The Inertia Tensor

The above calculation of the kinetic energy defines the inertia tensor.

$$T_{rot} = {1\over 2} I_{ij} \omega_i\omega_j$$

$$I_{ij} \equiv \sum\limits_\alpha m_\alpha\left[\delta_{ij}r_\alpha^2-r_{\alpha i} r_{\alpha j}\right]$$

Note that $I_{ij}$ is a symmetric tensor (under interchange of the two indices). We can also write the inertia tensor in matrix form.

$$\mathbb{I}=\sum\limits_\alpha m_\alpha \begin... ...lpha x_\alpha & -z_\alpha y_\alpha & r_\alpha^2-z_\alpha^2\end{pmatrix}$$

For a continuous mass distribution, we may use an integral rather than a sum over masses.

$$I_{ij}=\int\limits_V\rho(\vec{r})\left[\delta_{ij}r^2-r_ir_j\right] dV$$