Is there a physical system whose phase space is the torus?

Question

NOTE. This is not a question about mathematics and in particular it's not a question about whether one can endow the torus with a symplectic structure.

In an answer to the question

What kind of manifold can be the phase space of a Hamiltonian system?

I claimed that there exist (in a mathematical sense), Hamiltonian systems on the torus (and in fact on higher genus surfaces as well). However, when pressed to come up with a physical system in the real world (even an idealized one) whose dynamics could be modeled as a Hamiltonian system on the torus, I could not think of one.

Does such a system exist?

I would even be satisfied with a non-classical system which can somehow effectively be described by a Hamiltonian system on the torus, although I'm not sure that the OP of the other question I linked to above would be.

Answer 1

Consider a non-relativistic massless particle with charge $q$ on a 2D torus

$$\tag{1} x ~\sim~ x + L_x , \qquad y ~\sim~ y + L_y, $$

in a constant non-zero magnetic field $B$ along the $z$-axis.

Locally, we can choose a magnetic vector potential

$$\tag{2} A_x ~=~ \partial_x\Lambda, \qquad A_y ~=~ Bx +\partial_y\Lambda, $$

where $\Lambda(x,y)$ is an arbitrary gauge function. Locally, the Lagrangian (which encodes the Lorentz force) is given as

$$\tag{3} L~=~ q ( A_x\dot{x} + A_y\dot{y})~=~qB~x\dot{y}+ \text{(total time derivative)}. $$

[The ordinary kinetic term $T=\frac{m}{2}(\dot{x}^2+\dot{y}^2)$ is absent since the mass $m=0$. This implies that the characteristic cyclotron frequency of the system is infinite.] The Lagrangian momenta are

$$\tag{4} p_x ~=~ \frac{\partial L}{\partial\dot{x} }~=~A_x, \qquad p_y ~=~ \frac{\partial L}{\partial\dot{x} }~=~A_y. $$

Eq. (4) becomes second class constraints, so that the variables $p_x$ and $p_y$ can be eliminated. The Dirac bracket is non-degenerate in the $xy$-sector:

$$\tag{5} \{y,x\}_{DB}~=~\frac{1}{qB}. $$

[Alternatively, this can be seen using the Faddeev-Jackiw method.] In other words, the two periodic coordinates $x$ and $y$ become each others canonical variable with corresponding symplectic two-form

$$\tag{6} \omega_{DB}~=~qB ~\mathrm{d}x \wedge \mathrm{d}y. $$

The corresponding Hamiltonian $H=0$ vanishes. The classical eqs. of motion

$$\tag{7} \dot{x}~=0~=\dot{y} $$

imply a frozen particle.

Answer 2

$U(1)$ Chern-Simons theory with (physical) space a 2-torus is such an example. Its phase space is the gauge equivalence classes of flat connections on the 2-torus. These are specified by the holonomies around two 1-cycles forming a basis of $H_1(T^2)$. This is of course a 2-torus $U(1) \times U(1)$. Because of the form of the Chern-Simons action, these variables are in fact conjugate, and the symplectic volume of the phase space equals the Chern-Simons level.

I suspect there will only be ``topological" examples like this, since a compact phase space usually implies a finite dimensional Hilbert space (by Heisenberg uncertainty). If a system has local quantum observables, then the Hilbert space is automatically infinite dimensional, since the location of the observable is measurable.

This theory is actually realized in our reality as the long-range effective theory of certain quantum hall systems! (Of course we need to be considering the long-range theory to rid ourselves of local observables like electron correlators.)

Answer 3

In solid state physics, the bulk of a crystal is usually given periodic boundary conditions to avoid the sticky problem of what to do at the termination of the crystal. So the crystal is all bulk, no surface. This turns out to be a very good approximation to the bulk of a real crystal. It also gives the solid the topology of a 3-torus.

Answer 4

One example could be the Standard Map, which can be seen as a description of the Kicked rotor. Though it should be make clear that one is using the symmetry of the phase space in order to describe the dynamics as being in a torus (instead of in a cylinder).

The rotor "consists of a stick that is free of the gravitational force, which can rotate frictionlessly in a plane around an axis located in one of its tips, and which is periodically kicked on the other tip".

_{https://en.wikipedia.org/wiki/File:Std-map-0.6.png}

Answer 5

An isotropic 2D oscillator, when taken into action-angle variables by doing a canonical transformation of the Hamiltonian, $$ H(q_1, p_1, q_2, p_2 ) = \frac{q_1^2}{2m} + \frac{kq_1^2}{2} + \frac{q_2^2}{2m} + \frac{kq_2^2}{2} $$ will yield two constants of the motion, the actions , and two angles running from 0 to $2\pi$ which generates a torus. I'm not entirely sure of the process of doing the transformation, but this is what I came across recently when I was learning classical-physics.