[NOTES/ME-14015]-Moment of Inertia as Second Rank Tensor

Relative to a chosen set of axes, the moment of components of inertia tensor are defined to be \begin{equation} I_{jk}=\sum_\alpha m_\alpha(\delta_{jk} |\vec{x}_\alpha|^2 - x_{\alpha j}x_{\alpha k}). \end{equation} Here the rigid body is supposed to consist of masses \(m_\alpha\) at positions \(x_\alpha\). In case of a continuous body, the sum over \(\alpha\) should be replaced by an appropriate (volume) integral. \begin{equation} I_{jk}=\int d^3x \rho(\vec{x})(\delta_{jk} |\vec{x}_\alpha|^2 - x_{\alpha j}x_{\alpha k}). \end{equation} Frequently, we shall use a \(3\times3\) matrix notation for moment of inertia tensor and write \begin{equation} \underline{\sf \Ibb}=\begin{pmatrix} I_{11} & I_{12} & I_{13}\\ I_{21} & I_{22} & I_{23}\\ I_{31} & I_{32} & I_{33} \end{pmatrix} \end{equation} The inertia tensor transforms like a second rank tensor. In the matrix notation the transformation of the inertia tensor under a rotation \(R\) is given by \begin{equation} \underline{\sf \Ibb}{'} = R \, \underline{\sf \Ibb} \, R^T. \end{equation}  Question for you Verify the above equation.

1.Kinetic energy and angular momentum

If the angular velocity of rigid body is \(\vec{\omega}\), the kinetic energy and angular momentum are given by \begin{equation} \text{K.E.} = \omega_jI_{jk}\omega_k; \qquad L_j= I_{jk}\omega_k. \end{equation} In matrix form we shall write \begin{equation} \text{K.E.} = \underline{\pmb{\omega}}^T \,\underline{\mathbf I}\, \underline{\pmb{\omega}}; \qquad \underline{\mathbf L}= \underline{\mathbf I}\, \underline{\pmb{\omega}}. \end{equation} where a column vector notation \begin{equation} \underline{\pmb{\omega}}= \begin{pmatrix}\omega_1\\\omega_2\\\omega_3\end{pmatrix}, \qquad \underline{\mathbf L}= \begin{pmatrix}L_1\\L_2\\L_3\end{pmatrix}, \end{equation} for angular velocity and angular momentum has been used. Here \(\underline{\pmb \omega}^T\) means transpose of the matrix \(\underline{\pmb \omega}\). Using the known transformation properties of inertia tensor and vector nature of angular velocity it is straight forward to show that the kinetic energy is a scalar under rotations and the angular momentum transforms like a (pseudo)vector. We shall skip the proof here. Try it now How will you prove the above statement?}