I was refreshing my memory on Newtonian mechanics, and I ran across Torque in a non-inertial frame. I worked out the derivations for the rate of change of linear and angular momenta starting with the definition of linear and angular velocity. I arrived at the following
$$ \dot{\vec{L}} = m \vec{a}_{cm} = m[\vec{\alpha}\times\vec{r}+\vec{\omega}\times(\vec{\omega}\times\vec{r})] $$
$$ \dot{\vec{H_{c}}} = \vec{\omega}I_{c}\vec{\omega}+I_{c}\vec{\alpha} $$
However, something I took for granted is now confusing my mind. The first equation states that if the rigid body has zero angular velocity and acceleration, its linear momentum does not change over time. I thought $\vec{v}=\vec{\omega}\times\vec{r}$ is merely the linear velocity of a point on a rotating body.
Also, I understand that the rate of change of angular momentum about a given point is equal to the sum of moments about the same point. However, I often come across text writing $\dot{\vec{H_{c}}} = I_{c}\vec{\alpha}$. Why is the $\vec{\omega}I_{c}\vec{\omega}$ term not present in certain cases?
