Non-integrable systems 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? 

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.
