❓What is the final form of the generalized inertia tensor matrix M in generalized coordinates?

The final form of the generalized inertia tensor matrix $\mathbf{M}$ is: $\mathbf{M} = \begin{pmatrix} M_{rr} & M_{r\varphi} \\ M_{\varphi r} & M_{\varphi\varphi} \end{pmatrix} = \begin{pmatrix} m_1 + m_2 & 0 \\ 0 & m_1 r^2 \end{pmatrix}$

This matrix summarizes how the system's kinetic energy ($T$) depends on the generalized velocities ($\dot{r}$ and $\dot{\varphi}$), where $T = \frac{1}{2} (\dot{r} \ \dot{\varphi}) \mathbf{M} \begin{pmatrix} \dot{r} \\ \dot{\varphi} \end{pmatrix}$.

The components, have the following physical meanings:

$M_{rr} = m_1 + m_2$

• Meaning: This is the total mass of the system.

• Reason: Since both mass $m_1$ (on the plane) and mass $m_2$ (hanging below) move together with the radial velocity $\dot{r}$ (as $m_2$ moves vertically only when $r$ changes), they both contribute fully to the effective inertia for radial motion.

$M_{\varphi\varphi} = m_1 r^2$

• Meaning: This is the moment of inertia of mass $m_1$ alone.

• Reason: Only the mass $m_1$, which is free to move in the horizontal plane, rotates about the central $z$-axis (the hole). Mass $m_2$ is constrained to move purely vertically (radially) and does not contribute to the angular kinetic energy term ($\dot{\varphi}^2$), so it is excluded from $M_{\varphi\varphi}$.

$M_{r\varphi} = M_{\varphi r} = 0$

• Meaning: The cross-terms are zero, confirming that the generalized coordinates ( $r$ and $\varphi$) are uncoupled in the kinetic energy.

• Reason: As shown by the grouping of terms in the total kinetic energy expression, $T = \frac{1}{2} (m_1 + m_2) \dot{r}^2 + \frac{1}{2} (m_1 r^2) \dot{\varphi}^2$, there is no mixed-product term proportional to $\dot{r} \dot{\varphi}$. This zero result simplifies the subsequent Lagrangian dynamics equations, as radial acceleration is not directly dependent on angular velocity, and vice-versa (in terms of kinetic energy).

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