Eur. Phys. J. Plus (2020) 135: 267
https://doi.org/10.1140/epjp/s13360-020-00271-5

Regular Article

Newton’s discrete dynamics

Department of Science and Environment, Roskilde University, Postbox 260, 4000, Roskilde, Denmark

^* e-mail: st@ruc.dk

Received: 2 December 2019
Accepted: 8 February 2020
Published online: 20 February 2020

Abstract

In 1687, Isaac Newton published PHILOSOPHIÆ NATURALIS PRINCIPIA MATHEMATICA, where the classical analytic dynamics was formulated. But Newton also formulated a discrete dynamics, which is the central difference algorithm, known as the Verlet algorithm. In fact, Newton used the central difference to derive his second law. The central difference algorithm is used in computer simulations, where almost all Molecular Dynamics simulations are performed with the Verlet algorithm or other reformulations of the central difference algorithm. Here, we show that the discrete dynamics obtained by Newton’s algorithm for Kepler’s equation has the same solutions as the analytic dynamics. The discrete positions of a celestial body are located on an ellipse, which is the exact solution for a shadow Hamiltonian nearby the Hamiltonian for the analytic solution.