nForum - Search Results Feed (Tag: moment-of-inertia) 2022-06-29T05:22:24-04:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher n-dimensional moment of inertia https://nforum.ncatlab.org/discussion/3068/ 2011-08-29T07:23:56-04:00 2011-08-30T00:30:07-04:00 Mike Shulman https://nforum.ncatlab.org/account/3/ 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 ...

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

$\tau = I \alpha$

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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