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If a Hamiltonian system in $\mathbb{R}^{2n}$ has $n$ suitable first integrals, then it is called an integrable system, and the Arnold-Liouville theorem tells us all sorts of nice things about the system: In particular, if a flow is compact then the flow takes place on a torus $T^n$.

What means are there to show that a system, such as the three-body problem, is not integrable? Is there a generalisation of Arnold Liouville for these systems?

Jonathan Michael Foonlan Tsang
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    KAM theorem tells us how the nice properties of integrable systems are (or are not) preserved as you add a bit of perturbation. – nonlinearism May 27 '14 at 13:53

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There is something called Morales–Ramis theory which is (I've been told) the most powerful method for proving nonintegrability. There are preprint versions of various articles and even of a book (Differential Galois Theory and Non-integrability of Hamiltonian Systems) on the webpage of Juan Morales-Ruiz: http://www-ma2.upc.edu/juan/.

EDIT (Nov 2019): The old link is dead, but he has a page on ResearchGate instead: https://www.researchgate.net/profile/Juan_Morales-Ruiz.

Hans Lundmark
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