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*Astron. Astrophys. 331, 782-792 (1998)*
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## On the stability of motion of *N* -body systems:
the effect of the variation of particle number, softening and rotation
**
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 6 March 1997 / Accepted 6 October 1997*
**Abstract**
Using the Ricci and scalar curvatures of the configuration manifold
of gravitational *N* -body systems, we study the exponential
instability in their trajectories. It is found that the exponentiation
time-scale for isotropic Plummer spheres varies very little with
particle number if the softening is small. Large softening on the
other hand has a marked effect and, if large enough, can cause the
curvatures to become positive. This last result confirms the previous
observations for self gravitating sheets and suggests that the
*qualitative* behaviour of large-*N* and continuum systems
may be different, and that their equivalence is only obtained in the
limit of infinite *N* and finite softening. It is also found that
the presence of a large fraction of the kinetic energy in rotational
motion increases the exponentiation time-scales significantly - an
effect that should be expected given the regular nature of nearly
circular motion. In the light of the results of this and of previous
studies, it is suggested that the exponential instability may arise
from low order resonances between the period of the variation of the
gravitational field due to distant encounters and the orbital period
of a test particle. For periods long compared to the exponentiation
time but short compared to the diffusion time-scales of the action
variables, the standard picture of collisionless dynamics may be valid
in an averaged sense - nevertheless this time interval need not
coincide with that predicted by standard relaxation theory. Instead it
is suggested that, at least for systems with well defined final
states, the relaxation time should scale as .
**Key words:** stellar
dynamics
galaxies: kinematics and
dynamics
galaxies: evolution
© European Southern Observatory (ESO) 1998
Online publication: February 16, 1998
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