Symmetry and resonance in periodic FPU chains

The symmetry and resonance properties of the Fermi Pasta Ulam chain with pe riodic boundary conditions are exploited to construct a near-identity trans formation bringing this Hamiltonian system into a particularly simple form, This "Birkhoff-Gustavson normal form" retains the symmetries of the origin al system and we show that in most cases this allows us to view the periodi c FPU Hamiltonian as a perturbation of a nondegenerate Liouville integrable Hamiltonian. According to the KAM theorem this proves the existence of man y invariant tori on which motion is quasiperiodic. Experiments confirm this qualitative behaviour. We note that one can not expect this in lower-order resonant Hamiltonian systems, So the periodic FPU chain is an exception an d its special features are caused by a combination of special resonances an d symmtries.