Identity channel

Template:Expert-subject 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.

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:

${\begin{aligned}I_{1}{\dot {\omega }}_{1}&=(I_{2}-I_{3})\omega _{2}\omega _{3}~~~~~~~~~~~~~~~~~~~~{\text{(1)}}\\I_{2}{\dot {\omega }}_{2}&=(I_{3}-I_{1})\omega _{3}\omega _{1}~~~~~~~~~~~~~~~~~~~~{\text{(2)}}\\I_{3}{\dot {\omega }}_{3}&=(I_{1}-I_{2})\omega _{1}\omega _{2}~~~~~~~~~~~~~~~~~~~~{\text{(3)}}\end{aligned}}$

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),

${\begin{aligned}I_{2}I_{3}{\ddot {\omega }}_{2}&=(I_{3}-I_{1})(I_{1}-I_{2})\omega _{1}\omega _{2}\\{\text{i.e.}}~~~~{\ddot {\omega }}_{2}&={\text{(negative quantity)}}\times \omega _{2}\end{aligned}}$

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),

${\begin{aligned}I_{1}I_{3}{\ddot {\omega }}_{1}&=(I_{2}-I_{3})(I_{1}-I_{2})\omega _{1}\omega _{2}\\{\text{i.e.}}~~~~{\ddot {\omega }}_{1}&={\text{(positive quantity)}}\times \omega _{1}\end{aligned}}$

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

• Euler angles
• Moment of inertia
• Poinsot's ellipsoid
• Polhode

References

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• Mark S. Ashbaugh, Carmen C. Chicone and Richard H. Cushman, The Twisting Tennis Racket, Journal of Dynamics and Differential Equations, Volume 3, Number 1, 67-85 (1991).
• Dzhanibekov effect video demonstrated on the International Space Station