Simple tools to study global dynamics in non-axisymmetric galactic potentials – I

In a first part we discuss the well-known problem of the motion of a star in a general non-axisymmetric 2D galactic potential by means of a very simple but almost universal system: the pendulum model. It is shown that both loop and box families of orbits arise as a natural consequence of the dynamics of the pendulum. An approximate invariant of motion is derived. A critical value of the latter sharply separates the domains of loops and boxes and a very simple computation allows to get a clear picture of the distribution of orbits on a given energy surface. Besides, a geometrical representation of the global phase space using the natural surface of section for the problem, the 2D sphere, is presented. This provides a better visualization of the dynamics. In a second part we introduce a new indicator of the basic dynamics, the Mean Exponential Growth factor of Nearby Orbits (MEGNO), that is suitable to investigate the phase space structure associated to a general Hamiltonian. When applied to the 2D logarithmic potential it is shown to be effective to obtain a picture of the global dynamics and, also, to derive good estimates of the largest Lyapunov characteristic number in realistic physical times. Comparisons with other techniques reveal that the MEGNO provides more information about the dynamics in the phase space than other wide used tools. Finally, we discuss the structure of the phase space associated to the 2D logarithmic potential for several values of the semiaxis ratio and energy. We focus our attention on the stability analysis of the principal periodic orbits and on the chaotic component. We obtain critical energy values for which connections between the main stochastic zones take place. In any case, the whole chaotic domain appears to be always confined to narrow filaments, with a Lyapunov time about three characteristic periods.