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

In a rst part we discuss the well-known prob- lem of the motion of a star in a general non-axisymmetric 2D galactic potential by means of a very simple but al- most 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 approx- imate 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 pic- ture 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 prob- lem, 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 fac- tor of Nearby Orbits (MEGNO), that is suitable to inves- tigate the phase space structure associated to a general Hamiltonian. When applied to the 2D logarithmic poten- tial it is shown to be eective to obtain a picture of the global dynamics and, also, to derive good estimates of the largest Lyapunov characteristic number in realistic physi- cal times. Comparisons with other techniques reveal that the MEGNO provides more information about the dynam- ics in the phase space than other wide used tools. Finally, we discuss the structure of the phase space as- sociated 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 conned to narrow laments, with a Lyapunov time about three characteristic periods.