Numerical Precession in Variational Discretizations of the Kepler Problem
- 348 Downloads
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.
KeywordsVariational integrators Modified equations Kepler problem Orbital precession
MSC 2010:65L12 65P10 70F05 70H33
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.
- 3.Chin, S.A.: Physics of symplectic integrators: Perihelion advances and symplectic corrector algorithms. Phys. Rev. E 75(3), 036–701 (2007)Google Scholar
- 7.Goldstein, H.: Classical Mechanics. Addison-Wesley Pub, Co (1980)Google Scholar
- 8.Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, vol. 31. Springer (2006)Google Scholar
- 12.Noether, E.: Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, mathematisch-physikalische Klasse, Invariante Variationsprobleme 1918, 235–257 (1918)Google Scholar
- 13.Olver, P.J.: Applications of Lie Groups to Differential Equations, vol. 107. Springer Science & Business Media (2000)Google Scholar
- 16.Will, C.M.: Theory and experiment in gravitational physics, vol. 1. Cambridge University Press (1981)Google Scholar