Tennis Racket Theorem

This post is about the perplexities of rotating bodies and a famous theorem related to the rotation of rigid bodies called the Tennis Racket Theorem.

The Tennis Racket Theorem states that an object with three unique moments of inertia, rotation about the axis of intermediate moment of inertia is unstable, while rotation about the other two axis is stable.

At first glance, this theorem may seem complicated, and to those unfamiliar with rigid body dynamics, maybe even daunting. However, as with anything complicated, breaking down into smaller, more fundamental parts seems to elucidate what the theorem actually means. The next few parts of this article with discuss the various aspects of the theorem, and finally we will tie together everything, along with a demonstration.

Moment of Inertia

Moment of inertia can be interpreted as the rotational motion counterpart to mass. Let me explain: in linear motion, the motion is described by Newton’s Second Law

However, in rotational motion, the acceleration is replaced by angular acceleration and force is replaced by moment (or torque for the Americans). The replacement for mass is moment of inertia. So now Newton’s second Law is written as

Moment of inertia is written as I and for three dimensional objects, moment of inertia is usually an moment of inertia matrix.

Those familiar with matrices might wonder, what would it mean to diagonalise the moment of inertia matrix. If you diagonalise the moment of inertia matrix, you will get a matrix where the new coordinate system represents the principal axis of the object and the diagonal terms are the moment of inertia is every one of those axis. There are two ways to find the diagonalised moment of inertia matrix. The easy way is to observe the geometry of the object, as sometimes (in most common shapes) the principal axes are obvious. However, to be rigorous in the derivation, you can use a standard matrix diagonalisation from elementary linear algebra. The procedure will be as follows

Find the eigenvalues of inertia matrix.

Find eigenvector matrix P of inertia matrix.

1. Use P-1IP to get diagonalised inertia matrix.

The values in the diagonal matrix is going to allow us to understand the Tennis Racket Theorem. For those of you are sharp, you will notice that the axis of intermediate moment of inertia is the eigenvector of the intermediate eigenvalue.

Euler Equations

Newton’s second law for rotational motion gets very complicated to work with very fast. So Euler used the diagonalisation as a way to simplify and separate the three equations in Newton’s second law. The Euler equations are as follows

Breaking this component wise, in the principal axes

I will not go in depth into the derivation, but I will outline the chain of thought. Consider this alternate version of Newton’s law

Expanding the second equation will give you Euler’s equation of motion for rigid bodies.

Understanding the theorem

Now that we have our prerequisites, we can go on to understand the theorem. Consider an inertia matrix (diagonalised) with moment of inertia I1 and I2 and I3 such that I1 is the smallest and I3 is the biggest. Now consider motion about the axis of major moment of inertia, I3. Let the angular velocity vector be

where the epsilons are small perturbations in the other two principal axes.

Now plugging this into Euler equations, we obtain

Now we differentiate the second Euler equation

Substituting in our expression for omega 1 and omega 3, and since multiplying the epsilons makes it small enough to ignore,

This gives us a differential equation for omega 2 of the form

Whose solution is elementary

Therefore, we know that perturbation of rotation in the omega 1 axis is stable and undergoes periodic motion, or in the terminology of rigid body motion, it undergoes precession.

The perturbation in omega 3 follows a similar argument as above, and I shall leave it to you as an exercise to work it through.

For the intermediate axis, we have

Plugging into the Euler equations

Differentiating the third Euler equation

Substituting our derived expresssions

Now, when we rearrange, we derive the following differential equation

Notice that the coefficient is now positive, which therefore results in exponential solutions

This solution shows that omega 3 is unstable under perturbation of omega 2 along the intermediate axis.

Conclusion

Now, we can tie everything that we have derived and learnt to understand the theorem. Simply put, when the rotation along the intermediate axis is perturbed, it results in a differential equation that has exponential solutions. This results in unstable motion, contrary to the precessive motion observed in the other two axis.

This result is a rather surprising one. There is no intuitive backing to such a theorem as we cannot think of why the intermediate moment of inertia would result in unstable rotation. It seems as though it is purely mathematical in nature.

However, if we try to force some form of intuition into the reasoning of this theorem, we arrive at the following vague idea:

When there is a perturbation in the rotation along the major or minor axis, the system acts to decrease the perturbation, along with the perturbation in the two other axes forming a sine and cosine pair, negating each other. This causes precession. However, when there is a perturbation in the rotation along the intermediate axis, the system acts to increase the perturbation. In other words, the perturbation in the other two axes reinforce each other, and increase exponentially, creating unstable rotation.

If you want to see this theorem in action, pick up your phone, make sure you have a sturdy phone cover and start flipping your phone on all three of its axes. You will notice that in one of the axes, the rotation will be chaotic, and now you will know the reason why!