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博碩士論文 etd-0706115-131701 詳細資訊
Title page for etd-0706115-131701
論文名稱
Title
以保群結構法計算三星體運動軌跡
Group preserving scheme for three-body orbit
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
32
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2015-06-24
繳交日期
Date of Submission
2015-08-06
關鍵字
Keywords
週期性的牛頓系統、第二保群結構法、符號函數、非線性動態系統、保群結構法
non-linear dynamical system, group preserving schemes, second scheme GPS2, signum function, periodic Newtonian systems
統計
Statistics
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中文摘要
保群結構法是一種計算非線性動態系統軌跡的數值方法。此方法將非線性動態系統轉換成增廣動態系統,並在每一個迭代步驟維持群特性。第一保群結構法與適性步長尤拉法相似。第二保群結構法則可藉由簡單符號函數描述動態系統的複雜性。我們使用四種具週期性的牛頓系統以及一個不規則系統來測試數值方法彼此間的優異。
Abstract
The group preserving schemes are numerical methods to compute the orbit of non-linear dynamical systems. The non-linear dynamical system is converted to an augmented dynamical system, which preserves the group properties for every time increment. The first scheme GPS has the form similar to the first order Euler method with adaptive stepsize. The second scheme GPS2 is capable of depicting the complexity of the dynamical system by computing a simple signum function. We use four periodic Newtonian systems and an irregular system to test the performance of the schemes.
目次 Table of Contents
1. Introduction ‧‧‧‧‧ 1
1.1. Dynamic system of celestial mechanics ‧‧‧‧‧ 1
1.2. Three-body orbit ‧‧‧‧‧ 1
2. Preliminary ‧‧‧‧‧ 2
2.1. Collinear configuration ‧‧‧‧‧ 3
2.2. Regular triangular configuration ‧‧‧‧‧ 4
2.3. Figure eight orbits ‧‧‧‧‧ 5
3. Numerical methods ‧‧‧‧‧ 6
3.1 Group preserving schemes ‧‧‧‧‧ 6
3.2 Second group preserving schemes ‧‧‧‧‧ 8
4. Numerical results ‧‧‧‧‧ 15
4.1 Collinear configuration ‧‧‧‧‧ 15
4.2 Regular triangular configuration ‧‧‧‧‧ 17
4.3 Elliptical orbits ‧‧‧‧‧ 19
4.4 Figure eight orbits ‧‧‧‧‧ 21
4.5 Irregular orbits ‧‧‧‧‧ 23
5. Conclusions ‧‧‧‧‧ 25
Reference ‧‧‧‧‧ 26
參考文獻 References
[1] A. Albouy, Mutual distances in celestial mechanics, Nelin. Dinam., 2:3 (2006) 361–386.
[2] F. Calogero, Solution of a three body problem in one dimension, J. Math. Phys., 10 (1969) 2191.
[3] A. Cohan, A figure eight and other interesting solutions to the N-body problem, http://math.washington.edu, June 4, 2012.
[4] A. Chenciner and R. Montgomery, A remarkable periodic solution of the three body problem in the case of equal masses, Ann. Math., 152 (2000) 881-901.
[5] M. Hampton and R. Moeckel, Finiteness of relative equilibria of the four-body problem, Invent. Math., 163 (2006) 289-312.
[6] T.-L. Lee and M. Santoprete, Central configurations of the five-body problem with equal masses, Celestial Mech. Dynam. Astronom, 104 (2009) 369-381.
[7] T.-Y. Li and J. A. Yorke, Period Three Implies Chaos, Amer. Math. Monthly, 82:10 (1975) 985-992.
[8] C.-S. Liu, Cone of non-linear dynamical system and group preserving schemes, Internat. J. Non-Linear Mech., 36 (2001) 1047-1068.
[9] C.-S. Liu, A novel Lie-group theory and complexity of nonlinear dynamical systems, Commun. Nonlinear Sci. Numer. Simul., 20 (2015) 39–58.
[10] C. Simo, New families of solutions in N-body problems, European Congress of Mathematics, (2001) 101-115.
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