I'm using Goldstein's Classical Mechanics 3rd Ed. In Section 4.9 - Rate of Change of a Vector he derives the components of the angular velocity vector for the rate of vector change transformation between body axes to space axes.
He does this by using the angular velocity individually for each rotational axis $Z''-X'-Z$ (where $Z$ is in the body axes, $X'$ is in the intermediate axes frame and $Z"$ is in the space axes), projecting each of these using the Euler angle rotation matrices & then adding all these up.
These are the components for the angular velocity in body axes that he gets by doing so:
$$ \omega_{x'} = \dot{\phi}\sin\theta \sin\psi + \dot{\theta}\cos\psi $$ $$ \omega_{y'} = \dot{\phi}\sin\theta \cos\psi - \dot{\theta}\sin\psi $$ $$ \omega_{z'} = \dot{\phi}\cos\theta + \dot{\psi}. $$
Now, I'm not fully comfortable with this technique as it seems to go further off from the way he had originally derived angular velocity vector in the same section:
- He did so by starting off with a transformation matrix (projecting from the body axes to the space axes) & then applying differentiation to get an infinitesimal rotation matrix (that was shown to be skew-symmetric & which is shown to be equivalent to a vector with the cross product operation (this vector being the angular velocity)).
- In the preceding section (section 4.8, Infinitesimal Rotations) he had provided very different components for the infinitesimal Euler rotational matrix & hence the differential vector as $(d\theta, d\phi + d\psi,0).$
- In the current section (4.9) he then provides an explanation on why we cannot naively assume the rotational velocity to be $(\dot{\theta},\dot{\phi},\dot{\psi})$ given each rotation component occurs over a different set of axes (which is somewhat convincing but only raises mysteries about the expression in the preceding section).
Now I have three questions:
Why cannot we derive the angular velocity components by simply differentiating the Euler transformation matrix (in the Z"-X'-Z convention) & then assuming the $(\theta, \phi,\psi)$ angles to be 0 at that instantaneous time? It would give the same expression as what Goldstein had provided in Section 4.8 -- $(d\theta, d\phi + d\psi,0)$. Is there any further rigorous explanation apart from what he provided? If yes, is the preceding section wrong?
If the angular velocity cannot be derived as I described, is there any way I can get the angular velocity components by purely deriving a skew-symmetric matrix using the Euler rotation matrices? I don't want to rely on the technique of assuming angular velocity & then using projections as Goldstein did given it does not feel rigorous enough because of not working directly with the transformation matrices.
On a side-note question, according to this derivation, what does it even mean to say the angular vector lies in "body or space axes"? Can a transformation matrix be associated in that way?
