Newton's algorithm for discrete classical dynamics

Abstract

A recent article in J. Chem. Phys. argues that the two algorithms, the velocity-Verlet, and position-Verlet integrators, commonly used in Molecular Dynamics (MD) simulations, are different \cite{Ni2024}. But not only are the two algorithms just different formulations of the same discrete algorithm, but so are other simple discrete algorithms used in MD in the natural sciences. They are all reformulations of the discrete algorithm derived by Newton in 1687 in $\textit{Proposition I}$ in the very first part of his book $Principia$. The different reformulations of Newton's algorithm for discrete dynamics lead to identical discrete dynamics with the same invariances, momentum, angular momentum, and energy as Newton's analytical dynamics. Hundreds of thousands of MD simulations with Newton's discrete dynamics have appeared, but unfortunately with many recorded errors for energies, potential energies, temperatures, and heat capacities. The public software for MD should be corrected.