How do you connect angular velocity vector with the positions?

[rsq_a]

I've been working out a question about rigid body rotations, and though I can grasp the basic math, I'm completely lost about the physical quantities. In particular, I'm trying to relate the three components of the angular velocity vector that I found with the physical picture:

Suppose that you're studying the rotation of a rectangular book as it is thrown in the air. Assume that the center of mass of the book is fixed at the origin, so we are only interested in how the book is rotating.

You then define vectors $\mathbf{e}_1(t), \mathbf{e}_2(t), \mathbf{e}_3(t)$, which are functions of time, and which are pointed in the direction of the principle axes of the book (basically the length, width, and height). I define the angular velocity vector, $\mathbf{w}(t) = w_1\mathbf{e}_1 + w_2 \mathbf{e}_2 + w_3 \mathbf{e}_3$, which satisfies

$$\frac{d\mathbf{r}}{dt} = \mathbf{w} \times \mathbf{r}$$

where r(t) is a position vector defined with respect to the moving frame,

$$\mathbf{r}(t) = x_1\mathbf{e}_1 + x_2 \mathbf{e}_2 + x_3 \mathbf{e}_3$$

Suppose finally that I were to give you the exact values of the three components $w_1(t), w_2(t), w_3(t)$ that make up the angular velocity vector. Can you explain to me how we plot the principle axis vectors. That is, how do I figure out what $e_1(t), e_2(t), e_3(t)$ are in terms of the usual coordinates of (x, y, z)?

 

[LightHero]

The answer to this question lies in the Euler angles (or the Cardan angles). The Euler angles are a set of three angles (\phi, \theta, \psi) which describe the orientation of the book relative to a fixed coordinate system. The angular velocity vector can be expressed in terms of these angles as follows:\mathbf{w}(t) = \frac{d\phi}{dt}\hat{\phi} + \frac{d\theta}{dt}\hat{\theta} + \frac{d\psi}{dt}\hat{\psi}where \hat{\phi}, \hat{\theta}, and \hat{\psi} are unit vectors along the principle axes of the book. By taking derivatives of the Euler angles, we can then find the components of the angular velocity vector:w_1(t) = \frac{d\phi}{dt} \\w_2(t) = \frac{d\theta}{dt} \\w_3(t) = \frac{d\psi}{dt} \\Now, if we know the exact values of w_1(t), w_2(t), and w_3(t), then we can solve for the Euler angles, and thus determine the orientation of the book with respect to the fixed coordinate system. From there we can calculate the principle axis vectors \mathbf{e}_1(t), \mathbf{e}_2(t), \mathbf{e}_3(t).