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 , and be the principal moments of inertia of the book (the moments of inertia around it’s 3 principal axes), and , and 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

and its rotational energy T is given by

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.

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