Astron. Astrophys. 326, 113-129 (1997)

5. Conclusions

The assumptions of classical non-evolutionary galactic dynamics may not be satisfied in the presence of significant amount of chaos (Sect. 1). This makes it important to develop and to test methods characterizing such chaotic behaviour. Geometric methods have the double advantage of being local and comparing normal deviations between trajectories of dynamical systems and not the total divergence of temporal states. For higher dimensional systems these properties may be important in distinguishing between true phase space mixing which can be accompanied by changes in the physical characteristics of a system and phase mixing which conserves the action variables. Geometric methods are therefore better suited for studies of the short time evolution of galaxies than other (more traditional) measures of chaos (Sects. 2 and 3.1).

In large dimensional systems it may not be practical to determine the stability of a trajectory to all possible perturbations. The next best thing is to determine the average divergence of trajectories due to random perturbations. One way of doing this is by examining the Ricci curvature of the Lagrangian configuration manifold of a dynamical system (Sect. 2.3). To check the effectiveness of such an approach for N -body gravitational systems we have calculated the Ricci curvature for several small N -body systems integrated with high precision (Sect. 3.2 and 4). The results of these experiments show that:

1. When properly averaged to get rid of the contributions of close encounters the Ricci curvature is almost always negative, confirming that gravitational systems are unstable (e.g., Miller 1964; Goodman et al. 1993; Kandrup et al. 1994) and that the main mechanism of instability is the negativity of the curvature of the configuration manifold as predicted by Gurzadyan & Savvidy (1984, 1986) and Kandrup (1990a,1990b).
2. The Ricci curvature is more negative (hence predicts shorter evolutionary time-scales) when a system develops pronounced macroscopic instabilities (e.g., plasma type collective instabilities). In this case the rates of spatial macroscopic evolution of the different systems both relative to each other and in terms of evolution time-scales was reasonably well described by the time-scales derived on the basis of the Ricci curvature calculations.
3. When expressed in terms of dynamical times, evolutionary time-scales appeared to be slightly longer for systems starting from virial equilibrium than those starting from a virial ratio less than one. However, since when the kinetic energy varies significantly, chaotic behaviour cannot be described by the negativity of the Ricci curvature alone (Cerruti-Sola & Pettini 1995) this conclusion remains to be confirmed. (However in the case of large N -body systems near virial equilibrium the objections to the use of the Ricci curvature as outlined in the aforementioned paper are not likely to be important since the second and third term of their Eq. (26) are then very small. By using the Ricci curvature and eliminating large fluctuations due to close encounters we have therefore implicitly assumed that it is the instability in that (large-N) limit that interests us and not effects due to small scale fluctuations.)
4. In the presence of significant (but not very large) softening, the Ricci curvature becomes positive. This probably means that the phase-space structure of softened systems is radically different from that of point particles. This has consequences for the interpretation of results of numerical simulations. More fundamentally however this result may be interpreted to mean that there is no continuous transition from large- N discrete N -body systems to continuous ones. It may therefore explain why large N -body spherical systems are found to approximate exponentially unstable C-systems while it is known that motion in smooth spherical potentials is separable. More work however is needed to fully understand the meaning of this effect and its possible consequences.
5. Results derived on the basis of the scalar curvature agreed, in general, with those obtained from the evolution of the Ricci curvature. This shows the instability to be quite a robust phenomenon.

We have not looked at properties such as energy relaxation here. As was mentioned in the introduction this can have different time-scales from that of the instability of trajectories. To see how this may be the case we consider the change in the Hamiltonian of a test particle i in an N -body system. Using the Hamiltonian equations we find that along the motion this is given by

with the interaction potential given by

being a function of the positions of the remaining particles which are of course time dependent functions of the initial conditions. The change in energy of particle i along its path is then given by

which is the sum of the energy changes due to the individual interactions and therefore could proceed on time-scales similar to that given in (1). This does not of course mean that energy relaxation cannot be enhanced for systems consisting of particles with different masses or those out of virial equilibrium or where collective motion or large scale inhomogeneity or anisotropy occurs. In these cases chaotic behaviour can be important for energy relaxation. Indeed there is some evidence that in some situations two body relaxation estimates are inaccurate even for energy relaxation (El-Zant 1996a). In general however there may be phase-space "barriers" across which diffusion is slow. This may prevent some quantities from relaxing even when chaos is present (discussions of the issue of phase-space transport in higher dimensional Hamiltonian systems can be found in Wiggins 1991 or Benettin 1994). Nevertheless, chaos implies exponential divergence normal to the phase-space trajectory leading to diffusion in at least some of the action variables which determine the physical characteristics of a system (as opposed to phase mixing which conserves the action variables). It therefore has important consequences for the behaviour of dynamical systems as was stressed in the introductory section of this paper. This has long been generally recognised in various branches of Physics and Mathematics (e.g., Sagdeev et al. 1988) but not always in stellar dynamics. The question therefore is how widespread is the chaotic behaviour in realistic N -body realizations of the different galaxy types and what are the exponentiation time-scales. This is what we hope to find out using the method examined in this paper.

© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998
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