Investigation of nonlinear Hamiltonian systems by the normal form of the Birkhoff-Gustavson

Abstract

Results the paper are obtained with both classical and quantum approaches to autonomic 2D Hamiltonian system. Even if such system has negative Gaussian domains, the classical approach shows no dynamical chaos. For certain values of classical Hamilton function parameters, the motion integral is found. Direct numerical calculations for Poincare sections confirm the results. The classical normal Birkhof-Gustavson form and its quantum analogue according to Weyl rule are derived. On the base of this quantum analogue, the approximate formulas are obtained for evaluation of energy spectrum of the lowest states for the local minimum situated at the origin.