Angular momentum about a point or angular momentum about an axis? What is the difference between the definition of angular momentum about a point as $\vec L=\vec r\times \vec p$ and that about an axis? How are they related? Can someone explain when the first second definition is useful and when the second is useful and why?
 A: There seems to be a lot of confusion in the comments.

*

*Angular momentum is always defined with respect to a point of reference, not an axis.

*The motion of a single rigid body may be described by an angular velocity, which does not depend on a point of reference.

*Both angular momentum and angular velocity are vectors (or technically pseudovectors, though this has no bearing on the rest of the question). They are related by $\vec{L} = I \vec{\omega}$ where $I$ is the moment of inertia tensor. They are not necessarily parallel.

*When we talk about "the axis of rotation of a rigid body", we always mean the direction of the angular velocity, not the direction of the angular momentum.

*In very basic introductory physics courses, you will usually only consider rigid bodies with an axis of symmetry, rotating about that axis. In that case, the directions of $\vec{L}$ and $\vec{\omega}$ both coincide with that axis of symmetry, so sometimes people talk about "the angular momentum about that axis". But it's not a sensible concept in general, because the angular momentum is usually not even parallel to the axis of rotation.

