PHASE PORTRAITS OF THE HENON-HEILES POTENTIAL

Abstract

In this paper the Henon-Heiles potential is considered. In the second half of the 20th century, in astronomy the model of motion of stars in a cylindrically symmetric and time-independent potential was studied. Due to the symmetry of the potential, the three-dimensional problem reduces to a two-dimensional problem; nevertheless, finding the second integral of the obtained system in the analytical form turns out to be an unsolvable problem even for relatively simple polynomial potentials. In order to prove the existence of an unknown integral, the scientists Henon and Heiles carried out an analysis of research for trajectories in which the method of numerical integration of the equations of motion is used. The authors proposed the Hamiltonian of the system, which is fairly simple, which makes it easy to calculate trajectories, and is also complex enough that the resulting trajectories are far from trivial. 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. Equipotential curves, the momentum and coordinate dependences on time, and also the Poincaré section were obtained for this system. At the same time, with the increase in energy, many of these surfaces decay, which indicates the absence of the second integral. It is assumed that the obtained numerical results will serve as a basis for comparison with analytical solutions. Keywords: Henon-Heiles model, Poincaré section, numerical solutions.