Abstract
If one has to attain high accuracy over long timescales during the numerical computation of the N-body problem, the method called Lie-integration is one of the most effective algorithms. In this paper, we present a set of recurrence relations with which the coefficients needed by the Lie-integration of the orbital elements related to the spatial N-body problem can be derived up to arbitrary order. Similarly to the planar case, these formulae yield identically zero series in the case of no perturbations. In addition, the derivation of the formulae has two stages, analogously to the planar problem. Namely, the formulae are obtained to the first order, and then, higher-order relations are expanded by involving directly the multilinear and fractional properties of the Lie-operator.
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Notes
When \(z_i\rightarrow 0\) and \(\dot{z}_i\rightarrow 0\) for all \(1\le i\le N\).
Note that the symbol M represents the central mass while the symbols \(M_i\) (with a single index) denote the mean anomalies.
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Acknowledgments
The author would like to thank A. László for his helpful comments about the tensor rank analysis. The author also thanks the anonymous referees for their thorough reviews of the manuscript. The author thanks László Szabados for the careful proofreading. This work has been supported by the Hungarian Academy of Sciences via the Grant LP2012-31. Additional support is received from the Hungarian OTKA Grants K-109276 and K-104607.
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Pál, A. Lie-series for orbital elements: II. The spatial case. Celest Mech Dyn Astr 124, 97–107 (2016). https://doi.org/10.1007/s10569-015-9653-0
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DOI: https://doi.org/10.1007/s10569-015-9653-0