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

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.

• Comment to the question (v1): Torus in how many dimensions? Just 2 dimensions $S^1\times S^1$? Anyway, the $2n$-torus $(S^1)^{2n}$ can be endowed with a global symplectic structure in the obvious way. Jul 15, 2014 at 21:09
• Maybe billiard systems? I recall they may be modelled on toruses but I'm not completely sure... Jul 15, 2014 at 21:18
• Jul 15, 2014 at 21:25
• @pppqqq That is not the phase space of the double pendulum, that's its configuration space. Jul 15, 2014 at 21:32
• @yuggib Hmmm thanks. That's an interesting suggestion although it's definitely not clear to me how such examples would work. In particular, the issue I see is that a billiard's canonical momentum (at least in Cartesian coordinates) is not periodic. Jul 15, 2014 at 21:39

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.

• +1: Just a minor point: mathematically one may consider non-relativistic massless particles. But no such particles are found in nature. Jul 16, 2014 at 17:08
• This answer should be understood within the framework of non-relativistic Newtonian mechanics (where a massless particle is not confined to moved with the speed of light, but is merely a convenient idealization). Jul 16, 2014 at 17:18
• @Qmechanic Thanks for the link on the Faddeev-Jackiw method. On a tangential note, would you happen to know of any good references on constrained Hamiltonian systems besides Henneaux and Teitelboim? I find their presentation to be rather impenetrable. Jul 17, 2014 at 0:19
• @joshphysics: As an appetizer and introduction, you might enjoy reading P.A.M. Dirac, Lectures on QM, 1964. Beyond that it is hard to recommend a single source as comprehensive as Henneaux & Teitelboim. Jul 17, 2014 at 0:40
• @Qmechanic I see, yes I couldn't find anything else after much searching. I've also read Dirac; unfortunately I find it extremely hard to read both treatments. Their mathematics isn't particularly precise, but beyond that, I find the logic to be wanting. Perhaps I should write a sequence of SE questions that you will inevitably answer that will serve as it's own resource for those who share my frustration :) Jul 17, 2014 at 1:11

$U(1)$ 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.)

• Nice example, but is there any system in the real world world whose actual dynamics could be modeled by this guy? Jul 15, 2014 at 22:54
• Only long-range effective theories like I mention. By the way, I don't know any theories with higher genus surfaces as their phase space... Jul 15, 2014 at 23:46
• My apologies. I somehow missed that last paragraph. Jul 15, 2014 at 23:48
• @Ryan How do we see that the moduli space of U(1) flat connections on a 2-torus is itself a 2-torus ? Since there are 4 homology cycles with one relation between them, it seems that the dimension of the space should be three rather than two. Can you please point out what I am missing. Jul 15, 2014 at 23:58
• H_1 is Z plus Z. Flat connections are maps from this to U(1). Jul 16, 2014 at 0:21

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.

• But is the phase space a torus, too? Jul 15, 2014 at 21:42
• This is very interesting indeed, and my condensed matter theorist friend in fact already suggested something like this, but it's not the spatial topology of a system I'm wondering about. I'm looking for a system whose dynamics can be modeled as a Hamiltonian system whose phase space is a torus. Jul 15, 2014 at 21:42
• @joshphysics Whoops, I read too fast. I'm reluctant to pronounce on the phase space, but it's possible that the answer is yes, as the reciprocal space generated by the periodic potential is also periodic. Jul 15, 2014 at 21:49

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

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.

• Comment to the answer (v1): Note that the two angle variables Poisson commute, i.e. the Poisson structure in the angle torus sector is zero. Jul 23, 2014 at 7:57