I am a beginner in this field.
Just ask a simple question which confuses me.
Please consider the following:
- Conservation of angular momentum about fixed point $o$: $\dot{H}_o = M$.
$M$: the total external torque applied to the body about $o$. - Euler equation: $I\dot{\omega}+\omega\times I \omega = M$.
$I$: moment of inertia in matrix form (suppose diagonal $I$ for simplicity.)
My question: If there is no external torque ($M=0$), then from 1., we know $\dot{H}_o=0$ and by $H=I\omega$, we know $\dot{\omega}=0$ (Due to rigid body, $I$ is constant).
However, by 2., if $M=0$, $I\dot{\omega}=-\omega\times I \omega $. So $\dot{\omega}\ne 0$.
It confuses to me. Where am I wrong?