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

On the stability of motion of N -body systems: a geometric approach

A.A. El-Zant <sup>1, 2

1</sup> Astronomy Centre, University of Sussex, Brighton BN1 9QH, UK
² Physics Department, Technion - Israel Institute of Technology, Haifa 32000, Israel

Received 19 June 1996 / Accepted 18 March 1997

Abstract

Much of standard galaxy dynamics rests on the implicit assumption that the corresponding N -body problem is (near) integrable. This notion although leading to great simplification is by no means a fact. In particular, this assumption is unlikely to be satisfied for systems which display chaotic behaviour which manifests itself on short time-scales and for most initial conditions. It is therefore important to develop and test methods that can characterize this kind of behaviour in realistic situations. We examine here a method, pioneered by Krylov (1950) and first introduced to gravitational systems by Gurzadyan & Savvidy (1984, 1986). It involves a metric on the configuration manifold which is then used to find local quantification of the divergence of trajectories and therefore appears to be suitable for short time characterization of chaotic behaviour. We present results of high precision N -body simulations of the dynamics of systems of 231 point particles over a few dynamical times. The Ricci (or mean) curvature is calculated along the trajectories. Once fluctuations due to close encounters are removed this quantity is found to be almost always negative and therefore all systems studied display local instability to random perturbations along their trajectories. However it is found that when significant softening is present the Ricci curvature is no longer negative. This suggests that smoothing significantly changes the structure of the phase space of gravitational systems and casts doubts on the continuity of the transition from the large-N limit to the continuum limit. From the value of the negative curvature, evolution time-scales of systems displaying clear instabilities (for example collective instabilities or violent relaxation) are derived. We compare the predictions obtained from these calculations with the time-scales of the observed spatial evolution of the different systems and deduce that this is fairly well described. In all cases the results based on calculations of the scalar curvature qualitatively agree. These results suggest that future applications of these methods to realistic systems may be useful in characterizing their stability properties. One has to be careful however in relating the time-scales obtained to the time-scales of energy relaxation since different dynamical quantities may relax at different rates.

Key words: instabilities celestial mechanics, stellar dynamics galaxies: evolution

Contents

• 1. Introduction and motivation
  • 1.1. On the assumptions of standard galaxy dynamics
  • 1.2. How chaos drives evolution
• 2. Characterization of chaos and the Ricci criterion
  • 2.1. Difficulties with commonly used methods
  • 2.2. The study of motion on Lagrangian manifolds
  • 2.3. Ricci curvature and the corresponding criterion
• 3. Applications
  • 3.1. Some applications that take advantage of the geometric setting
  • 3.2. Specific application and model parameters
• 4. Results of numerical experiments
  • 4.1. Averaging
  • 4.2. Rotating versus Non-rotating models
  • 4.3. Varying the energy and virial ratio
  • 4.4. Systems starting in virial equilibrium
  • 4.5. The effect of softening
  • 4.6. Scalar curvature
• 5. Conclusions
• Acknowledgements
• References

© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998
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