Angular speed

• Speed of \(p_i\;\): \(\;\;p'_i(t)=v(t)+\mathrm{R}'(t)\,r_0\)

• Introduce angular speed \(\omega\in\mathbb{R}^3\) such that
• • \(p'_i(t)=v(t) + \omega(t) \times r(t)\)
  • \(\omega\) \(\simeq\) vector expressing the instantaneous rotation of \(r(t)\)

• By identification \(\mathrm{R}'(t)\,r_0 = \omega(t)\times r(t)\)
• \(\Rightarrow \mathrm{R}'(t)\,r_0 = \omega(t)\times (\mathrm{R}(t)\,r_0)\)

• Matrix expression of \(\omega=(\omega_x,\omega_y,\omega_z)\)
• • \(\hat \omega=\left(\begin{array}{c}0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \end{array}\right)\)

• \(\Rightarrow \mathrm{R}'(t) = \hat\omega(t) \, \mathrm{R}(t)\)