What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "integrable" and "chaotic"? (There is an interesting Wikipedia article, but I don't find it completely satisfying.)
Update (Dec 2010): Thanks for the many excellent answers. I came across another quote from Nigel Hitchin:
"Integrability of a system of differential equations should manifest
itself through some generally recognizable features:
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the existence of many conserved quantities
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the presence of algebraic geometry
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the ability to give explicit solutions.
These guidelines would be interpreted in a very broad sense."
(If there are some aspects mentioned by Hitchin not addressed by the current answers, additions are welcome…)
Closely related questions:
Best Answer
This is, of course, a very good question. I should preface with the disclaimer that despite having worked on some aspects of integrability, I do not consider myself an expert. However I have thought about this question on and (mostly) off.
I will restrict myself to integrability in classical (i.e., hamiltonian) mechanics, since quantum integrability has to my mind a very different flavour.
The standard definition, which you can find in the wikipedia article you linked to, is that of Liouville. Given a Poisson manifold $P$ parametrising the states of a mechanical system, a hamiltonian function $H \in C^\infty(P)$ defines a vector field $\lbrace H,-\rbrace$, whose flows are the classical trajectories of the system. A function $f \in C^\infty(P)$ which Poisson-commutes with $H$ is constant along the classical trajectories and hence is called a conserved quantity. The Jacobi identity for the Poisson bracket says that if $f,g \in C^\infty(P)$ are conserved quantities so is their Poisson bracket $\lbrace f,g\rbrace$. Two conserved quantities are said to be in involution if they Poisson-commute. The system is said to be classically integrable if it admits "as many as possible" independent conserved quantities $f_1,f_2,\dots$ in involution. Independence means that the set of points of $P$ where their derivatives $df_1,df_2,\dots$ are linearly independent is dense.
I'm being purposefully vague above. If $P$ is a finite-dimensional and symplectic, hence of even dimension $2n$, then "as many as possible" means $n$. (One can include $H$ among the conserved quantities.) However there are interesting infinite-dimensional examples (e.g., KdV hierarchy and its cousins) where $P$ is only Poisson and "as many as possible" means in practice an infinite number of conserved quantities. Also it is not strictly necessary for the conserved quantities to be in involution, but one can allow the Lie subalgebra of $C^\infty(P)$ they span to be solvable but nonabelian.
Now the reason that integrability seems to be such a slippery notion is that one can argue that "locally" any reasonable hamiltonian system is integrable in this sense. The hallmark of integrability, according to the practitioners anyway, seems to be coordinate-dependent. I mean this in the sense that $P$ is not usually given abstractly as a manifold, but comes with a given coordinate chart. Integrability then requires the conserved quantities to be written as local expressions (e.g., differential polynomials,...) of the given coordinates.