Symmetry and resonance in periodic FPU chains

Abstract

The symmetry and resonance properties of the Fermi Pasta Ulam chain with periodic boundary conditions are exploited to construct a nearidentity transformation bring ing this Hamiltonian system into a particularly simple form This BirkhoGustavson normal form retains the symmetries of the original system and we show that in most cases this allows us to view the periodic FPU Hamiltonian as a perturbation of a nonde generate Liouville integrable Hamiltonian According to the KAM theorem this proves the existence of many invariant tori on which motion is quasiperiodic Experiments conrm this qualitative behaviour We note that one can not expect it in lowerorder resonant Hamiltonian systems So the FPU chain is an exception and its special fea tures are caused by a combination of special resonances and symmetries