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.