This term shows up in euler’s equations of rigid body dynamics… what is the meaning of this term… why are angular momentum and velocity not collinear
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Let’s start with a more formal mathematical reason and then give an example.
For a rigid body, linear momentum is given by mass × velocity, where velocity and momentum are vectors and mass is a scalar. Since mass is a scalar, velocity and momentum are trivially always colinear.
*Angular momentum*, on the other hand, is defined as L = I 𝜔, where I is the moment of inertia and 𝜔 is angular momentum. L and 𝜔 are once again vector quantities, but unlike mass, moment of inertia is no longer scalar! Rather, it is a rank 2 tensor, which you can think of as a more complicated “proportionality” represented by a 3 by 3 matrix.
Moment of inertia is a fundamentally more complicated object than mass, which is reflected in the more complicated relationship between angular velocity and angular momentum.
Now to see physically *why* this should be the case, think about a “wobbly” rotating object, [like this one](https://i.stack.imgur.com/0rkLC.png). What is the direction of angular velocity? It is, as usual, normal to the plane of rotation, and therefore in the z-direction.
What about the angular momentum? This isn’t quite as simple. Angular momentum is by definition orthogonal to the plane of the position vector and the linear momentum, which *is not*, in this case, flat to the xy plane.
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