Physical and chemical aspects of the study of clusters, nanostructures and nanomaterials
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Normalization of classical hamiltonian systems with two degrees of freedom

I.N. Belyaeva1, I.K. Kirichenko2, O.D. Ptashnyi3, N.N. Chekanova4, T.A. Yarkho3

1 Belgorod National Research University
2 National University of Civil Defence of Ukraine
3 Kharkiv National Automobile and Highway University
4 V. N. Karazin Kharkiv National University

DOI: 10.26456/pcascnn/2020.12.348

Original article

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

  • Irina N. Belyaeva – Ph. D, Docent, Department of Computer Science, Natural Sciences and Teaching Methods, Belgorod National Research University
  • Igor K. Kirichenko – Dr. Sc., Professor, Department of physical and mathematical sciences, National University of Civil Defence of Ukraine
  • Oleg D. Ptashnyi – Ph. D, Docent, Department of Higher Mathematics, Kharkiv National Automobile and Highway University
  • Natalia N. Chekanova – Ph. D, Docent, Department of Information Technologies and Mathematical Modeling, Educational and Scientific Institute «Karazin Banking Institute», V. N. Karazin Kharkiv National University
  • Tatyana A. Yarkho – Dr. Sc., Docent, Department of Higher Mathematics, Kharkiv National Automobile and Highway University

Reference:

Belyaeva, I.N. Normalization of classical hamiltonian systems with two degrees of freedom / I.N. Belyaeva, I.K. Kirichenko, O.D. Ptashnyi, N.N. Chekanova, T.A. Yarkho // Physical and chemical aspects of the study of clusters, nanostructures and nanomaterials. – Tver: TSU, 2020. — I. 12. — P. 348-355. DOI: 10.26456/pcascnn/2020.12.348. (In Russian).

Full article (in Russian): download PDF file

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