Lie-series for orbital elements: I. The planar case
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Abstract
Lie-integration is one of the most efficient algorithms for numerical integration of ordinary differential equations if high precision is needed for longer terms. The method is based on the computation of the Taylor coefficients of the solution as a set of recurrence relations. In this paper, we present these recurrence formulae for orbital elements and other integrals of motion for the planar \(N\)-body problem. We show that if the reference frame is fixed to one of the bodies—for instance to the Sun in the case of the Solar System—the higher order coefficients for all orbital elements and integrals of motion depend only on the mutual terms corresponding to the orbiting bodies.
Keywords
\(N\)-body problems Numerical methods Lie-integration Planetary systems Recurrence relations Taylor coefficientsNotes
Acknowledgments
The author would like to thank the anonymous referees for their valuable comments. The author also thanks László Szabados for the careful proofreading. This work has been supported by the Hungarian Academy of Sciences via the Grant LP2012-31.
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