I have just recently studied Euler's Equations for Rigid body dynamics. Following through the proofs and equations, there is one thing that I can't seem to make a sense of.
in
$$ \dot{\textbf{L}} + \boldsymbol{\omega} \times {\textbf{L}} = \mathbf{\Gamma} $$
Where angular momentum $\textbf{L}$ is calculated in a rotating reference frame with coordinate axes as principal axes. $$ \textbf{L} = (\lambda_1\omega_1 ,\lambda_2\omega_2, \lambda_3\omega_3) $$
My question is if we measure the angular velocity in the rotating frame or the 'body frame' (i.e., attached to the body). How can the angular velocity of the body in the 'body frame' be anything but zero? And if $\boldsymbol{\omega}$ is zero, so should be the Angular momentum in the rotating frame.
