1

Consider the Hamiltonian for the classic planar harmonic oscillator:

$$H = H_x + H_y$$

where $$H_x~=~\frac{1}{2}(p_x^2+x^2), \qquad H_y~=~\frac{1}{2}(p_y^2+y^2).$$

So it is possible to obtain a set of action-angle variables $(H_x, H_y, \phi_x, \phi_y)$ .

My question is, how does the trajectory look in such coordinates?

I'm thinking of a (1:1) 1-torus on a 2-torus, i.e. a closed line that on a 2-torus manifold makes one trip around the first circumference while it does one around the other, something looking like this (imagine part of the track in red in front of the torus, and part behind):

enter image description here

Am I right?

Qmechanic
  • 201,751
Lo Scrondo
  • 627

1 Answers1

1

Yes, your idea of these trajectories is correct. The equations for constant $H_y$ and $H_x$ determine a torus in the same way $x^2+y^2=r^2$ determines a circle.

An interesting point to note about classical mechanics: so long as the system is integrable (meaning we have a complete set of linearly independent conserved quantities at every point in phase space), the trajectories for fixed values of the conserved quantities are locally diffeomorphic to a torus almost everywhere completely independent of the other details of the system.

I believe this theorem was proven by Arnold and some others. But the interesting thing is that once the harmonic oscillator is understood well, the only things left to understands are more global properties of the phase space and non-integrable (essentially, chaotic) systems.

Richard Myers
  • 4,889