Birkhoff normal form for the periodic Toda lattice

with potential V (x) = γe + V1x + V2 and γ, δ, V1, V2 ∈ R constants. The Toda lattice has been introduced by Toda [12] and studied extensively in the sequel. It is an FPU lattice, i.e. a Hamiltonian system of particles in one space dimension with nearest neighbor interaction. Models of this type have been studied by Fermi-Pasta-Ulam [FPU]. In numerical experiments they found recurrent features for the lattices they considered. Despite an enormous effort from the physics and mathematics community, some of these numerical experiments still defy an explanation. For a recent account of the fascinating history of the FPU problem, see e.g. [1]. At least in the case of the periodic Toda Supported in part by the Swiss National Science Foundation Supported in part by the Swiss National Science Foundation, the programme SPECT and the European Community through the FP6 Marie Curie RTN ENIGMA (MRTN-CT-20045652) 2000 Mathematics Subject Classification: 37J35, 37J40, 70H08