Shameful confession: I have felt inclined to this kind of objection when classical mechanics is redone with cotangent bundles.
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Replying to @stevenstrogatz @JSEllenberg and
Re: cotangent bundles: Another question is whether the new mathematical clothing makes ideas easier to understand or harder to understand.
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Replying to @DynamicsSIAM @stevenstrogatz and
These ideas did eventually lead to nice advances in topics like symplectic integration that benefited from the increased formalism.
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Replying to @stevenstrogatz @DynamicsSIAM and
But the question remains: why cotangent bundles (and not tangent bundles, on which certain natural tangent vector fields are 2nd-ord. ODEs)?
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Replying to @MathPrinceps @stevenstrogatz and
The answer is profoundly interesting, but very seldom given. First appeared in the work of Lie, but traces all the way back to Jacobi.
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Replying to @MathPrinceps @stevenstrogatz and
Not even remotely enough to say that cotangent bundles alone have contact 1-forms with symplectic exterior derivatives. True, but so what?
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Replying to @MathPrinceps @stevenstrogatz and
It's all about the link between symmetries and conserved quantities. Coadjoint orbits, and not cotangent bundles, are the things to ponder.
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Replying to @MathPrinceps @stevenstrogatz and
A phase space should have exactly the structure needed to ensure that each symmetry correspond to a conserved quantity. That's the key idea.
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Replying to @MathPrinceps @stevenstrogatz and
Twitter isn't a great format for how you are commenting. You could write this up in a blog entry and put a link to it in a Twitter response.
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Thanks so much for the helpful comment.
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Replying to @MathPrinceps @stevenstrogatz and
No problem! Please do write the blog entry. Even the books on geometric mechanics have a very high barrier to entry, so it would be helpful.
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