对数哈密顿方法及其应用

发布时间：2018-11-05 12:12

【摘要】：天体力学数值方法作为天体力学的重要领域之一在辛算法的提出后得到长足发展,辛算法保持哈密顿系统辛结构且计算过程中系统没有能量和角动量的长期误差累积。辛算法适用于哈密顿系统的长期定性演化研究同时也具有数值精度不高、显辛算法要求固定步长的不足。通常积分计算天体紧密交汇问题或大偏心率轨道运动都需缩短步长来克服天体受引力过大而剧增的加速度,直接变步长将丢失辛算法保持辛结构的优势,考虑时间变换的思路,原时间变量取变步长而新的时间变量仍为固定步长,则既能调节步长又能保持辛算法固有优势。本文的主要内容为构造针对不同哈密顿系统的对数哈密顿算法及论证其在具有更高的数值精度和保证获得有效的混沌判别结果方面的优势。针对不同的哈密顿系统结构构造不同形式的时间变换辛算法。对于可分解为分别只含状态量广义动量和广义坐标的动能部分和势能部分的哈密顿函数,可构造取时间变换函数为形式不同但等价的两个函数得到显式对数哈密顿方法,其中时间变换作用于哈密顿函数,本文构造了由三个二阶蛙跳算子构成的显式对数哈密顿Yoshida四阶方法。对于动能部分具有广义动量和广义坐标的交叉项而势能部分仅含位置变量的系统构造显隐式混合对数哈密顿方法,对于动能部分应用隐式中点法。而对于更一般的系统则构造隐式对数哈密顿方法。隐式方法具有更广泛的应用但也由于算法构造中包括迭代需耗费更多的计算机时间降低计算效率。本文详细论证了显式对数哈密顿方法在应用于牛顿圆型限制性三体问题及相对论圆型限制性三体问题时较于非时间变换辛算法更具数值精度优势。且在前一系统的精度优势独立于轨道偏心率的变化。对于后一系统这一现象未能发生但数值精度也明显优越于常规辛算法。特别对于高偏心率轨道,非时间变换算法得到的虚假的混沌判别指标,如Lyapunov指标和快速Lyapunov指数(FLI)。而通过对数哈密顿方法则可获得可靠地定性分析结果,彻底地解决后牛顿圆型限制性三体问题的高偏心率轨道Lyapunov指数的过度估计和FLI快速增大的问题。在得到论证后本文应用对数哈密顿方法讨论了动力学参数两主天体间距离的变化对动力学系统有序和混沌转化的影响。本文通过数值模拟验证了对数哈密顿方法具有更高的数值精度及可得到可靠的定性研究成果的优势。适用于定性研究和定量计算高偏心率问题,为天体力学研究开拓了新思路。在实际的天体紧密交汇处的动力学演化提供反映动力学实质的积分工具。
[Abstract]:As one of the important fields of celestial mechanics, the numerical method of celestial mechanics has been developed rapidly since the symplectic algorithm was put forward. The symplectic algorithm keeps the symplectic structure of Hamiltonian system and there is no long-term error accumulation of energy and angular momentum in the calculation process. The symplectic algorithm is suitable for the long-term qualitative evolution of Hamiltonian systems. In order to overcome the acceleration caused by excessive gravity, the direct variable step size will lose the advantage of symplectic algorithm to keep the symplectic structure, and consider the idea of time transformation, so as to solve the problem of close intersection of celestial bodies or the motion of orbit with large eccentricity in order to overcome the acceleration caused by excessive gravity. If the original time variable takes variable step size and the new time variable is fixed step size, it can not only adjust the step size but also maintain the inherent advantage of the symplectic algorithm. The main content of this paper is to construct logarithmic Hamiltonian algorithm for different Hamiltonian systems and demonstrate its advantages in obtaining higher numerical accuracy and ensuring effective chaos discrimination results. Different symplectic algorithms of time transformation are constructed for different Hamiltonian systems. For Hamiltonian functions which can be decomposed into kinetic energy parts and potential energy parts containing only generalized momentum and generalized coordinates of state variables, two functions with different form but equivalent form can be constructed to obtain explicit logarithmic Hamiltonian method. In this paper, an explicit logarithmic Hamiltonian Yoshida fourth order method consisting of three second-order leapfrog operators is constructed. The implicit mixed logarithmic Hamiltonian method is used for the system with generalized momentum and generalized coordinates in the kinetic energy part and the potential energy part with only the position variable. The implicit midpoint method is applied to the kinetic energy part. For more general systems, an implicit logarithmic Hamiltonian method is constructed. Implicit methods are more widely used, but they also cost more computer time to reduce computational efficiency due to the need of iteration in the construction of the algorithm. In this paper, it is demonstrated in detail that the explicit logarithmic Hamiltonian method is more accurate than the non-time transformation symplectic algorithm when it is applied to Newtonian circular restricted three-body problem and relativistic circular restricted three-body problem. Moreover, the accuracy advantage of the former system is independent of the variation of orbit eccentricity. For the latter system, this phenomenon does not occur, but the numerical accuracy is obviously superior to that of the conventional symplectic algorithm. Especially for high eccentricity orbits, false chaotic discriminant indexes, such as Lyapunov index and fast Lyapunov exponent (FLI)., are obtained by non-time transformation algorithm. By means of the number Hamiltonian method, the reliable qualitative analysis results can be obtained, and the problem of overestimation of Lyapunov exponent of high eccentricity orbit and rapid increase of FLI in post-Newton circular restricted three-body problem can be solved thoroughly. In this paper, the logarithmic Hamiltonian method is used to discuss the influence of the distance between the two main objects on the order and chaos transformation of the dynamical system. In this paper, it is proved by numerical simulation that the logarithmic Hamiltonian method has higher numerical accuracy and can obtain reliable qualitative research results. It is suitable for qualitative research and quantitative calculation of high eccentricity. The dynamic evolution of the actual celestial bodies at close junctions provides an integral tool to reflect the essence of dynamics.
【学位授予单位】：南昌大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：P13

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