Having fun rereading Rota's "Indiscrete Thoughts". These aphorisms give its flavor. cc @nntalebpic.twitter.com/clgsYRD7yq
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But the question remains: why cotangent bundles (and not tangent bundles, on which certain natural tangent vector fields are 2nd-ord. ODEs)?
The answer is profoundly interesting, but very seldom given. First appeared in the work of Lie, but traces all the way back to Jacobi.
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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