Rigid body kinematics

Similarily to particles

- Position of the com: \(p(t)\)
- Velocity of the com: \(v(t)=p'(t)\)
- Linear Momentum:
  \(P(t) = m\,v(t)\)
  \(\displaystyle \;\;\left(= \int_{\Omega} \rho\,v_i(t)\;\mathrm{d}\Omega) \right)\)

Specific to rigid body

- Orientation of the body: \(\mathrm{R}(t)\in\mathbb{R}^{3\times 3}\)
- Angular velocity of the body: \(\omega(t)\) such that \(\hat \omega = \mathrm{R}'(t) \, \mathrm{R}^T(t)\)
- Angular Momentum:
  \(L(t)=\mathrm{I}(t)\,\omega(t)\)

- - with \(\mathrm{I}(t)\): Inertia tensor
  - - \(\displaystyle \mathrm{I}(t)=\mathrm{R}(t)\,\mathrm{I}_0\,\mathrm{R}^T(t) \)
    - \(\displaystyle \mathrm{I}_0=\int_{r\in\Omega} \rho(r)\,(r^T\,r\,\mathrm{I}_d-r\,r^T)\,\mathrm{d}\Omega \)

Mass: resistance to change of speed (of the com)
Inertia: resistance to change of angular speed