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