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

= 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. --Leon (talk) 09:25, 17 July 2018 (UTC)
 * 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?