Simple tools to study global dynamics in non--axisymmetric galactic potentials -- I P.~M. Cincotta, Facultad de Ciencias Astron\'omicas y Geof\'{\i}sicas, Universidad Nacional de La Plata, Paseo del Bosque, 1900 La Plata, Argentina Present address (until August 1999): Departament de Matem\`atica Aplicada i An\`alisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain. E-mail: pablo@maia.ub.es -- pmc@fcaglp.unlp.edu.ar and C. Sim\'o, Departament de Matem\`atica Aplicada i An\`alisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain E-mail: carles@maia.ub.es Abstract In the first part of this paper we discuss the well--known problem of the motion of a star in a general non--axisymmetric non--rotating 2D galactic potential by means of a very simple but almost universal system: the pendulum model. It is shown how both families of orbits, loop and box, arise naturally as a consequence of the dynamics of the pendulum. An approximate invariant of motion is then derived where a critical value of the latter separates sharply the domains of loop and box orbits. So a very simple computation allows to get a clear picture of the distribution of orbits on a given energy surface. The second part is devoted to introduce a geometrical representation of the global phase space using the natural manifold for the problem, the 2D sphere. A surface of section displayed on the sphere provides a better visualization of the dynamics. In the third part we introduce a simple tool, derived from the definition of the largest Lyapunov characteristic number, that appears to be suitable to investigate the phase space structure associated to a general Hamiltonian. The results of its application to the 2D logarithmic potential show that this technique is very effective to obtain a picture of the global and local dynamics and, simultaneously, to derive a good estimation of the largest Lyapunov characteristic number but in realistic physical times. The required computational effort is comparatively small, almost the same needed to get the latter number but in much shorter time intervals. Comparisons with other techniques reveal that this simple method provides more information about the phase space structure than other wide used tools. We include a fourth part where we mainly discuss the structure of the phase space associated to the 2D logarithmic potential for several values of the semiaxis ratio and energy, focusing the attention on the stability analysis of the principal periodic orbits and on the chaotic component. We derive 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 but with a very short Lyapunov time, about 3 characteristic periods. Some mathematical results are gathered on an Appendix. Keywords: galaxies: dynamics -- stellar dynamics -- Lyapunov characteristic number -- global phase portrait -- chaos