# 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.

#### Linearization

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.

#### Interpretation

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