Classical theory of nonlinear vibrational systems; local modes

Abstract

In this Chapter we review briefly the basics of nonlinear Hamilton systems with a finite number of degrees of freedom. The representation in the variables ‘action-angle’ is introduced for harmonic and Morse oscillators. The perturbation theory, based on the method of canonical transformation is provided for Hamiltonians in the variables action-angle. The method of canonical transformation is closely related to the method of averaging, which has a clear physical meaning. The problems in the perturbation approaches due to small denominators are described. The theorem of Kolmogorov - Arnold- Moser (KAM) on the stability of trajectories for integrable Hamilton systems under the action of small perturbations is formulated. The KAM theorem is applied to the study of vibrations in the system of nonlinear weakly interacting oscillators.