The KAM theory of systems with short range interactions. I (Q1070614)

The Kolmogorov, Arnol'd, Moser (KAM) theory proves that small perturbations of integrable Hamiltonian systems possess a ''large'' set of initial conditions for which the trajectories remain quasiperiodic. Classical estimates for a general analytic perturbation of strength \(\epsilon\) require \(\epsilon <C(N!)^{-\alpha}\) to ensure that the KAM applies, where N is the number of degrees of freedom. The author investigates the Hamiltonian systems consisting of long chains of nearly identical subsystems, with interactions which decay rapidly with increasing distance between the interacting components. In this case the existence of the quasiperiodic trajectories is proved for \(\epsilon <CN^{-160}\). The results are stated and partly proved.