# Integrable dynamical system - relation to elliptic curves

From seminar on kdV equation I know that for integrable dynamical system its trajectory in phase space lays on tori. In wikipedia article You may read (http://en.wikipedia.org/wiki/Integrable_system):

When a finite dimensional Hamiltonian system is completely integrable in the Liouville sense, and the energy level sets are compact, the flows are complete, and the leaves of the invariant foliation are tori. There then exist, as mentioned above, special sets of canonical coordinates on the phase space known as action-angle variables, such that the invariant tori are the joint level sets of the action variables. These thus provide a complete set of invariants of the Hamiltonian flow (constants of motion), and the angle variables are the natural periodic coordinates on the torus. The motion on the invariant tori, expressed in terms of these canonical coordinates, is linear in the angle variables.

As I also know that elliptic curve is in fact some kind of tori, then there natural question arises: Are tori for quasi-periodic motion in action-angle variables of some dynamical systems related in any way to algebraic structure like elliptic curve? Maybe some small dynamical systems and some elliptic curves are related in some way?

The most interesting in this matter is for me the size of space of elliptic functions: its quite small, every elliptic curve is rational function of Weiestrass function, and its derivative. Has this property any analogy in integrable dynamical systems theory?

As isomorphic elliptic curves shares some invariants, it is also interesting it they have any "dynamical meaning".

-
This is a great question, but I do want to point out that KdV is an infinite-dimensional integrable system, so I am not sure if the tori interpretation makes sense in this setting. In particular, KdV admits infinitely many conservation laws! Drazin and Johnson's "Solitons: an introduction" expands on this. I realize this has no real bearing on the question on hand. –  Justin Curry Feb 8 '10 at 17:19
As You are right, kdV has infinitely many conservation laws, it also is quasi-periodic motion! So probably it may be case of "infinite dimensional" tori, which obviously is not related to elliptic curves. I am physicist by education ( but not working as scientist, I treat math as fun and hobby), so kdV is something which is example of integrable system with nontrivial equations of motion. –  kakaz Feb 8 '10 at 19:00

If your system is algebraic, then you bet! More generally, you can get abelian varieties as the fibers for many interesting integrable systems. Google the following for more: algebraic complete integrable Hamiltonian system, Calogero-Moser System, Hitchin System.

As for elliptic curves, they'll only pop out in low dimensional cases, because otherwise, the fibers have to have larger dimension.

As for the latter, it depends what you might want. I've seen the definition of integrable given by "can be solved by a sequence of quadratures" and in this terminology, you can check that an algebraic system you're always working with the global section of the theta function on the abelian variety, which is the unique (up to scaling) global section of the theta divisor on the abelian variety, which for an elliptic curve, is just the Weierstrass function.

-
If in low dimensional case, elliptic curves appear in this question, is it true that curves which are isomorphic arises in equivalent algebraic dynamical systems? Is this kind of homomorphism between structures? –  kakaz Feb 8 '10 at 16:31
Well, really what's going on is that you have families of elliptic curves. I'm not an expert, really, but isomorphic families give equivalent integrable systems, and I'm not sure about what notion of equivalent systems you're using (and even if I did, I'm not sure I'd be able to say much). –  Charles Siegel Feb 8 '10 at 17:30
I thought on case when moving from one curve to another in equivalence class gives for example change of variables in equation of motion or something like that - some kind of reparametrization which is not quite formal but also has some kind of "physical meaning". –  kakaz Feb 8 '10 at 19:03
Perhaps. I don't know enough to say for certain. –  Charles Siegel Feb 8 '10 at 20:08

"The de-geometrisation of mathematical education and the divorce from physics sever these ties. For example, not only students but also modern algebro-geometers on the whole do not know about the Jacobi fact mentioned here: an elliptic integral of first kind expresses the time of motion along an elliptic phase curve in the corresponding Hamiltonian system. "

From A.I.Arnold, here: http://pauli.uni-muenster.de/~munsteg/arnold.html Definitely I should learn more in this area....

-