力梯度辛算法的拓广与应用

发布时间：2018-03-12 07:25

本文选题：天体力学　切入点：力梯度辛算法　出处：《南昌大学》2011年硕士论文　论文类型：学位论文

【摘要】：在天体力学和非线性动力学的研究过程中,数值方法以及混沌的识别方法是研究天体力学和非线性动力学的主要研究方法和工具,所以寻找可靠而且高效的数值方法和混沌识别方法是目前天体力学和非线性动力学研究的重要课题,我们在本文中的主要研究工作是关于数值方法的拓广与应用。力梯度辛算法在精度上高于非力梯度辛算法,1997年Chin等提出了四阶力梯度辛算法。我们在此基础上进一步构造了新型的四阶力梯度辛算法并把它们应用于Henon-Heiles系统和四极矩核-壳模型进行模拟比较,发现它们具有较好的数值性能；此外,我们还运用力梯度辛算法研究了三维限制性三体问题的动力学。下面分别简述这些工作。首先,在能够分解为动能T部分和势能V部分的可分离哈密顿系统中,对势能V部分所对应的Lie算子加入力梯度算子在内的有关算子,使其包含一阶导数、二阶导数和三阶导数项,从而成功构造出新型的四阶力梯度辛算法,其中Chin等所提出的力梯度辛算法也是我们所构造的辛算法的一种特殊形式。把所推广的新型辛算法拓广应用于Henon-Heiles系统和四极矩核-壳模型,分别使用所构造的新型辛算法对有序轨道和混沌轨道进行数值模拟,数值结果表明无论是在能量误差方面还是在位置误差方面,新构造的辛算法精度远远优越于Forest-Ruth的非力梯度四阶辛算法,最优化辛算法具有良好的能量精度。新型辛算法可以推荐到实际计算中。其次,限制性三体问题是天体力学非常重要的模型之一,扁率项对限制性三体问题的平动点具有一定的影响作用。我们运用分析近似方法研究了含扁率J2和J3项的三维限制性三体问题在赤道平面外的平动点位置和稳定性；然后,我把该限制性三体问题的哈密顿分解为包含动量与坐标交叉项的动能部分和势能部分这两个可积部分,探讨了力梯度辛算法应用的可能性,并且用能量误差评估了力梯度辛算法的效果。最后研究了该问题的有序与混沌性质以及与动力学参数的依赖关系。总之,本学位论文的主要工作就是将已有四阶力梯度辛算法推广构造新型力梯度辛算法,并探讨了力梯度辛算法应用于限制性三体问题的可能性以及动力学参数与混沌的依赖关系。
[Abstract]:In the course of the study of celestial mechanics and nonlinear dynamics, numerical methods and chaotic identification methods are the main research methods and tools for the study of celestial mechanics and nonlinear dynamics. Therefore, finding reliable and efficient numerical methods and chaotic identification methods is an important subject in the research of celestial mechanics and nonlinear dynamics. Our main research work in this paper is on the extension and application of numerical methods. The force gradient symplectic algorithm is more accurate than the non-force gradient symplectic algorithm. In 1997, Chin et al proposed the fourth order force gradient symplectic algorithm. On this basis, we further constructed a new fourth-order force gradient symplectic algorithm and applied it to Henon-Heiles system. Compared with the four-pole moment core-shell model, It is found that they have good numerical performance, in addition, we use the force gradient symplectic algorithm to study the dynamics of the three-dimensional restricted three-body problem. Firstly, in a separable Hamiltonian system which can be decomposed into kinetic energy T part and potential energy V part, the Lie operator corresponding to the potential energy V part is added to some related operators, including the force gradient operator, so that it contains the first order derivative. Second order derivative and third order derivative term, thus successfully construct a new fourth-order force gradient symplectic algorithm. The force gradient symplectic algorithm proposed by Chin et al is also a special form of symplectic algorithm constructed by us. The generalized new symplectic algorithm is extended to Henon-Heiles system and quadrupole moment core-shell model. The new symplectic algorithm is used to simulate the ordered orbit and chaotic orbit respectively. The numerical results show that both the energy error and the position error are obtained. The new symplectic algorithm is far superior to Forest-Ruth 's non-force gradient fourth-order symplectic algorithm, and the optimization symplectic algorithm has good energy accuracy. The new symplectic algorithm can be recommended for practical calculation. Second, the restricted three-body problem is one of the most important models of celestial mechanics. The flattening term has a certain influence on the translational point of the restricted three-body problem. We use the analytical approximation method to study the position and stability of the translational point outside the equatorial plane of the three-dimensional restricted three-body problem with flattening ratio J _ 2 and J _ 3. I decompose the Hamiltonian of the restricted three-body problem into two integrable parts, which include the kinetic energy part and the potential energy part of the intersection of momentum and coordinate, and discuss the possibility of the application of the force gradient symplectic algorithm. The effect of force gradient symplectic algorithm is evaluated by energy error. Finally, the ordered and chaotic properties of the problem and its dependence on dynamic parameters are studied. In a word, the main work of this dissertation is to extend the fourth-order force gradient symplectic algorithm to construct a new force gradient symplectic algorithm. The possibility of applying the force gradient symplectic algorithm to the restricted three-body problem and the dependence of dynamic parameters on chaos are discussed.
【学位授予单位】：南昌大学
【学位级别】：硕士
【学位授予年份】：2011
【分类号】：P14

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