This is a bit off the beaten path for what we discuss around here, but I know there are people here who think about physics in nice abstract ways, so maybe someone knows the answer. My understanding of the concept of *moment of inertia* is that for any rigid body, there’s supposed to be an equation

parallelling $F=m a$, where $\tau$ is torque, $\alpha$ is angular acceleration, and $I$ is this “moment of inertia” thing. But what type of thing *is* $I$?

In oriented Euclidean 2-space, $\tau$ and $\alpha$ can be identified with scalars, and likewise so can $I$.

In oriented Euclidean 3-space, $\tau$ and $\alpha$ can be identified with vectors, and $I$ becomes a symmetric rank-2 tensor.

(In oriented Euclidean 1-space, there is no room for rotational motion.)

In oriented Euclidean n-space, my best guess is that $\tau$ and $\alpha$ should both lie in $so(n) \cong \Lambda^2 \mathbb{R}^n$. So what type of beast is $I$? Is there a good reference for rotational motion in n dimensions?

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