noncommutative geometry in leaf spaces of classical physics
I was trying to think of natural physical reasons to want to study noncommutative geometry, and it seems to me that it should be relevant even in a classical scenario. Let me know what you think of this as I am no expert. I'm not being terribly precise, either, so let me know if I should elaborate.
Let a classical particle be confined to a 2-dimensional torus V. Then if you flick the particle in a direction so that it starts moving, the particle's orbit and translates of the orbit will foliate the space, with the particle confined to a certain leaf of the foliation. Then it should be interesting to look at the leaf space under the quotient topology of this foliation, no? If the angle at which the particle flicked was rational, then the leaf space is just the circle S^1. But if the angle is irrational then the orbits are no longer diffeomorphic to the circle S^1 but are instead diffeomorphic to the real line, and the leaf space is now singular! Then in order to say anything interesting about the leaf space one must turn to noncommutative geometry (von Neumann algebras)
Leaf spaces in classical mechanics seem a natural object to study since let's say that the particle started on a uniformly-picked random leaf of the foliation and we want to know the probability that the particle is on a given subset of leaves. Then we measure how much of the leaf space that subset of leaves takes up to get the probability that the particle is on that subset of leaves. In the irrational angle case, the leaf space is singular and so the question cannot be answered with the classical measure theory of the leaf space. But it still seems like a very physical question to ask, so it seems to me that this is a natural manifestation of noncommutative geometry in classical physics.
If this really is an example of noncommutative geometry appearing in classical physics, what are some other examples? And any other comments on this topic are appreciated.