保持Runge-Lenz向量的数值方法  

A Note on the Numerical Method for Conserving the Runge-Lenz Vector

作　　者：刘福窑[1] 伍歆[2] 陆本魁[1] 

机构地区：[1]中国科学院紫金山天文台 [2]南昌大学理学院

出　　处：《天文学报》2005年第3期294-306,共13页Acta Astronomica Sinica

基　　金：国家自然科学基金(10303001)专项基金(10447112)资助

摘　　要：对孤立积分和能够保持Runge-Lenz向量的梯形公式进行详尽讨论.孤立积分就是限制粒子运动区域的不变量.具有n个自由度的自治可积哈密顿系统且只有n个互相对合的独立孤立积分,并且其他孤立积分的存在对粒子的运动是有意义的.Kepler二体系统存在能量积分、角动量积分和Runge-Lenz向量.对于平面运动情况,这三类积分中只有3个独立孤立积分;而对于三维空间情形,该三类积分仅有5个是独立的.就前者而言,Kepler二体平面运动积分构成该系统中的对称群SO(3),经过Levi-Civita变换,它可以转化为二维各向同性谐振子系统中的对称群,而该对称群能够被梯形公式准确保持.另一方面,对于后者梯形公式对这三类积分的严格保持还可以在5个Kepler轨道根数a、e、i、Ω和ω上得到体现.An intensive discussion is given here on an isolating integral and the trapezoidal rule applied to the preservation of the Runge-Lenz vector introduced by Minesaki and Nakamura. The isolating integral is an integral of a dynamical system which can further restrict the motion region of a particle in the system. It is well known that an integrable autonomous Hamiltonian system with n degrees of freedom must hold n independent isolating integrals in involution each other. If there exist other independent isolating integrals in this system, these isolating integrals have significance. It is clear that a bound Kepler problem contains energy integral, angular momentum integral and the Runge-Lenz vector. It is found that a symmetry group SO(3) formed by three independent isolating integrals in a dynamical system of two-dimensional Kepler motion is to be identified with a group of two-dimensional isotropic harmonic oscillator derived from the orginal Kepler system in terms of Levi-Civita transformation. As a result, the group of the isotropic harmonic oscillator can be strictly conserved by the trapezoidal rule. In addition, the trapezoidal rule can preserve exactly five orbital elements a, e, i, Ω and ω in a three-dimensional Kepler motion with five independent isolating integrals.

关 键 词：天体力学 孤立积分 Runge-Lenz向量 辛方法 

分 类 号：P138[天文地球—天体力学]