\[\newcommand{\u}[1]{\boldsymbol{\mathsf{#1}}} \renewcommand{\b}[1]{\boldsymbol{#1}} \newcommand{\t}[1]{\textsf{#1}} \newcommand{\m}[1]{\mathbb{#1}} \def\RR{\bf R} \def\bold#1{\bf #1} \def\mbf#1{\mathbf #1} \def\uv#1{\hat{\usf {#1}}} \def\dl#1{\underline{\underline{#1}}} \newcommand{\usf}[1]{\boldsymbol{\mathsf{#1}}} \def\bs#1{\usf #1}\] \[\renewcommand{\u}[1]{\boldsymbol{#1}} \newcommand{\usf}[1]{\mathsf{\u{\mathsf{#1}}}} \newcommand{\pr}[1]{\left( #1 \right)}\]

Angular velocities: Axial Vectors

Corresponding to the skew symmetric linear operator $\usf{W}(\tau)\in \mathcal{M}_{3,3}(\mathbb{R})$ there exists a vector $\usf{w}(\tau)=(w_i(\tau))\in \mathbb{R}^3$ where

\[\begin{equation} w_{i}(\tau):=-\frac{1}{2}\epsilon_{ijk}W_{jk}(\tau) \end{equation}\]

It has the property that for all vectors $\usf{x}\in \mathbb{R}^3$

\[\begin{align} \usf{W}(\tau)\usf{x}=\usf{w}(\tau)\times \usf{x} \end{align}\]

Proof:

\[\begin{align} \pr{\usf{w}(\tau)\times \usf{x}}_i&=\epsilon_{ijk}w_{j}(\tau)x_k\\ &=\epsilon_{ijk} \left( -\frac{1}{2} \epsilon_{jpq}W_{pq}(\tau) \right) x_k\\ &=-\frac{1}{2}\epsilon_{ijk}\epsilon_{jpq}W_{pq}(\tau)x_k\\ &=\frac{1}{2}\epsilon_{jik}\epsilon_{jpq}W_{pq}(\tau)x_k\\ &= \frac{1}{2} \left( \delta_{ip}\delta_{kq}-\delta_{iq}\delta_{kp} \right) W_{pq}(\tau)x_k\\ &= \frac{1}{2} \left( \delta_{ip}\delta_{kq}W_{pq}(\tau)x_k-\delta_{iq}\delta_{kp}W_{pq}(\tau)x_k \right) \\ &= \frac{1}{2} \left( W_{ik}(\tau)x_k-W_{ki}(\tau)x_k \right) \\ &= \frac{1}{2} \left( W_{ik}(\tau)x_k+W_{ik}(\tau)x_k \right) \\ &= W_{ik}(\tau)x_k \end{align}\]

The vector $\usf{w}(\tau)$ is called the linear operator $\usf{W}(\tau)$’s axial vector. Specifically, if $\usf{W}(\tau)=\usf{R}’(\tau)\usf{R}^{\sf T}(\tau)$ then $\usf{w}(\tau)$ is called the angular velocity vector at the time instance $\tau$.

The magnitude of $\usf{w}(\tau)$, i.e., $\lVert\usf{w}(\tau) \rVert_{2}=(w_{i}(\tau)w_{i}(\tau))^{1/2}$ is called the angular velocity. It’s units are $\rm T^{-1}$.

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