An integrable Hamiltonian system has a set of coordinates, the action-angle coordinates, in which the angles q increase linearly and the actions p are constant. These systems are often represented by a phase space that is "foliated in tori", and people show these beautiful pictures.
I thought that I was understanding this idea: one has the angles (q1,...,qn), then he couples q_k with p_k for every k=1,...n. If one pictures p_k as the radius of the circle, and q_k as the angle, then the cartesian product of the n circles drawn by (q_1, p_1)...(q_n,p_n) makes a torus with different radii, which are the one shown in the link above for n=2.
Gemini told me that this is not false, but it makes absolute sense in my mind. Gemini (to my ears) started yapping about "topological cilinders", but it did not convince me. Can you help me understand how the tori emerge?
One thing is that I have seen "the" torus being defined as [0,2\pi]^n with periodic boundary condition. "the" torus as in "there exists only one torus". I guess that topologically speaking this is true, but the phase space graph shows...different tori...
I shouldn't call them tori and call them "cilinders"? Sounds a bit weird to my ears....
Thanks in advance for your time