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

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July 17

Integrable systems and tori

I understand the concept of Hamiltonian systems having $2n$ phase space coordinates, and that when they are integrable (in the strongest sense) they have $n$ constants of motion.

I'm also aware that with integrable systems trajectories lie on tori. But, whilst I think that I know what an $n$-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)