In this article, we will find that the inertia matrix naturally appears when we calculate the kinetic energy $T$ or the angular momentum $\mathbf M$ of a rigid body. Then, we introduce the concept of principal inertia $\mathbf J_{\mathrm{pri}}$. We also study how the inertia matrix changes under translations and rotations and how those transformations may lead to conclusions that can help us simplify the calculation of inertia matrices.
https://ulysseszh.github.io/physics/2022/11/12/rigid-energy-momentum.html
In this article, we will find that the inertia matrix naturally appears when we calculate the kinetic energy $T$ or the angular momentum $\mathbf M$ of a rigid body. Then, we introduce the concept of principal inertia $\mathbf J_{\mathrm{pri}}$. We also study how the inertia matrix changes under translations and rotations and how those transformations may lead to conclusions that can help us simplify the calculation of inertia matrices.