First, it is acknowledged that a number of scientists do not believe that
molecular dynamics does work, including some who accept the validity of
molecular mechanics. Second, this presentation does not address the
accuracy of classical force fields as an approximation to quantum
mechanics, nor is it primarily concerned with defects in the ergodic
hypothesis. Rather, the concern is that computed trajectories are
overwhelmed by the effect of finite step size (and finite precision) due
to the chaotic nature of the Hamiltonian systems and the very long
integration times. The best-behaved numerical simulations are generally
those that employ symplectic integrators. For these it can be proved
that the numerical solution is very nearly the exact solution of a
modified Hamiltonian system on a limited time interval. However, from
examining the numerical trajectories of one-dimentional systems, it is
not apparent that this result extends to very lon! g time intervals. The
aim of the presentation is to give a plausible mathematical basis for
long time numerical integration using the concept of weak convergence.
