  Computer simulation of the time evolution in a classical system is a standard
numerical method, used in numerous scientific articles in Natural Science.
Almost all the simulations are performed by discrete Molecular Dynamics (MD).
The algorithm used in MD was originally formulated by I. Newton at the
beginning of his book $Principia$. Newton's discrete dynamics is exact in the
same sense as Newton's analytic counterpart Classical Mechanics. Both dynamics
are time-reversible, symplectic, and have the same dynamic invariances. There
is no qualitative difference between the two kinds of dynamics. This is due to
the fact, that there exists a ''shadow Hamiltonian'' nearby the Hamiltonian
$H(\textbf{q},\textbf{p})$ for the analytic dynamics, and where its dynamics
can be obtained by an asymptotic expansion from $H(\textbf{q},\textbf{p})$, and
where the positions generated by MD are located on the analytic trajectories
for the shadow Hamiltonian.
  It is only possible to obtain the solution of Newton's classical differential
equations for a few simple systems, but the exact discrete Newtonian dynamics
can be obtained for almost all complex classical systems. Some examples are
given here: The emergence and evolution of a planetary system. The emergence
and evolution of planetary systems with inverse forces. The emergence and
evolution of galaxies in the expanding Universe.
