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Owen Lynch (Feb 15 2022 at 21:08):This seems to be saying that there is a use for non-integrable Dirac structures; i.e. they give an example where the Dirac structure is non-integrable and it seems like a physically important structure:
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John Baez (Feb 15 2022 at 21:16):That's interesting. "Non-holonomic constraint" means precisely that some integrability conditions fail: you can move along a long path obeying the constraints to get from a state to some arbitrarily nearby state that you can't get to using a short path that obeys the constraints. So, this makes sense.
Owen Lynch (Feb 15 2022 at 21:17):What does that correspond to in the euro example?
Owen Lynch (Feb 15 2022 at 21:19):Oh, I guess you can't shimmy the coin perpendicularly to its movement
John Baez (Feb 15 2022 at 21:28):Right. I usually use the example of a ball "rolling without slipping" on a table. You can get from any state to any other state, but if you have two nearby states they may not be connected by a short path with no slipping.
John Baez (Feb 15 2022 at 21:30):I wrote a paper once with John Huerta on how the exceptional Lie group and the 'split octonions' arise from the problem of a ball rolling without slipping or twisting on a ball 3 times as big across. There's something magic that happens at precisely this ratio of radii.
There's an animation of it in these web pages:
John Baez (Feb 15 2022 at 21:32):Hmm, I have to update the MathJax on these pages. :angry: They broke all my pages.