I Different invariant tori in the case of a 2D harmonic oscillator

[Lo Scrondo]

Hi everyone!

Both sources I'm currently reading (page 291 of Mathematical Methods of Classical Mechanics by Arnol'd - get it here - and page 202 of Classical Mechanics by Shapiro - here) say that, in the case of the planar harmonic oscillator, using polar or cartesian coordinate systems leads to different action-angle variables (and I'm ok with that) and different invariant tori.

I think I've understood in what sense those tori could be different (i.e. not diffeomorphic at all), but I'd be very glad to see a proof (which I found nowhere) or even an insight...

 

[jambaugh]

These tori ($S^1\times S^1$) reflect the decoupling of the two dimensional (counting configuration variables) simple harmonic oscillator (SHO). In the general case your elasticity will be a 2-tensor but it can be diagonalized via a choice of oblique coordinates which will decouple the system...

This is to say we can treat the 2-dim SHO as two independent 1-dim SHO's, and the composite system description will then be the Cartesian product of the two independent descriptions (initial conditions plus factored component dynamics).

Recalling that the orbit of a single 1-dim SHO in phase space (1+1 dim) is a circle (topologically speaking) then the orbit of the 2-dim SHO becomes a circle $\times$ circle i.e. a torus in the 2+2 dimensional composite phase space.

One can then also infer that under small perturbations from the 2-dim SHO via introduction of small coupling dynamics between the otherwise independent components, the orbits shold not significantly alter and preserve their topology although the geometric shape will be deformed in the 4-dim phase space.

 

[Lo Scrondo]

Many thanks for your insight @jambaugh!
My concern was about the following fact. Let's say, in the case of the 2D SHO, we got a couple of action-angle variables, $I_1, \omega_1$ and $I_2, \omega_2$.
Now, let $$det (\frac{\partial \omega_n}{\partial I_n}) = 0$$This means that the torus created by the trajectories is in fact 1-dimensional.
If we got different action-angle variables, will such torus be the same or at least homotopically equivalent?