The Spinning Book – State Space Approach


One really cool and tangible application of the dynamical systems theory we have been learning is developing an understanding of the rotational dynamics of a spinning object. It’s not something that can be explained in just a few minutes. This is going to take a few steps. Each step will provide a little more illumination.

A Glimpse of the Phenomenon

Usually, when I do this experiment, I do it here on earth, where gravity limits you to just a second or so of observation at a time. Here is a really interesting implementation of the phenomenon on the International Space Station.

Finding Equilibria

Here, “equilibria” refer to equilibria of Euler’s rotational dynamics equations. Therefore, an “equilibrium” corresponts to a state of constant rotation.


Next, we linearize the dynamics in the vicinity of the equilibria to investigate stability.

 A Nonlinear Look at the BIG Picture

Because of a lack of hyperbolicity, the linearization has limited value.  Next, I’m going to show you how to apply a nonlinear approach that will give you a beautiful view of the BIG picture. Here, we leverage the fact this particular problem has two conserved quantities.


One more video is coming. It will provide an interpretive framework for all the geometry we just saw.