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I was looking into the moment of inertia expression for angular momentum. The angular momentum of a group of particles can be expressed as a linear transformation of the angular velocity vector. This produces a simple form for the whole body when the angular velocity is constant. Moreover, assuming we would like a simpler diagonal form for moment of inertia, the matrix must be diagonalisable so also must have real eigenvalues. The rotation vector must also be parallel to an eigenvector if we further want a constant moment of inertia

My question is, are these conditions necessary for moment of inertia to make sense? Moreover what shortcuts do we know a priori of calculating the linear transformation, to tell us that these conditions do in fact hold?

Constant angular velocity, for I to exist Diagonalisable Matrix with angular velocity parallel to one eigenvector

If so, what are some shortcuts we can take to observe when these conditions hold?

Sorry if my question was unclear

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I might be missing something in your question, but the only condition you've listed that I can parse is whether or not the moment of inertia matrix need be diagonal- and to that I say, emphatically, no!

Moment of Inertia matrices are given by real, symmetric 3x3 matrices, and they need not be diagonal. One may always diagonalize such a matrix, as symmetric matrices are orthogonally diagonalizable.

That orthogonality is key- it is what describes the "principal axes of rotation" in a body: the directions in which rotations may be made that are orthogonal to each other.

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  • $\begingroup$ ah, but don't you need a constant angular velocity vector in order to produce a single matrix that transforms w into L over the whole body? $\endgroup$ – lucky-guess Apr 20 '19 at 14:46
  • $\begingroup$ One does not require any particular angular velocity vector to construct the moment of inertia matrix- one only needs to assign bounds to use while integrating for the components of the moment of inertia matrix. That's how the matrix is produced- and any selection of where on the body to do rotation may be computed from there using the parallel axis theorem. $\endgroup$ – swickrotation Apr 20 '19 at 19:24

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