Is there a simple physical interpretation for the off-diagonal entries of the moment of inertia tensor?
I know that those entries are de facto necessary to use the tensor to calculate quanties like angular momentum or rotational kinetic energy, and that they can be dispsensed of by switching to principal axes, but that's more of a tautology than a way of understanding them, at least in my opinion.
Another way to phrase this question (I guess): why isn't any set of (linearly independent) axes a set of principal axes for a rigid body?
Answer
Those terms represent a coupling between the orthogonal components of momentum and rotation. It means the motion along one axis, affects the angular momentum on another axis.
If rotating not about an axis of symmetry material has to move in and out of the plane of motion for each particle and the manifests itself as a change in momentum in a direction perpendicular to the rotation axis.
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