本文選題：天體力學(xué)　切入點(diǎn)：力梯度辛算法　出處：《南昌大學(xué)》2011年碩士論文　論文類型：學(xué)位論文

【摘要】：在天體力學(xué)和非線性動(dòng)力學(xué)的研究過程中,數(shù)值方法以及混沌的識(shí)別方法是研究天體力學(xué)和非線性動(dòng)力學(xué)的主要研究方法和工具,所以尋找可靠而且高效的數(shù)值方法和混沌識(shí)別方法是目前天體力學(xué)和非線性動(dòng)力學(xué)研究的重要課題,我們?cè)诒疚闹械闹饕芯抗ぷ魇顷P(guān)于數(shù)值方法的拓廣與應(yīng)用。 力梯度辛算法在精度上高于非力梯度辛算法,1997年Chin等提出了四階力梯度辛算法。我們?cè)诖嘶A(chǔ)上進(jìn)一步構(gòu)造了新型的四階力梯度辛算法并把它們應(yīng)用于Henon-Heiles系統(tǒng)和四極矩核-殼模型進(jìn)行模擬比較,發(fā)現(xiàn)它們具有較好的數(shù)值性能；此外,我們還運(yùn)用力梯度辛算法研究了三維限制性三體問題的動(dòng)力學(xué)。下面分別簡述這些工作。 首先,在能夠分解為動(dòng)能T部分和勢(shì)能V部分的可分離哈密頓系統(tǒng)中,對(duì)勢(shì)能V部分所對(duì)應(yīng)的Lie算子加入力梯度算子在內(nèi)的有關(guān)算子,使其包含一階導(dǎo)數(shù)、二階導(dǎo)數(shù)和三階導(dǎo)數(shù)項(xiàng),從而成功構(gòu)造出新型的四階力梯度辛算法,其中Chin等所提出的力梯度辛算法也是我們所構(gòu)造的辛算法的一種特殊形式。把所推廣的新型辛算法拓廣應(yīng)用于Henon-Heiles系統(tǒng)和四極矩核-殼模型,分別使用所構(gòu)造的新型辛算法對(duì)有序軌道和混沌軌道進(jìn)行數(shù)值模擬,數(shù)值結(jié)果表明無論是在能量誤差方面還是在位置誤差方面,新構(gòu)造的辛算法精度遠(yuǎn)遠(yuǎn)優(yōu)越于Forest-Ruth的非力梯度四階辛算法,最優(yōu)化辛算法具有良好的能量精度。新型辛算法可以推薦到實(shí)際計(jì)算中。 其次,限制性三體問題是天體力學(xué)非常重要的模型之一,扁率項(xiàng)對(duì)限制性三體問題的平動(dòng)點(diǎn)具有一定的影響作用。我們運(yùn)用分析近似方法研究了含扁率J2和J3項(xiàng)的三維限制性三體問題在赤道平面外的平動(dòng)點(diǎn)位置和穩(wěn)定性；然后,我把該限制性三體問題的哈密頓分解為包含動(dòng)量與坐標(biāo)交叉項(xiàng)的動(dòng)能部分和勢(shì)能部分這兩個(gè)可積部分,探討了力梯度辛算法應(yīng)用的可能性,并且用能量誤差評(píng)估了力梯度辛算法的效果。最后研究了該問題的有序與混沌性質(zhì)以及與動(dòng)力學(xué)參數(shù)的依賴關(guān)系。 總之,本學(xué)位論文的主要工作就是將已有四階力梯度辛算法推廣構(gòu)造新型力梯度辛算法,并探討了力梯度辛算法應(yīng)用于限制性三體問題的可能性以及動(dòng)力學(xué)參數(shù)與混沌的依賴關(guān)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.
【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】：2011
【分類號(hào)】：P14

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本文編號(hào)：1600571