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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?

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marked as duplicate by Emilio Pisanty, Qmechanic Jun 10 '16 at 2:07

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ Presumably you mean the moment of inertia? (It does matter - there isn't really an "inertia tensor".) $\endgroup$ – Emilio Pisanty Jun 10 '16 at 0:49
  • $\begingroup$ I thought it was called the inertia tensor for short , I'm not sure though -- I decided that maybe moment of inertia only referred to the components of the tensor, and that inertia tensor was unambiguous because there is no such thing besides a moment of inertia tensor $\endgroup$ – Chill2Macht Jun 10 '16 at 0:52
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    $\begingroup$ No. 'Moment' describes the fact that the inertia (i.e. the mass) is being multiplied by a power of the position, and it is never omitted. There is indeed no "inertia tensor", but that doesn't mean you can drop the 'moment'. Also, tensor calculus doesn't really apply. $\endgroup$ – Emilio Pisanty Jun 10 '16 at 1:00
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    $\begingroup$ In any case, this is very close to a duplicate of What is the physical significance of the off-diagonal moment of inertia matrix elements? - if your question isn't covered there, you should specify why not. $\endgroup$ – Emilio Pisanty Jun 10 '16 at 1:01
  • $\begingroup$ It probably is a duplicate - my bad. Anyway I made the appropriate changes to my post. $\endgroup$ – Chill2Macht Jun 10 '16 at 1:03
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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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