Why can I interpret this as angular velocity?

Question

I've been trying to derive the usual mathematics behind relative motion in kinematics, and specifically proving the transport theorem: $$\frac{d\mathbf p}{dt}_{/A}(t_0) = \frac{d\mathbf p}{dt}_{/B}(t_0) + \mathsf \Omega_{B/A}(t_0)\mathbf p(t_0) $$ where $t_0$ is some instant in time, $A$ is the fixed reference frame, $B$ is the free reference frame, $\mathbf p$ is the map describing the position of a vector in $\mathbb R^3$ as a function of time, and $\mathsf \Omega_{B/A}$ is a matrix that encapsulates how $B$ moves with respect to $A$. It is evident that such matrix is of the form $$\mathsf \Omega_{B/A}(t_0) = \begin{bmatrix} 0 & \omega_3(t_0) & -\omega_2(t_0) \\ -\omega_3(t_0) & 0 & \omega_1(t_0) \\ \omega_2(t_0) & -\omega_1(t_0) & 0\ \end{bmatrix} $$ Now, I see everywhere that the three scalars contained in the matrix can be arranged in a vector $$\mathbf \omega(t_0) = \begin{bmatrix} \omega_1(t_0) \\ \omega_2(t_0) \\ \omega_3(t_0)\end{bmatrix}$$ that is interpreted as the angular velocity vector at time $t_0$ of the free reference frame with respect to the fixed frame. However, I can't see how this denomination could be reconciled with the definition of angular velocity as the time derivative of angular displacement. If the naming were justified, I'd expect there to be some matrix $\mathsf \Theta_{B/A}(t_0)$ that holds all information about "spatial angular displacement", whatever that is, such that $$\mathsf \Omega_{B/A}(t_0) = \frac{d}{dt}_{/A} \mathsf \Theta_{B/A}(t_0)$$ but then it isn't really clear to me how that connects with time-differentiating the unit vectors of frame $B$ within $A$. I'm guessing the answer should be hiding behind what are the entries of $\mathsf \Theta_{B/A}(t_0)$ with respect to the unit vectors of $B$?

Answer 1

However, I can't see how this denomination could be reconciled with the definition of (scalar) angular velocity as the time derivative of angular displacement.

Angular velocity was never a scalar quantity to begin with. However, in introductory Newtonian mechanics, one often works with the sizes (norms) of what are really vectors; just think of the endless times you've done this (if you've taken such a course) with forces decomposed in their respective components along the $x,y,z$-axes. This could be the source of your confusion. Let me know if not.

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EDIT: I see what you're going for now. I'd recommend you look at the derivation found in the second chapter of Theoretical Mechanics of Particles and Continua by Fetter & Walecka - the whole derivation is too long to post here, but their coverage is excellent. The idea is that you look at the time derivative of the unit vectors of the rotating coordinate system. If $\hat e_1,\hat e_2,\hat e_3$ are the unit vectors of the rotating coordinate system, you can show that $\frac{d\hat e_i}{dt}=\frac{d\Omega}{dt}\times\hat e_i\equiv\omega \times \hat e_i$. The cross-product is the reason for the interpretation of angular velocity.