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Tie a book closed with a rubber band and toss it into the air. It wobbles around in some manner, but we know two things about it: it conserves angular momentum, and it conserves rotational kinetic energy.

Let I_{1}, I_{2} and I_{3} be the principal moments of inertia of the book (the moments of inertia around it’s 3 principal axes), and \omega_{1}, \omega_{2} and \omega_{3} be the angular velocities about these axes. (If you’re unfamiliar with moments of inertia, it’s basically jut a measure of resistance to turning. A small I corresponds to something that’s light and compact, while a big I corresponds to something that’s heavy or spread out.)

Then the square of its angular momentum L is given by

L^2 = (I_{1} \omega_{1})^2 +  (I_{2} \omega_{2})^2 +  (I_{3} \omega_{3})^2

and its rotational energy T is given by

T = \frac{1}{2} (I_{1} \omega_{1}^2 + I_{2} \omega_{2}^2 + I_{3} \omega_{3}^2)

Both these quantities are conserved throughout the motion.

Poinsot’s Construction

Now if we want to understand the exact motion of the body, we’d need to bust out the Euler equations of motion, a set of differential equations that govern the angular velocities. But as long as we’re just look for a qualitative idea, there’s a nice geometric method due to Poinsot, that’s good for building insight about rotations.

The idea is that the above two equations both describe ellipsoids in phase space (\omega_{1}, \omega_{2}, \omega_{3} space). In this space the object’s trajectory is constrained to these ellipsoids, and the only trajectory that satisfies both equations is the intersection between the two ellipsoids.

Let I_{1} < I_{2} < I_{3}, as is the case for a book or a cell phone.

Then if we spin the book around the 1 axis (the one with the minimum moment of inertia), the two  ellipsoids look like this (same figure shown with 2 different opacities):

If we spin it around the 3 axis (the one with the maximum moment of inertia), the ellipsoids look like this:

The red ellipsoid is the surface of constant angular momentum, the green one is the surface of constant energy. The allowed trajectory occurs at the intersection of red and green, so we can see that the object keeps rotating around the axis we originally spin it in. The more accurate your original spin, the more this trajectory shrinks to a point.

However, if we spin the book about the 2 axis (with the intermediate moment of inertia), its constrained to the following trajectory (the red-green line):

Even if the particle is initially spun perfectly along the 2 axis, the trajectory is unstable and traces out a path called a polhode. From the figure below you can see these trajectories are closed and so the particle repeats the motion.

Poinsot’s trick: Possible trajectories in phase space
Figure taken from David Tong’s lecture notes

Try it! You’ll find that when you toss something into the air about the axis with the middle moment of inertia, it’ll wobble in an unstable pattern. Spinning about the other two axes should give smooth, stable rotational motion.

What if you throw a bag of peas into the air? The peas rumble around inside the bag, causing friction and dissipating energy. The energy decreases and so the green energy ellipsoid shrinks in size. The trajectory goes from something like

to to

The bag starts to wobble but eventually the wobbles dies out and the bags ends up spinning about the axis with the largest moment of inertia. Poinsot’s trick gives us a lot of insight for very little work!

Instability of a rigid body spinning freely in space (includes movies): Link
David Tong’s excellent classical mechanics lecture notes, sec. 3 on rigid bodies: Link


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