【摘要】：天體力學(xué)數(shù)值方法作為天體力學(xué)的重要領(lǐng)域之一在辛算法的提出后得到長(zhǎng)足發(fā)展,辛算法保持哈密頓系統(tǒng)辛結(jié)構(gòu)且計(jì)算過(guò)程中系統(tǒng)沒(méi)有能量和角動(dòng)量的長(zhǎng)期誤差累積。辛算法適用于哈密頓系統(tǒng)的長(zhǎng)期定性演化研究同時(shí)也具有數(shù)值精度不高、顯辛算法要求固定步長(zhǎng)的不足。通常積分計(jì)算天體緊密交匯問(wèn)題或大偏心率軌道運(yùn)動(dòng)都需縮短步長(zhǎng)來(lái)克服天體受引力過(guò)大而劇增的加速度,直接變步長(zhǎng)將丟失辛算法保持辛結(jié)構(gòu)的優(yōu)勢(shì),考慮時(shí)間變換的思路,原時(shí)間變量取變步長(zhǎng)而新的時(shí)間變量仍為固定步長(zhǎng),則既能調(diào)節(jié)步長(zhǎng)又能保持辛算法固有優(yōu)勢(shì)。本文的主要內(nèi)容為構(gòu)造針對(duì)不同哈密頓系統(tǒng)的對(duì)數(shù)哈密頓算法及論證其在具有更高的數(shù)值精度和保證獲得有效的混沌判別結(jié)果方面的優(yōu)勢(shì)。針對(duì)不同的哈密頓系統(tǒng)結(jié)構(gòu)構(gòu)造不同形式的時(shí)間變換辛算法。對(duì)于可分解為分別只含狀態(tài)量廣義動(dòng)量和廣義坐標(biāo)的動(dòng)能部分和勢(shì)能部分的哈密頓函數(shù),可構(gòu)造取時(shí)間變換函數(shù)為形式不同但等價(jià)的兩個(gè)函數(shù)得到顯式對(duì)數(shù)哈密頓方法,其中時(shí)間變換作用于哈密頓函數(shù),本文構(gòu)造了由三個(gè)二階蛙跳算子構(gòu)成的顯式對(duì)數(shù)哈密頓Yoshida四階方法。對(duì)于動(dòng)能部分具有廣義動(dòng)量和廣義坐標(biāo)的交叉項(xiàng)而勢(shì)能部分僅含位置變量的系統(tǒng)構(gòu)造顯隱式混合對(duì)數(shù)哈密頓方法,對(duì)于動(dòng)能部分應(yīng)用隱式中點(diǎn)法。而對(duì)于更一般的系統(tǒng)則構(gòu)造隱式對(duì)數(shù)哈密頓方法。隱式方法具有更廣泛的應(yīng)用但也由于算法構(gòu)造中包括迭代需耗費(fèi)更多的計(jì)算機(jī)時(shí)間降低計(jì)算效率。本文詳細(xì)論證了顯式對(duì)數(shù)哈密頓方法在應(yīng)用于牛頓圓型限制性三體問(wèn)題及相對(duì)論圓型限制性三體問(wèn)題時(shí)較于非時(shí)間變換辛算法更具數(shù)值精度優(yōu)勢(shì)。且在前一系統(tǒng)的精度優(yōu)勢(shì)獨(dú)立于軌道偏心率的變化。對(duì)于后一系統(tǒng)這一現(xiàn)象未能發(fā)生但數(shù)值精度也明顯優(yōu)越于常規(guī)辛算法。特別對(duì)于高偏心率軌道,非時(shí)間變換算法得到的虛假的混沌判別指標(biāo),如Lyapunov指標(biāo)和快速Lyapunov指數(shù)(FLI)。而通過(guò)對(duì)數(shù)哈密頓方法則可獲得可靠地定性分析結(jié)果,徹底地解決后牛頓圓型限制性三體問(wèn)題的高偏心率軌道Lyapunov指數(shù)的過(guò)度估計(jì)和FLI快速增大的問(wèn)題。在得到論證后本文應(yīng)用對(duì)數(shù)哈密頓方法討論了動(dòng)力學(xué)參數(shù)兩主天體間距離的變化對(duì)動(dòng)力學(xué)系統(tǒng)有序和混沌轉(zhuǎn)化的影響。本文通過(guò)數(shù)值模擬驗(yàn)證了對(duì)數(shù)哈密頓方法具有更高的數(shù)值精度及可得到可靠的定性研究成果的優(yōu)勢(shì)。適用于定性研究和定量計(jì)算高偏心率問(wèn)題,為天體力學(xué)研究開拓了新思路。在實(shí)際的天體緊密交匯處的動(dòng)力學(xué)演化提供反映動(dòng)力學(xué)實(shí)質(zhì)的積分工具。
[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.
【學(xué)位授予單位】：南昌大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2016
【分類號(hào)】：P13

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