Numerical integration of dynamical systems with Lie series
Numerical integration of dynamical systems with Lie series
复制标题
动力系统与李级数的数值积分
DOI：
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发表时间：
2012
影响因子：
1.6
通讯作者：
W. Thuillot
中科院分区：
文献类型：
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作者：
D. Bancelin;D. Hestroffer;W. Thuillot
太阳系和太阳系外行星系统的引力动力学系统的运动方程在整体情况下是不可积的，通常采用数值积分的方法进行积分。在求解常微分方程的各种数值方法中，利用李级数进行数值积分显示出了一定的优势。在其最初的形式（Hanslmeier和Dvorak，Astron Astrophys 132，203 1984），它仅限于N体问题，其中只考虑引力相互作用。在本文中，我们提出了一个概括的方法，推导出一个表达式的李项时，考虑其他主要力量。事实上，以前的研究已经完成，但只针对在引力作用下运动的物体。如果加入其他扰动，则必须重建李积分器。在目前的工作中，我们考虑两种情况下，涉及位置和位置-速度相关的扰动：相对论加速度的广义相对论的框架和简化的力的Yarkovsky效应。一个一般的迭代程序适用于导出李级数的任何顺序和精度。然后，我们给出了一个应用程序的典型近地天体和水星的运动方程的集成。
The integration of the equations of motion in gravitational dynamical systems—either in our Solar System or for extra-solar planetary systems—being non integrable in the global case, is usually performed by means of numerical integration. Among the different numerical techniques available for solving ordinary differential equations, the numerical integration using Lie series has shown some advantages. In its original form (Hanslmeier and Dvorak, Astron Astrophys 132, 203 1984), it was limited to the N-body problem where only gravitational interactions are taken into account. We present in this paper a generalisation of the method by deriving an expression of the Lie terms when other major forces are considered. As a matter of fact, previous studies have been done but only for objects moving under gravitational attraction. If other perturbations are added, the Lie integrator has to be re-built. In the present work we consider two cases involving position and position-velocity dependent perturbations: relativistic acceleration in the framework of General Relativity and a simplified force for the Yarkovsky effect. A general iteration procedure is applied to derive the Lie series to any order and precision. We then give an application to the integration of the equation of motions for typical Near-Earth objects and planet Mercury.