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.
Non Constant Inertia and Dynamic Coupling #audio
The Power of Generalized Coordinates
By choosing the generalized coordinates r and $\varphi$, we completely decoupled the kinetic energy expression. The total kinetic energy $T$ became a simple sum of terms, $\frac{1}{2} M_{r r} \dot{r}^2+\frac{1}{2} M_{\varphi \varphi} \dot{\varphi}^2$, with no cross-term involving $\dot{r} \dot{\varphi}$.
Physical Interpretation of Components
The components of the tensor reveal the effective inertia for each type of motion:
Coordinate Dependence (Non-Constant Inertia)
Notice that the $M$ tensor is not a constant matrix. The $M_{\varphi \varphi}$ component is $m_1 r^2$, meaning the angular inertia of the system depends on the mass $m_1$ 's current radial position $r$. This is the mathematical backbone of the angular momentum conservation effect we saw in the animation: as $r$ changes, the system's resistance to angular acceleration changes dynamically.
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