The stability of gravitational systems with 3 or more bodies
is one of the great open problems of mathematics.
It is not known, for instance, if there are initial conditions
for three bodies such that, for all small perturbations,
orbits remain at a uniformly bounded distance
from the center of mass forever.
It *is* know, however, that such a "stable" initial condition
exists if everything (positions and initial velocities)
is restricted to a plane. This follows from the KAM theorems
(KAM = Kolmogorv-Arnold-Moser).
Whether this has any relationship with the fact that
the planets of our solar system lie nearly in a plane
I don't know.
The amount of time for which our particular solar system
can be shown to be stable (bar, of course, the possibility
of some large body from outside the solar system entering
the scene and ruining all predictions) is today of the order
of billions of years, which roughly coincides with the life time
of the Sun. Curiously, this coincidence remains since Newton's
time. According to astronomers at Newton's time, the solar system
was a few thousand years old (this may or may not have something
to do with the Bible) and Newton himself knew that the solar
system was stable for about that long
(but apparently thought that God needed to intervene
in order to keep things in order for longer periods).
At the 19th century, Laplace and others had shown
that the solar system was stable "up to first order"
and the estimated age of the solar system was then of several
million years (since nuclear fusion was then unimaginable,
scientists calculated that the Sun could only burn for about
that long).
