Does every ODE comes from something in physics? - MathOverflow

Does every ODE comes from something in physics?

2009-12-09T23:38:58Z

Not sure if this is appropriate to Math Overflow, but I think there's some way to make this precise, even if I'm not sure how to do it myself.

Say I have a nasty ODE, nonlinear, maybe extremely singular. It showed up naturally mathematically (I'm actually thinking of Painleve VI, which comes from isomonodromy representations) but I've got a bit of a physicist inside me, so here's the question. Can I construct, in every case, a physical system modeled by this equation? Maybe even just some weird system of coupled harmonic oscillators, something. There are a few physical systems whose models are well understood, and I'm basically asking if there's a construction that takes an ODE and constructs some combination of these systems that it controls the dynamics of.

Any input would be helpful, even if it's just "No." though in that case, a reason would be nice.

Answer by Dan Piponi for Does every ODE comes from something in physics?

2009-12-09T23:54:49Z

It is possible to solve a large class of ODEs by means of analog computers. Each of the pieces of the differential equation corresponds to an electronic component and if you wire them up the right way you get a circuit described by the ODE. Wikipedia has lots of information on the subject and a link like this one gives explicit examples of circuits. It's not hard to build circuits for things like the Lorenz equation and see a nice Lorenz attractors on an oscilloscope display.

Answer by José Figueroa-O'Farrill for Does every ODE comes from something in physics?

2009-12-10T00:08:47Z

This may not be the answer that you are looking for, but I believe that you should be able to write the Painlevé VI equation as a hamiltonian system, in which case it would govern the dynamics of some "physical" system. The reason for the double quotes is that this is perhaps not a system arises in nature. Most likely -- although I don't know for sure -- it will not be just coupled harmonic oscillators.

Less directly, Painlevé equations arise in the study of integrable hierarchies, some of which (e.g., KdV, nonlinear Schrödinger,...) are used to model natural phenomena.

Edit

An explicit form for the Hamiltonian of Painlevé VI can be found here, right after Theorem 2.1. Although it is polynomial, it does depend explicit on 'time'. Hence as a hamiltonian system it is certainly not very natural. For one thing, energy is not conserved. This is to be expected, since Painlevé VI is itself not integrable, which it would have to be if you could find a conserved quantity as it is a one-dimensional system.

Answer by jc for Does every ODE comes from something in physics?

2009-12-10T00:16:24Z

A fairly silly answer is that the answer is obviously "Yes", since one can build a computer to integrate numerical solutions to your ODE. (now that I think about it, Sigfpe's answer is essentially the same as mine.)

Going along these lines I guess one can find more "physical" models of my suggestion (in the sense that doing physics is often finding toy models that contain the essence of the phenomena etc) by proposing various lattice models or cellular automata which are known to have universal constructors. Or by designing circuits made out of balls and springs.

I spent a little bit of time trying to put down the right words which would make your question more precise and more in line with your intent, but I think ultimately it boils down to what kind of physical models you'd be satisfied with.

As much of physics can be described in terms of ODEs, any sufficiently powerful type of model is going to contain the sort of answer I described above. I think the right question is what's the "simplest" (or perhaps "weakest") known physical model for Painlevé VI.

One kind of answer to that question would be finding a physical system for which some solution of Painlevé VI gives some physically measurable function - along these lines, I know that Painlevé functions are highly useful in various integrable models / lattice models, e.g. famously, the spin-spin correlation function in the 2D Ising model in the scaling limit is a solution to Painlevé III - thus, my guess is that Painlevé VI shows up in one of these contexts, but the literature is pretty vast.

Answer by S. Carnahan for Does every ODE comes from something in physics?

2009-12-10T01:27:53Z

I vote no. Let n be a large enough positive integer that larger numbers can't be reasonably written using primitive recursion. Take a generic nonlinear ODE involving about n terms with derivatives of order around n. I'd claim that this doesn't model anything physical, and it can't be integrated by any device that fits in the universe.

If you demand that the ODE be reasonably small (e.g., mathematically interesting), then it tautologically models the behavior of a device set up to integrate it. I don't really have an answer to the broader question of why certain differential equations show up in areas of mathematics close to physics where you don't really expect them.

Answer by David Speyer for Does every ODE comes from something in physics?

2009-12-10T02:18:36Z

This is very far from my expertise, but I believe that the point of Universal Differential Equations is that any behavior exhibited by any differential equation can be found in a UDE for certain choices of parameter. (Think by analogy to a universal Turing machine, which can mimic any Turing machine.) So, if you are willing to idealize a lot, it would be enough to find a physical model for a UDE.

Answer by jvp for Does every ODE comes from something in physics?

2009-12-10T02:37:53Z

For the general question I vote no. As you formulated I see it, potentially, as a question of differential algebra. Although there is no definition of physical proccess I imagine it can be formalised through constraints on the field extensions allowed to solve your equation. The prototypical result I have in mind is Liouville's Theorem.

If instead you specialize the general question to Painleve's equation then I would bet that the answer is yes. Painlevé equation is one of these ubiquitous objects in Mathematics and I would not be surprised if it models a physical phenomena. I already crossed with a Springer Lecture Notes relating it to the geometry of surfaces.

As a side remark let me notice that Painlevé's equations were originally found not as equations governing isomonodromic deformations but instead as non-linear second order equations having the so called Painlevé property(absence of movable singular points).

Answer by Francis for Does every ODE comes from something in physics?

2010-08-13T12:23:25Z

No. Let us take an ODE with a blowing up solution (x'=x^2,x(0)=1) o let us take an ODE without uniqueness of solutions (x'=x^(1/3),x(0)=0). There is no possible physical interpretation for such kinds of equations. Obviously, the addition of new terms in such equations can yield a physically implementable problem, but in such a case the resulting ODE is not the same as the original one.