NORMALIZATION OF CLASSICAL HAMILTONIAN SYSTEMS WITH TWO DEGREES OF FREEDOM

Abstract

The family of the Hamiltonian systems with two degrees of freedom was investigated. The calculations of the Poincaré sections show that with arbitrary values of the parameters of the Hamilton function, the system is non-integrable and dynamic chaos is realized in it. For the three parameter sets, the system in question was found to be integrable, but shows that in one integrable case on the potential energy surface (PES) there are regions with the negative Gaussian curvature. It was found that in one integrable case for the same values of the parameters, the potential energy surface has a region with the negative Gaussian curvature. At the same time, in the other two cases, the domains with negative Gaussian curvature are not integrable for the corresponding values of the parameters. Thus, the presence of regions with negative Gaussian curvature on the potential energy surface is not enough for the development of the global chaos in the system. The classical normal form for arbitrary parameter values is obtained.

Keywords

• hamiltonian systems
• normalization
• computer modeling
• potential energy surface