Abstract. 
KAM theory seeks quasi-periodic solutions in
dynamical systems.

When the systems preserve a geometric structure
(symplectic, volume, conformal symplectic) or
admit a variational formulation, there are
some identities.

We use these identities a) to obtain very
efficient algorithms (they are quasi-newton
steps, with small storage or operation counts)
b) to obtain "a posteriori" theorems
which say that near computed solutions (which
have reasonable condition numbers) there are
true solutions.

The algorithms have been implemented and they lead to
some conjectures about breakdown.

Joint work with: R. Calleja, E. Fontich, A. Gonzalez-Enriquez,
G. Huguet, Y. Sire