Poincare sections for two fixed centers problem and Henon-Heiles potential

Abstract

In this paper, we study the Henon-Heiles potential and the problem of two fixed centers. In studies of nonlinear systems for which  exact solutions are unknown, the Poincare section method is used. For the Henon-Heiles potential, Poincare sections were obtained. At low energies, the Henon-Heiles system looks integrable, since independently of the initial conditions, the trajectories obtained with the help of numerical integration lie on two-dimensional surfaces, i.e. as if there existed a second independent integral. Next, the potential of two fixed centers was investigated.  It was shown on the basis of the Poincare section that, in the case μ1 = μ2 = 1 the internal cross-sectional structure decomposes from the values H = –1.7, but the internal cross-sectional structure is preserved in the interval , in the case μ1 = 0.9 and μ1 = 0.1 the internal cross-sectional structure decomposes from the values  but the internal cross-sectional structure is preserved in the interval , in the case of μ1 = 0.7 and μ1 = 0.3 the internal cross-sectional structure decomposes from the values , but the internal cross-sectional structure is preserved in the interval .  With increasing energy, many of these surfaces decay. It is assumed that the numerical results obtained will serve as the basis for comparison with analytical solutions.

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Published
2020-03-26
How to Cite
MALKOV, E.A. et al. Poincare sections for two fixed centers problem and Henon-Heiles potential. Recent Contributions to Physics (Rec.Contr.Phys.), [S.l.], v. 72, n. 1, p. 4-10, mar. 2020. ISSN 2663-2276. Available at: <https://bph.kaznu.kz/index.php/zhuzhu/article/view/1239>. Date accessed: 22 sep. 2020. doi: https://doi.org/10.26577/RCPh.2020.v72.i1.01.
Section
Theoretical Physics. Nuclear and Elementary Particle Physics. Astrophysics

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