[NOTES/CM-09009] General comments on Motion of a Rigid Body

Introduction
The kinetic energy of a rigid body can be decomposed into two parts. One of the parts corresponds to translational motion of the centre of mass. The second part corresponds to rotational motion about the centre of mass of the body. The equations of motion in an inertial frame take the form
\begin{equation}
\dd[{\mathbf P}]{t}= {\mathbf F}, \qquad \dd[{\mathbf L}]{t} = \pmb{\tau}.
\end{equation}

In order to set up the Lagrangian, we need to choose a set of six generalized coordinates. Three generalized coordinates corresponding to the translational motion are taken to be the position \({\mathbf X}_{\text{cm}} \)of the center of mass of the body.

For rotational motion, we choose a set of coordinate axes \(K\) fixed in the body. In case the interest lies in motion of a body with one of its points fixed, the fixed point must be chosen as the origin of the body fixed axes.

In all cases the body fixed axes are chosen to coincide with the principal axes. This leads to a lot of simplification. The Euler angles \(\psi,\theta, \phi\) of the body axes relative to a space fixed axes with origin at centre of mass may be chosen to give the three generalized coordinates corresponding to the rotational motion. If \(\vec{\omega}=(\omega_1,\omega_12\omega_3)\) denote the components of the angular velocity relative to the body axes, the rotational kinetic energy is given by
\begin{equation}
T = \frac{1}{2} I_1 \omega_1^2 + \frac{1}{2} I_3 \omega_2^2 +\frac{1}{2} I_3 \omega_3^2
\end{equation}
Here \(I_1, I_2,I_3\) are the principal moments of inertia and the angular velocities are given by
\begin{eqnarray} \omega_1&=&\dot{\theta} \cos \psi +\dot{\phi} \sin \theta \sin \psi \\ \omega_2&=& \dot{\phi} \sin \theta \cos \psi -\dot{\theta} \sin \psi \\ \omega_3&=&\dot{ \phi} \cos \theta +\dot{\psi} \end{eqnarray}