I’ve lately been doing some research in the general area of geometric PDEs inspired by the intricate theory of minimal and constant mean curvature surfaces. This has given me the chance to apply differential-geometric techniques to problems which I used to believe could only be approached analytically. To introduce some of these ideas, I had started to write a post on the parallel parking problem — but I got scooped by that sneaky upstart blogger Charles over at Rigorous Trivialities!
Still, I think parallel parking is a great way to starting thinking about the geometry which governs differential equations. So let’s go down the rabbit hole…
To understand how we might think of a differential equation geometrically, let’s take a trip back to Calculus 1. We often give students problems like “given such-and-such a graph for , sketch a graph of “. Hopefully, we remember to give the students a new set of axes to draw this graph on — if maps times to positions (say, to ), then maps times to velocities (say, to for “pmomentum”). So rather than draw a pair of graphs to represent both
we could combine them into a single curve in :
This curve is called the graph of the 1-jet of , and it lets us easily read off the values of both and .
Of course, already was totally determined by , so it must be that only some of the curves that we draw in -space correspond to graphs of 1-jets of functions. Can we figure out exactly which ones come from functions?
Sure, it is easy enough: a curve is going to be the graph of a 1-jet of a function exactly when
Manipulating this expression (yes, Virginia, there is a quantity ), we get
This differential form is usually denoted and is called the contact form on . A curve corresponds to the graph of a 1-jet exactly when . We can visualize this by drawing a field of planes in -space which correspond to the kernel of (sorry for the bad planefield):
looking down the p-axis at the (x,y)-plane:
looking down the x-axis at the (p,y)-plane:
Now we can translate the problem of parallel parking into a question about moving around in this space. I’m going to deal with parallel parking a unicycle here so that we don’t have to get too deep into modelling a car’s motion — the result is the same, but would involve twisting our planefield somewhat. Now, suppose our unicycle is sitting at the origin of the (x,y)-plane and pointing down the x-axis. In other words, we are at the origin of -space. At any given time, we can either rotate ourselves (changing our coordinate) or we can move in the direction we are pointing. The first motion corresponds to the vector
while the second corresponds to
You can easily verify that and = 0, so our unicycle is nicely adapted to the planefield we drew above: from any point, we may freely move in the whatever directions are along the planefield at that point.
To parallel park our unicycle, we want to move a big distance in the y-direction while only moving an infinitesimal distance in the -plane. That is to say, we want to move sideways without bumping into the nearby parked unicycles and without turning our unicycle very much from the horizontal. Now, a puzzle for the reader. By looking at that planefield can you figure out how to move yourself up the y-axis without moving more than a tiny distance away from it?
So if you want to parallel park but you don’t want to think about group theory, you can just imagine yourself happily trying to move vertically in a contact field. Easy as could be!
Now, what does all of this have to do with differential equations? Well, let us take the most general sort of first order ODE that we could think of. This is going to be some equation involving , , and so it gives us a variety
in -space. For example, the differential equation corresponds to the spherical variety . Any curve which is the 1-jet of a solution must satisfy this equation, in addition to the equation which says that came from an honest function. So now we can break the problem of solving the ODE into two different parts: find all of the “formal” solutions with , then try to cull these formal solutions down to only the honest / holonomic / ones.
Let us see how these ideas can be used to more easily prove things about solutions to some differential equation. Though we have only talked about ODEs in one variable so far, the whole theory works for arbitrary PDEs as well. The central problem is this: suppose we can easily find formal solutions to our differential equation. How can we promote these formal solutions to actual holonomic solutions?
One thing we could try is to flow along the surface in such a way that the “energy” decreases as quickly as possible. This gives us a gradient descent on the space of formal solutions to our differential equation. If we are lucky, we might even be able to show that every formal solution will eventually go to a global minimum of this energy — a point where . Question for the readers: does this particular idea have a name? Is it studied?
Another thing we could try, similar to but slightly weaker than the last idea, is to take formal solutions and deform them continuously to real solutions. This is the idea behind Gromov’s h-principle. Differential equations (or differential inequalities) which satisfy the h-principle have the wonderful property that every formal solution is homotopic to an honest solution. The h-principle is a vast generalization of Smale’s proof of the sphere eversion phenomenon. More about this soon…
Closely related to parallel parking and stronger than just the h-principle, there is also the holonomic approximation property. Scroll back up and look at that contact field again. Now imagine taking any curve drawn in . Using the parallel parking example as inspiration, can you see how to approximate the curve arbitrarily well (in the topology) by a curve which stays tangent to the contact field? Congratulations, you just proved that curves in have the holonomic approximation property: they are all arbitrarily close to holonomic curves! This means that if you have an open ODE (defined by an open subset of then every formal solution is -close to a holonomic solution of the equation — a very strange idea, considering how hard it is to come by solutions to differential equations. In particular, this means that if you take an ODE and replace it with the differential inequality , every formal solution to the ODE can be perturbed to an honest solution (warning: there is a subtlety involving the dependent variable being swept under the rug). So the holonomic approximation principle which you proved when you learned to parallel park means that you know how to -solve any ODE!
Coming up in the not-too-distant future: what all this has to do with sphere eversions, symmetry, and the geometrization conjecture…