[NOTES/CM-09004] Kinetic Energy of Rigid Body

The kinetic energy Of a rigid body can be expressed in terms of moment of inertia tensor. Let $\vec x_\alpha$ denote position of a point $\alpha$ on the body a seen in the space fixed frame.The kinetic energy for a system of point particles is given by\begin{equation}\text{KE}=\sum_{\alpha}m_{\alpha_\alpha}\dot{\vec{x}}_\alpha^2\end{equation}The expression of the kinetic energy for a rigid body with continuous mass distribution becomes the volume integral\begin{equation}\text{KE}=\int d^3x \rho(x)\dot{\vec{x}}\cdot\dot{\vec{x}},\end{equation}where \(\rho\) is the density of the body. The velocity \(\vec x\) can be expressed in terms of the position vector \(\vec X\) w.r.t the body frame.\begin{equation}\dot{\vec{x}}=\vec{\omega}\times\vec{X}\end{equation}Therefore, the kinetic energy is given by\begin{eqnarray} \text{KE} &=& \int dv \rho(X)(\vec{\omega}\times\vec{X})(\vec{\omega}\times\vec{X})\\ &=&\int dv \rho(X)\vec{\omega}\cdot\vec{X}(\vec{\omega}\times\vec{X})\\ &=& \int dv \rho(X)\vec{\omega}(\vec{x}^2\vec{\omega}-(\vec{\omega}\cdot\vec{X}\vec{X})\\ &=&\sum_{ij}\omega_i I^b_{ij} \bar{\omega_j}. \end{eqnarray}where\begin{equation}I^b_{ij}=\int dv \rho(X)(X^{2}\delta_{ij}-X_i X_j)\end{equation}is the moment of inertia tensor w.r.t. the body fixed axes. We therefore get\begin{equation}\text{KE}=\sum_{ij} \omega_{bi} I_{ij}^{(b)} \omega_{ij}\end{equation}where $b$ means the components are taken with reference to the body axes.

Comments
For many applications, using of energy conservation, it turns out to be sufficient to have the expression \eqRef{EQ10} w.r.t. a set of preferred axes, for example principle axes.For several applications we need to start with the Lagrangian equations of motion. So far we have not chosen any particular set of generalized coordinates. In order to set up the Lagrangian formalism it becomes necessary to write the angular velocities in terms of generalized coordinates and generalized velocities. To achieve this one must express the angular velocities in terms of three generalized coordinates, for example Euler angles, and corresponding generalized velocities.