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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.

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I'm wondering where your interest in Painleve VI comes from. Frobenius manifolds, perhaps? Is this question related to anything else, or is it just something you're wondering about? –  Kevin H. Lin Dec 10 '09 at 0:37
    
Well, the question of physical systems was just idle curiosity. My interest in Painleve VI is coming from a couple of papers by Boalch (one of which I linked to below) that talk about it, irregular connections, etc, and some relations of that stuff to Higgs bundles and the Hitchin System –  Charles Siegel Dec 10 '09 at 0:44
    
You know,this is a good question and I never thought of it before. I suppose in principle,sure-but it's kind of like building an amusement park on the ocean floor that people couldn't actually use.In principle,you certainly could.But why would you? –  Andrew L Aug 13 '10 at 22:04
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7 Answers

up vote 14 down vote accepted

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.

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From the setup, it seems like it can only handle certain types of nonlinearity, and it seems with constant coefficients. The specific equation I'm looking at is on page three of this paper: arxiv.org/pdf/math/0308221v3 and it seems nastier than any of the examples by a rather large amount. It might still work, but I don't see it. –  Charles Siegel Dec 10 '09 at 0:11
    
I just pointed you to the simplest example of how to derive a circuit from an ODE, but sophisticated analog computers can add, subtract, multiply, integrate, differentiate, and even divide (using logarithms), so your differential equation is in the closure of this set of possible operations, apart from the singularities. If you expect to pass through the singularities then you'll probably have to transform coordinates and it might get messy. –  Dan Piponi Dec 10 '09 at 0:19
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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.

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I'm having trouble putting into words quite what I'm looking for, but I think it's roughly an algorithm to generate a "physical" system (in physics language, not just a Hamiltonian system) from the ODE. –  Charles Siegel Dec 10 '09 at 0:12
    
I think I understand what you mean, even though I agree it is difficult to formalise. My guess is that not every ODE arises out of a "physical" (in your sense) system. –  José Figueroa-O'Farrill Dec 10 '09 at 0:18
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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).

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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.

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For particular parameters, Painlevé VI describes the spin-spin correlations of the square lattice Ising model with diagonal separation between spins. (No scaling limit has been taken.) See "Studies on holonomic quantum fields, XVII". Michio Jimbo and Tetsuji Miwa, Proc. Jpn. Acad., Ser. A, Vol. 56, 405-410 (1980). A crucial erratum appears as an appendix to "The $\tau$ function of the Kadomtsev-Petviashvili equation transformation groups for soliton equations, I". Masaki Kashiwara and Tetsuji Miwa, Proc. Jpn. Acad., Ser. A, Vol. 57, 342-347 (1981). –  Will Orrick Dec 10 '09 at 1:24
    
Nice find, though I don't have access. Amazing that the series of papers goes all the way from I to XVII. –  j.c. Dec 10 '09 at 1:46
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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.

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I'd argue that the notion of "physical modeling" is independent of whether a physical model can be realized in the real world. After all, physicists talk about infinite system sizes all the time, or idealized behavior of particles and matter in regimes which can never be tested. From Charles's question, it seems he's willing to accept at least some degree of artificiality in the models. –  j.c. Dec 10 '09 at 1:39
    
In that case, I suppose one can always construct a device in an infinite-volume universe in which information propagates at infinite speed (and there is no gravity). –  S. Carnahan Dec 10 '09 at 1:47
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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.

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If you're going to do that then we could just use zeta function universality, and the fact that it is known how to build an analogue computer to compute the zeta function, to claim we can solve any differential equation. :-) –  Dan Piponi Dec 10 '09 at 2:27
    
I am not sure it would be enough. Indeed I suspect that the generic differential equation (in any reasonable sense) is an universal differential equation. There are results which ensures that any germ of holomorphic map f:(0)(0) can be approximated by a holonomy map of a rational differential equations of the form dydx=P(xy)Q(xy) . See "The generic rational differential equation dw/dz=P(z,w)/Q(z,w) on $\mathbb{C}\mathbb{P}^2$ carries no interesting transverse structure" by Belliart, Liousse, and Loray. Erg. Theory and Dyn. Syst., 21 (2001), p.1599-1607. –  jvp Dec 10 '09 at 2:55
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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.

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There are physical situations where uniqueness of solutions does not hold because of symmetry breaking, for example, the buckling of a beam under the force directed along its axis. –  Victor Protsak Aug 14 '10 at 4:44
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