Applied Mathematics and Mechanics >
A geometric algorithm for orbital dynamics based on Lie derivative
Received date: 2025-11-23
Revised date: 2026-02-24
Online published: 2026-05-06
Supported by
Project supported by the National Natural Science Foundation of China (Nos. 12232009 and 12372002) and the 2024 Fundamental Research Program of Liaoning University (Scientific Research Platform Construction Project)
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Orbital dynamics is a fundamental problem in celestial mechanics, yet its governing equations are characterized by irrational terms and denominator-type nonlinearities. Traditional numerical methods, which are locally discrete, may lead to ambiguities near singularities (e.g., x=y=z=0), thereby limiting the numerical stability and accuracy. To address these challenges, we propose a Lie derivative algorithm that constructs discrete iterative schemes based on the Lie series expansion. Unlike local schemes, this approach discretizes the vector field in a global manner, effectively avoiding singular inconsistencies while ensuring stable long-term integration. Numerical experiments demonstrate that, when compared with a high-accuracy reference solution under uniform step-size settings, the proposed approach not only achieves higher accuracy but also improves computational efficiency by up to 47% in the two-body problem and 67% in the circular restricted three-body problem, relative to classical second-order schemes. These results indicate that the Lie derivative algorithm provides an efficient and practical alternative for high-precision orbital dynamics computations.
Yuhan SONG , Shixing LIU , Wenan JIANG , Yongxin GUO . A geometric algorithm for orbital dynamics based on Lie derivative[J]. Applied Mathematics and Mechanics, 2026 , 47(5) : 1157 -1176 . DOI: 10.1007/s10483-026-3377-7
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