nForum - Search Results Feed (Tag: tensor) 2019-10-20T06:59:08-04:00 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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Schur functors https://nforum.ncatlab.org/discussion/1507/ 2010-06-24T19:25:52-04:00 2010-07-09T15:31:49-04:00 John Baez https://nforum.ncatlab.org/account/17/ I’m struggling to further develop the page on Schur functors, which Todd and I were building. But so far I’ve only done a tiny bit of polishing. I deleted the discussion Todd and I were having ...

I’m struggling to further develop the page on Schur functors, which Todd and I were building. But so far I’ve only done a tiny bit of polishing. I deleted the discussion Todd and I were having near the top of the page, replacing it by a short warning that the definition of Schur functors given here needs to be checked to see if it matches the standard one. I created a page on linear functor and a page on tensor power, so people could learn what those are. And, I wound up spending a lot of time polishing the page on exterior algebra. I would like to do the same thing for tensor algebra and symmetric algebra, but I got worn out.

In that page, I switched Alt to $\Lambda$ as the default notation for exterior algebra. I hope that’s okay. I think it would be nice to be consistent, and I think $\Lambda$ is most widely used. Some people prefer $\bigwedge$.

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