*Astron. Astrophys. 326, 113-129 (1997)*
## On the stability of motion of *N* -body systems: a geometric approach
**
A.A. El-Zant **^{ 1, 2}
^{1} 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
© European Southern Observatory (ESO) 1997
Online publication: April 20, 1998
helpdesk.link@springer.de |