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Wikipedia:Reference desk/Archives/Mathematics/2018 July 17

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July 17

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Integrable systems and tori

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I understand the concept of Hamiltonian systems having phase space coordinates, and that when they are integrable (in the strongest sense) they have constants of motion.

I'm also aware that with integrable systems trajectories lie on tori. But, whilst I think that I know what an -dimensional torus is, I don't know what it means for a trajectory to lie on one.

  1. Do some/all/no trajectories uniquely define tori?
  2. Are the tori merely topologically tori, or actually tori?
  3. If a trajectory is near another but lies on a different torus, is its torus necessarily near the torus of the other?

--Leon (talk) 09:25, 17 July 2018 (UTC)[reply]