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.
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References
I. Newton, PHILOSOPHIÆ NATURALIS PRINCIPIA MATHEMATICA. LONDINI, Anno MDCLXXXVII
L.J. Garay, J. Mod. Phys. A 10, 145 (1995)
S. Toxvaerd, Phys. Rev. E 47, 343 (1993)
S. Toxvaerd, J. Chem. Phys. 140, 044102 (2014)
S. Toxvaerd, Phys Rev. E 50, 2271 (1994)
I.B. Cohen, A. Whitman, U. California Press, Berkeley (1999)
L. Verlet, Phys. Rev. 159, 98 (1967)
L. Levesque, L. Verlet, Eur. Phys. J. H 44, 37 (2019)
R.W Hockney, J.W Eastwood, Computer Simulation Using Particles; Chapter 4 and 11. ISBN-13: 978-0852743928
M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids. Second Edition 2017, (2017). https://doi.org/10.1093/oso/9780198803195.001.0001
D. Frenkel, B. Smit, Molecular Simulatiom; Academic Press, London (2002); ISBN-13: 978-0122673511
A. Cromer, Am. J. Phys. 49, 455 (1981)
J.N. Tokis, IJAA 4, 683 (2014)
Newtons PROPOSITION I. THEOREM I., Figure 1 and Eq. (5) are for a fixed force center at S and before Newton formulated the third law. For two body central force dynamics the masses must be replaced by reduced masses
J.M. Sanz-Serna, Acta Numer. 1, 243 (1992)
E. Hairer, Ann. Numer. Math. 1, 107 (1994)
S. Reich, SIAM J. Numer. Anal. 36, 1549 (1999)
J. Gans, D. Shalloway, Phys. Rev. E 61, 4587 (2000)
S. Toxvaerd, O.J. Heilmann, J.C. Dyre, J. Chem. Phys. 136, 224106 (2012)
S. Toxvaerd, J. Chem. Phys. 137, 214102 (2012)
It is convenient to express the energy and length in units of a given planet energy, e.g. \(E^*=gMm^*/r_p^*\), and length \(r_p^*\) from the sun (e. g. the planet Earth). The corresponding time unit is \(t^*=r_p^*\sqrt{m^*/E^*}\). The relations below are given in these reduced units
M. Nauenberg, Am. J. Phys. 86, 765 (2018)
T.D. Lee, Phys. Lett. 122B, 217 (1983)
R. Friedberg, T.D. Lee, Nucl. Phys. B 225 [FS9], 1 (1983)
T.D. Lee, J. Stat. Phys. 46, 843 (1987)
M. Milgrom, Astrophys. J. 270, 371 (1983)
Acknowledgements
Ole J. Heilmann, Niccolõ Guicciardini and Jeppe C Dyre are gratefully acknowledged. This work was supported by the VILLUM Foundation’s Matter project, Grant No. 16515.
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Toxvaerd, S. Newton’s discrete dynamics. Eur. Phys. J. Plus 135, 267 (2020). https://doi.org/10.1140/epjp/s13360-020-00271-5
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DOI: https://doi.org/10.1140/epjp/s13360-020-00271-5