The two pages explain why two separate terms vanish during the proof of the rotational identity for the moment of inertia tensor, $\dot{I}{ij} \omega_j = \epsilon{ijk} \omega_j I_{k \ell} \omega_{\ell}$. The first page explains that the term containing the time derivative of the inertia tensor contracted with the angular velocity ($\dot{I}{ij} \omega_j$) vanishes because the integral $\int_V \rho x_q dV$ it contains represents the first moment of mass, and this integral is zero when the rigid body's rotation is defined to be about its center of mass. The second page clarifies that the term $\epsilon{ijk} \omega_j \omega_k x_m x_m$ disappears due to a tensor algebra rule: the contraction (summation) of the antisymmetric Levi-Civita symbol ($\epsilon_{ijk}$) with the symmetric product of angular velocities ($\omega_j \omega_k$) is always zero.

Cue Columns

<aside> ❓

  1. Why does the third term in the expression (the contraction between the time derivative of the inertia tensor and the angular velocity) vanish?
  2. Why does the first term in the final expression for the Right-Hand Side vanish? </aside>