What Are the Correct Euler's Angles for Rotating a Cube?

[Cosmossos]

Homework Statement

What are the Euler's angles corresponding to the rotations of a cube in \theta radians around each of its principal axes



around the x: \theta=\theta
\phi=\psi=0

around z:\psi=\theta=0
\phi=\theta

around y:\phi=\theta=\theta
\psi=-\theta


is it correct? how can I make it more clear? It's very confusing...

 

[Cosmossos]

Help me please...

 

[Maxim Zh]

In order to formalize the solution you can use the rotation matrices: \hat{R}_x(\alpha), \hat{R}_y(\alpha), \hat{R}_z(\alpha) and the matrix which describes the whole Euler's transform:

<br /> \hat{R}(\theta, \phi, \psi) =<br /> \hat{R}_z(\phi) \hat{R}_x(\theta) \hat{R}_z(\psi) \quad (1)<br />

It's easy to get rotation around the x and z axes from (1) and your answers for these cases are right.

As for y-axis the condition

<br /> \hat{R}(\theta, \phi, \psi) = \hat{R}_y(\alpha)<br />

yields

<br /> \theta = -\frac{\pi}{2} - \alpha;<br />

<br /> \phi = \psi = -\frac{\pi}{2}.<br />