Newton's discrete dynamics

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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.
Original languageEnglish
Article number267
JournalThe European Physical Journal Plus
Issue number2
Number of pages10
Publication statusPublished - 1 Feb 2020

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