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[ognik]

I'm not 100% confident of my approach to the 2 exercises below:

Orbital angular momentum of i'th element is $\vec{L_i} = \vec{r_i} \times \vec{p_i} = m_i \vec{r_i} \times (\omega \times \vec{r_i}) $

a) Find the inertia matrix $I$ such that (omitting vector signs from here on) $L = I \omega, |L \rangle = I|\omega \rangle$

$=> L_i = - m_i[r_i \times r_i \times \omega]$ (cyclic anti-commute)
$ =-m_i[r_i(r_i \cdot \omega) - \omega (r_i \cdot r_i)] $ (BAC-CAB)
$ = m_i\omega r_i^2 $ ($r_i$ and $\omega$ orthogonal)
$ L = \sum L_i = \sum (m_i r_i^2)\omega =\sum I_i \omega = I \omega, i.e. |L \rangle = I| \omega \rangle$

b)Similarly derive I starting with Kinetic Energy $T_i = \frac{1}{2}m_i(\omega \times r_i)^2 $

$ = \frac{1}{2}m_i [|\omega|^2 |r_i|^2 - (\omega \cdot r_i)^2] $ (identity)
$ = \frac{1}{2} \omega (m_i r_i^2) \omega $ ($r_i$ and $\omega$ orthogonal)
$ \therefore T= \sum T_i = \frac{1}{2} \omega \sum(m_i r_i^2) \omega = \frac{1}{2} \omega I \omega = \frac{1}{2}\langle \omega |I| \omega \rangle$

 

[jvicens]

Thank you for sharing your approach to finding the inertia matrix in these exercises. I would like to offer some feedback and clarification on your work.

Firstly, in part a), you have correctly calculated the orbital angular momentum of the i'th element using the given equation. However, your final expression for the total angular momentum, $L$, is missing a vector sign. It should be $L = \sum \vec{L_i}$, where each $\vec{L_i}$ is a vector quantity. This is important because the inertia matrix, $I$, is also a vector quantity and should be represented as such in your final expression.

Additionally, in part b), you have correctly calculated the kinetic energy of the i'th element. However, your final expression for the total kinetic energy, $T$, is missing a vector sign and a factor of 1/2. It should be $T = \frac{1}{2} \sum \vec{T_i} = \frac{1}{2} \langle \vec{\omega} | I | \vec{\omega} \rangle$.

In both parts, it would also be helpful to explicitly state that $I$ is a diagonal matrix, with each element $I_i$ representing the moment of inertia of the i'th element.

Overall, your approach is sound and your derivations are correct. However, in order to fully convey your ideas and solutions, it is important to include all necessary vector signs and factors in your final expressions. Keep up the good work and don't hesitate to reach out if you have any further questions or concerns.