🧄Finding the Generalized Inertia Tensor for the Coupled Mass System

The derivation of the generalized inertia tensor highlights how constraints simplify complex mechanics: the diagonal structure confirms that the kinetic energy is instantaneously decoupled into independent radial ( $\dot{r}$ ) and angular ( $\dot{\varphi}$ ) velocity terms. The radial inertia ( $M_{r r}$ ) simplifies to the total mass ( $m_1+m_2$ ) because both particles move with the same radial speed. Conversely, the angular inertia ( $M_{\varphi \varphi}$ ) is simply the moment of inertia of $m_1$ alone ( $m_1 r^2$ ), as $m_2$ does not rotate. Crucially, this tensor is non-constant because the angular component depends on the current radius $r$, which is the exact mathematical foundation for the strong coupling and oscillation we observed in the animation through the conservation of angular momentum.

Finding the Generalized Inertia Tensor for the Coupled Mass System | Proof and Derivationviadean on Notion