  A long-term numerical integration of the classical Newtonian approximation to
the planetary orbital motions of the full Solar System (sun + 8 planets),
spanning 20 Gyr, was performed. The results showed no severe instability
arising over this time interval. Subsequently, utilizing a bifurcation method
described by Jacques Laskar, two numerical experiments were performed with the
goal of determining dynamically allowed evolutions for the Solar System in
which the planetary orbits become unstable. The experiments yielded one
evolution in which Mercury falls onto the Sun at ~1.261Gyr from now, and
another in which Mercury and Venus collide in ~862Myr. In the latter solution,
as a result of Mercury's unstable behavior, Mars was ejected from the Solar
System at ~822Myr. We have performed a number of numerical tests that confirm
these results, and indicate that they are not numerical artifacts. Using
synthetic secular perturbation theory, we find that Mercury is destabilized via
an entrance into a linear secular resonance with Jupiter in which their
corresponding eigenfrequencies experience extended periods of commensurability.
The effects of general relativity on the dynamical stability are discussed. An
application of the bifurcation method to the outer Solar System (Jupiter,
Saturn, Uranus, and Neptune) showed no sign of instability during the course of
24Gyr of integrations, in keeping with an expected Uranian dynamical lifetime
of 10^(18) years.
