Abstract
Kepler’s first law states that the orbit of a point mass with negative energy in a classical gravitational potential is an ellipse with one of its foci at the gravitational center. In numerical simulations of this system one often observes a slight precession of the ellipse around the gravitational center. Using the Lagrangian structure of modified equations and a perturbative version of Noether’s theorem, we provide leading order estimates of this precession for the implicit MidPoint rule (MP) and the Störmer-Verlet method (SV). Based on those estimates we construct some new numerical integrators that perform significantly better than MP and SV on the Kepler problem.
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References
Chartier, P., Hairer, E., Vilmart, G.: Numerical integrators based on modified differential equations. Math. Comput. 76(260), 1941–1953 (2007)
Chin, S.A.: Symplectic integrators from composite operator factorizations. Phys. Lett. A 226(6), 344–348 (1997)
Chin, S.A.: Physics of symplectic integrators: Perihelion advances and symplectic corrector algorithms. Phys. Rev. E 75(3), 036–701 (2007)
Chin, S.A., Kidwell, D.W.: Higher-order force gradient symplectic algorithms. Phys. Rev. E 62(6), 8746 (2000)
Curtis, L.J., Haar, R.R., Kummer, M.: An expectation value formulation of the perturbed Kepler problem. Am. J. Phys. 55(7), 627–631 (1987)
Forest, E., Ruth, R.D.: Fourth order symplectic integration. Physica 43(LBL-27662), 105–117 (1989)
Goldstein, H.: Classical Mechanics. Addison-Wesley Pub, Co (1980)
Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, vol. 31. Springer (2006)
Lévy-Leblond, J.M.: Conservation laws for gauge-variant Lagrangians in classical mechanics. Am. J. Phys. 39(5), 502–506 (1971)
Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Num. 2001(10), 357–514 (2001)
Morehead, J.: Visualizing the extra symmetry of the Kepler problem. Am. J. Phys. 73(3), 234–239 (2005)
Noether, E.: Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, mathematisch-physikalische Klasse, Invariante Variationsprobleme 1918, 235–257 (1918)
Olver, P.J.: Applications of Lie Groups to Differential Equations, vol. 107. Springer Science & Business Media (2000)
Rogers, H.H.: Symmetry transformations of the classical Kepler problem. J. Math. Phys. 14(8), 1125–1129 (1973)
Vermeeren, M.: Modified Equations for Variational Integrators Numer. Math. 137(4), 1007–1037 (2017)
Will, C.M.: Theory and experiment in gravitational physics, vol. 1. Cambridge University Press (1981)
Acknowledgements
This research is supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”. The author would like to thank the organizers and participants of the Brainstorming Workshop on New Developments in Discrete Mechanics, Geometric Integration and Lie-Butcher Series for making it an inspiring event.
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Vermeeren, M. (2018). Numerical Precession in Variational Discretizations of the Kepler Problem. In: Ebrahimi-Fard, K., Barbero Liñán, M. (eds) Discrete Mechanics, Geometric Integration and Lie–Butcher Series. Springer Proceedings in Mathematics & Statistics, vol 267. Springer, Cham. https://doi.org/10.1007/978-3-030-01397-4_10
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DOI: https://doi.org/10.1007/978-3-030-01397-4_10
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