Undergrad Understanding 3D Coordinate Rotations with Euler Angles

[QuasarBoy543298]

I'm trying to wrap my head around the concept. we use 3 rotations to transfer our regular cartesian coordinates (3 x,y,z unit vectors) to other 3 unit vectors. each rotation is associated with an angle. so far I'm good.
but now I saw in Landau's and Lifshitz's "mechanics" book this thing (picture attached).
I couldn't understand where omega is and what we are trying to do.
Is omega the rotation of the new system in the old system, or just some rotation vector who is constant in the new system that we are trying to express in the old system ? (so far that makes more sense since we got a result who is dependent only on Euler angles and their time derivatives)

 

[FactChecker]

The angular rates are not the same as the rotated Euler angles. In the rotated Euler angles, the second and third rotations are in the intermediate axis systems that are the result of the prior rotations. That is not true of the angular rotation rates. They are all instantaneous rates measured in the original axis system (because all the theoretical rotations are infinitesimal and no coordinate change has actually happened).

 

[vanhees71]

You have the rotation matrix $\hat{D}$ as a function of the Euler Angles, which themselves are functions of time. My convention here is that
$$\vec{e}_k'=\vec{e}_j D_{jk},$$
where $\vec{e}_j$ are the three Cartesian basis vectors fixed in an inertial frame of reference ("lab frame") and $\vec{e}_k'$ are the three Cartesian basis vectors of the body-fixed reference frame.

Then the components of the angular velocity with respect to the body-fixed frame is given by the antisymmetric matrix $\hat{\Omega}'=\hat{D}^{\text{T}} \dot{\hat{D}}$. The matrix elements are related to $\vec{\omega}'$ by $\Omega_{jk}'=-\epsilon_{jkl} \omega_l'$. Since $\vec{\omega}$ is a vector its components wrt. the lab frame are given by $\vec{\omega}=\hat{D} \vec{\omega}'$.

To evaluate $\vec{\omega}'$ (or $\vec{\omega}$, depending on which one you like to know), I recommend the use of a computer algebra system like Mathematica since it's really a cumbersome calculation doing just boring derivatives and matrix multiplications. I did it once in the theory lecture on mechanics, and then I disliked the rigid-body theory although it's a fascinating topic ;-)).