} =\frac{(I_2-I_1)(I_3-I_2)}{I_2I_3} \omega_1^2 \omega_2 \end{equation} This frequency \(\nu{'}\) becomes imaginary, this signals that and the other components of angular velocities will not remain small at later times.
% For more details see Harald Ira \(\S11.3\) \cite{Ira}.
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4727:Diamond Point
