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 

