  Simulations of objects with classical dynamics are in fact a particular
version of discrete dynamics since almost all the classical dynamics
simulations in natural science are performed with the use of the simple
''Leapfrog" or ''Verlet" algorithm. It was, however, Newton who in $Principia$,
$ Proposition$ $I$ in 1687 first formulated the discrete algorithm, which much
later in 1967 was rederived by L. Verlet. Verlet also formulated a first-order
approximation for the velocity $\textbf{v}(t)$ at time $t$, which has been used
in simulations since then. The approximated expressions for $\textbf{v}(t)$ and
the kinetic energy lead to severe errors in the thermodynamics at high
densities, temperatures, strong repulsive forces, or for large discrete time
increments used in the discrete ''Molecular Dynamics" (MD) simulations. Here we
derive the exact expressions for the discrete dynamics, and show by simulations
of a Lennard-Jones system, that these expressions now result in equality
between temperatures determined from the kinetic energies, and the
corresponding configurational temperatures determined from the expression of
Landau and Lifshitz, derived from the forces.
