I'm working on analytical mechanics for a rigid rotating body and I'm a bit confused about when we're allowed to express the inertia tensor as a matrix proportional to the identity matrix ie
$$\begin{pmatrix}
I_{1} & 0 & 0\\
0 & I_{2} & 0\\
0 & 0 & I_{3}
\end{pmatrix}
$$
Where $I_{1}, I_{2}, I_{3}$ are the principle moments of inertia.
From my class notes, for a freely rotating rigid body, it says "this case (referring to the case of a diagonal inertia tensor) occurs when the body is regular: sphere, cube, tetrahedron, etc."
However, from this website, it seems that when the body is rotating about one of the principle axis $\{\bf{e}_{i}'\}$ and also given
we reorient our Cartesian coordinate axes so the they coincide with the mutually orthogonal principal axes of rotation. In this new reference frame, the eigenvectors of $\tilde{\bf I}$ are the unit vectors, ${\bf e}_x$, ${\bf e}_y$, and ${\bf e}_z$, and the eigenvalues are the moments of inertia about these axes, $I_{xx}$, $I_{yy}$, and $I_{zz}$, respectively
then the resulting inertia tensor is diagonal. It doesn't mention anything about requiring a "regular shaped body", so I'm wondering then if these conditions mentioned from the website are met can a rigid body of any shape have an inertia tensor expressed as a diagonal matrix?
Best Answer
It is true that for any shape you could make the inertia tensor diagonal by choosing appropriate coordinate axes. Since the matrix $I$ is symmetric and real, it can always be diagonalized by a basis change. Physically this basis change corresponds to rotating the coordinate axes until they coincide with the principal axes of rotation.
What the notes of your course probably mean is that for symmetric objects our natural choice of coordinate axes usually coincide with the principal axes of rotation for this object and you get a diagonal inertia tensor right away. The sphere is a separate story, since every axis is equivalent for it - so you would get a diagonal inertia tensor (proportional to the unit matrix, which does not change under change of the basis) for all choices of axes.