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<p>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., <i>x</i> = <i>y</i> = <i>z</i> = 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.</p>

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A geometric algorithm for orbital dynamics based on Lie derivative

  • 期刊论文

  • 发表日期：

    2026年5月6日

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  • Yuhan Song,
  • Shixing Liu,
  • Wenan Jiang,
  • Yongxin Guo

摘要

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.