太阳系动力学中几种辛和类辛方法的比较  

Comparison of Several Symplectic and Quasi-Symplectic Methods in Solar System Dynamics

作　　者：万晓生[1] 黄天衣[1] 

机构地区：[1]南京大学天文学系,南京210093

出　　处：《南京大学学报（自然科学版）》2002年第4期462-470,共9页Journal of Nanjing University（Natural Science）

基　　金：国家自然科学基金 (10 1730 0 7) ;紫金山天文台小行星基金会

摘　　要：辛方法是目前研究太阳系天体轨道长期演化的最佳数值方法 .它能保持哈密顿系统的主要性质而不会引入人为的耗散 ,因而可以用大步长进行长时期的数值积分 .国际文献中用于太阳系动力学定性研究的辛方法主要是Wisdom和Holman的SYA方法 ,它是SYP方法的一种近似 .它们都采用雅可比坐标系 .在质心坐标系中建立的辛方法有传统的将动能和势能分离的SYS方法 .提出一种在质心坐标系中构造的类辛方法SYQ ,并对这 4种方法进行了较为详尽的效率分析和误差比较 ,结论是SYA和SYQ是比较好的方法 。Symplectic methods are so far the best numerical methods for qualitative exploration in solar system dynamics. They maintain the symplectic structure and key properties of Hamiltonian systems and do not bring in any artificial dissipation, making possible long_term numerical integrations with a large step size. The symplectic method that has been widely adopted in references on qualitative studies of solar system dynamics is the method worked out by Wisdom and Holman (abbreviated as SYA hereafter). It is built in the Jacobian coordinate system and takes an approximation of the hamiltonian. The Wisdom and Holman's method for an exact hamiltonian is abbreviated as SYP. Actually a symplectic integrator can be built in the barycentric coordinate system (abbreviated as SYS), which separates the hamiltonian into two parts, the potential energy and the kinetic energy. Here we propose a quasi_symplectic method SYQ in the barycentric coordinate system. An extensive comparative study of these four types of methods is given, especially on their computation effciency and error accumulation. This research draws the following conclusions. Considering that symplectic integrators are mainly used in exploring the qualitative evolution of dynamical systems and a high precision is not required, SYS should not be recommended in solar system dynamics for its low effciency. During a 10\+8 years integration, SYP methods cause almost the same errors on the positions of the planets but they take almost 40% more computing time. We thus believe that SYP can not compete with SYA or SYQ, but it is hard to tell SYA or SYQ is better. Our research has also shown that resonances play a role in keeping the orbit configuration of a planetary system during long_term numerical integrations.

关 键 词：类辛方法 太阳系动力学 数值方法 辛方法 SYA方法 SYQ方法 天体轨道 质心坐标系 

分 类 号：P135[天文地球—天体力学]