Yesterday (July 26) I gave a talk at “Poisson 2010” at IMPA. The talk was about the entropy of Poisson manifolds, normal forms (of Hamiltonian systems on Poisson manifolds), resonances, and (non)integrability, and was dedicated to the memory of JJ Duistermaat (who is well-known for Fourier operators and symplecitc geometry, but also did a pioneering work on the theme “resonance implies non-existence of first integrals”) and VI Arnold (of Arnold-Liouville-Mineur theorem about action-angle variables).

The following theorem is in the spirit of “resonance implies non-existence of first integrals”, and is in contrast with the symplectic case (where generic Hamiltonian systems are formally integrable, by Birkhoff):

If g is a simple Lie algebra of rank $n$ and H is a generic function on g^* (with the linear Poisson structure), then the Hamiltonian system of H on g^* has only n+1 formal first integrals (the first n come from a Cartan subalgebra of g after a normalization obtained by Monnier-Z, and the last one is H itself). In other word, any other 1st integral will be functionally dependent of these n+1 functions and the Casimir functions of g. As a corollary, when g is big enough then the system is not formally integrable (because the dimension of the symplectic leaves will be greater than 2(n+1) ).

The significance of this theorem is that it shows that “Symmetry leads to Non-Integrability” !!! (A system on g^* is usually a reduced system with G-symmetry). Sounds a bit surprising, because we usually think that symmetry leads to integrability, by Noether.

Why are you surprise? I mean one does not need to find first integrals in order to solve an ODE by quadrature. Liouville theorem is a red-herring: it tells one to look for first integrals while integrating the eqs is what should be pursued.

Lie symmetries are the well-known unknown…