Algebraic Curves and Phase Diagrams of Physical Systems

Lots of low degree curves arise naturally as the phase spaces of physical systems (that is, the curve parameterized by $(q,p)$ where $q$ is a generalized position variable and $p$ is a generalized momentum variable (such that $p=\dot{q}$, etc).

For instance, if degree is 1, then we can construct the curve as the phase space of a particle moving with constant velocity, and it is parameterized by $(x,0)$, so is given by the equation $y=0$, up to a choice of coordinates.

For $d=2$, a conic, we can use, for instance, a spring, whose position is $\cos(x)$, and so $(\cos x,\sin x)$ parameterizes a circle (ellipses are also fairly easy to see how to do this way, and parabolas are linearly accelerated particles, though I don't see hyperbolas immediately)

For $d=3$, it is well known that a simple pendulum has phase space an elliptic curve with roots the initial height, the length of the pendulum, and minus the length.

Are there other nice examples like this? Is there a natural physical system that realizes hyperbolas? Does every real elliptic curve arise in this way? How about quartic or higher curves? I assume there are physical systems that work, as every ODE is modeling SOME system, but are there well-known examples that arise naturally in physics?

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 Oh, I'd also be interested in natural physical systems whose phase curve consists of irreducible singular curves. – Charles Siegel Nov 4 2010 at 3:42 I'd written up a long explanation here of algebraic curves in phase diagrams in the sense of "boundaries in a parameter space separating phases of matter" before I realized that was not really what you were after. – jc Nov 4 2010 at 9:48

The phase space of Kovalevskaya's top is an Abelian surface. If you fix some natural invariant you usually get a curve of small genus. If memory serves right, fixing the energy gives you a genus 2 curve.

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Does the group law play nicely with the physics on the surface? – Charles Siegel Nov 5 2010 at 12:35
@Charles: sadly I can only answer from my sieve-like-memory, but to some extent yes - see e.g. onlinelibrary.wiley.com/doi/10.1002/… If I recall correctly the fact that the Abelian surface in question is not principally polarized was a huge issue for years, because people tried various principally polarized ones, which all give "equally good" invariant, but not the ideal ones (it's detailed in the linked paper). – David Lehavi Nov 5 2010 at 13:52

You get (approximately) elliptic trajectories in 2D phase space whenever you are close to a nondegenerate stable equilibrium. This is one way to explain why harmonic oscillators are so universal. Similarly, you get hyperbolic trajectories when you are close to a nondegenerate unstable equilibrium. The standard example is a ball on top of a smooth frictionless hill or an upside-down pendulum, in the limit of small deviations.

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I have a nice example concerning the electric field in two dimensions, which I'm not sure if physicists know about. One can easily phrase it in terms of ODE.

The electric field lines of a system of point charges in two dimensions form a pencil of (real) algebraic curves, the degree of which equals the sum of (the absolute values of) the charges (which have to be integers to begin with, and the system only has finite number of charges). You get a lot of singular ones in particular, and irreducible singular ones.

I'd be happy to see a reference of it. It's so natural that one would expect that the 19th century physicists/mathematicians know about them.

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 This is pretty, but could you explain the connection to trajectories in a phase space of position and momentum? – jc Nov 4 2010 at 15:35