The three pages collectively explain the role of the Generalized Inertia Tensor ($M$) in the Lagrangian mechanics of a coupled mass system (where one mass moves radially and the other angularly). The system's configuration is described by the two generalized coordinates $\mathbf{q}=\{r, \phi\}$ (radial distance and angle), which are the minimum set of variables needed to account for the constraint of a fixed string length. The components of the tensor $\mathbf{M}$ are found using the second partial derivatives of the kinetic energy $T$ with respect to the generalized velocities, $M_{ij} = \frac{\partial^2 T}{\partial \dot{q}i \partial \dot{q}j}$. The most critical takeaway is why the cross-terms $M{r\phi}$ and $M{\phi r}$ are zero: because the total kinetic energy $T$ contains no mixed product terms like $\dot{r}\dot{\phi}$, which mathematically ensures that the radial and angular motions are inertially decoupled.
<aside> ❓