# 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?

• 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