  Almost all Molecular Dynamics (MD) simulations are discrete dynamics with
Newton's algorithm first published in 1687, and much later by L. Verlet in
1967. Discrete Newtonian dynamics has the same qualities as Newton's classical
analytic dynamics. Verlet also published a first-order expression for the
instant temperature which is inaccurate but presumably used in most MD
simulations. One of the motivations for the present article is to correct this
unnecessary inaccuracy in $NVT$ MD dynamics. Another motivation is to derive
simple algorithms for the Nos\'{e}-Hoover $NVT$ dynamics (NH) with the correct
temperature constraint. The simulations with NH discrete Newtonian dynamics
show that the NH works excellent for a wide range of the response time $\tau$
of the NH thermostat, but NH simulations favor a choice of a short response
time, even shorter than the discrete-time increment $\delta t$ used in MD, to
avoid large oscillations of the temperature.
