Why Isn't Angular Momentum Aligned with Angular Velocity in 3-D Dynamics?

[beserk]

In 3-D dynamics why is the angular momentum not aligned with angular velocity?
Does this mean H = Iw is wrong in 3-D ?

 

[Tide]

How do you know the angular momentum is not aligned with the angular velocity?

Generally, in 3D, the angular velocity is a vector and the moment of inertia is a tensor (matrix).

 

[BobG]

In 3-D dynamics why is the angular momentum not aligned with angular velocity?
Does this mean H = Iw is wrong in 3-D ?

The angular momentum vector can be aligned with the angular velocity vector. It's just not necessarily the case. If the two vectors are aligned, H=Iw is perfectly valid as a scalar equation.

As Tide said, the moment of inertia is a matrix and the angular velocity is a vector. If the angular momentum vector and the angular velocity vector are aligned, all the elements of the matrix except the diagonals will be zero. As a result, the scalar equation can be used for each axis of rotation.

Generally, if one were planning something, one would probably plan to have the geometric axes align with the object's principal axes (axes based on mass distribution) and would plan to keep the angular momentum vector and angular velocity vector aligned. In practice, the outside environment will cause the two vectors to become misaligned with each other (nutation).