Thursday, February 14, 2008

skaters and flying saucers

This one isn’t about Lent; it’s about math. I’m going to describe some excellent examples that came up in Thurston’s seminar this morning. To do so, I’m going to have to assume some pretty sophisticated manifold theory, so this isn’t really a “general interest” post. For those who have some knowledge of manifolds, however, these examples may prove enlightening.

We were talking about different kinds of admissible structures on a manifold $M$—specifically, what kinds of local homeomorphisms can be used to define structures. The last two we discussed were symplectic and contact structures. The former is fairly well-known: you have a non-degenerate, closed 2-form on $M$ that lets you do things like Hamiltonian mechanics. Thurston’s examples of symplectic transformations were flinging a chair around (position + momentum of an object gives a manifold, called the phase space of the object, with a canonical symplectic form) and light passing through lenses (presumably also a phase space-type manifold, but my grasp of Hamiltonian mechanics is relatively weak, having been acquired almost entirely in symplectic geometry classes).

Contact manifolds are less well-known. They’re often brought up to illustrate a “sister” geometry to symplectic geometry: symplectic manifolds are always even-dimensional, while contact manifolds are always odd-dimensional. Here’s the definition: a contact manifold $M$ has a non-degenerate 1-form $a$ whose exterior derivative $da$ is non-degenerate on the kernel of $a$ in each tangent space to $M$. Okay, flung that all out there at once. Here’s the geometry: since a 1-form restricts to a linear functional on each tangent space, its kernel (if it is non-degenerate) is a codimension 1 subspace of the tangent space—i.e., a hyperplane. So a contact manifold has a special collection of “tangent hyperplanes”, and the condition on $da$ tells in what way this field of hyperplanes is special. Here I’m not really interested in the technical reasons this definition is chosen. I just want to give the examples Thurston described.

The first was of an ice skater. On a skating rink, one has both position and direction: that gives a three-dimension manifold $M$, which can be thought of as (rink)x(circle of directions), or can be unfolded into $R^3$ if, as Thurston put it, you keep track of the winding number of the skater. However, at a given position and direction, you can’t move arbitrarily in $M$; the skate can move forward and backward in the direction it’s facing, or it can change direction. So you have a hyperplane $H_x$—i.e., a plane—in the tangent space to $M$ at $x$, which describes these possibilities of movement: one direction in $H$ points in the direction you’re currently facing, and one points in the “direction of changing direction”. This is a contact manifold. Skating a path around the ice rink means tracing a curve in the contact manifold that always remains tangent to the hyperplanes.

The second example shows how the first might be generalized. Suppose you have a jet, or a flying saucer, which can be at any point in $R^3$ and can take any orientation at any point, but can only move to a different point in a direction of the plane of its current orientation. The position-orientation manifold is a product of $R^3$ and $RP^2$ (the real projective plane)—or $S^2$ if you keep track of which way is “up” for the flying saucer—and hence 5-dimensional. The contact structure at a particular pair (position,orientation) is the product of the plane in $R^3$ corresponding to the current orientation and the tangent space to $RP^2$/$S^2$, which corresponds to the fact that you can roll either up-and-down or side-to-side.

Thurston went on to remark that many physical systems with some sort of dynamical possibilities can be described by contact structures, and the dynamics of the system are represented by diffeomorphisms that preserve the contact structure. I know more can be found in his book Three-Dimensional Geometry and Topology. I’m going to have to go look that book up soon.

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