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 三體問題(three-body problem)，是指空間中三個天體以萬有引力互相作用；而限制性三體問題是指三個天體中其中一個的質量小到可以忽略不計下，互相作用下的行為‧我們較感興趣的是三體運動的穩定性‧因實際天體中，如太陽、地球、月亮就是一個標準的三體問題；而木星與特洛依(trojian)小行星群以正三角形的方式繞著太陽轉…等皆是重要的天體問題‧ 本文以漢密頓力學由多體問題指向三體問題，定性的分析向心型與共線性，再建立數學模型，並討論三體運動的穩定性，最後再用電腦程式模擬三體在不同起始條件下的運動行為，以確認是否與數學模型相符‧
 The problem considers the mutual gravitational interaction of three-body in the space. In the restricted three-body problem, one of the masses is taken to be small enough so that it does not influence the motion of the interaction. Thus, we are much interested in the stability motion of three-body. As the real heavenly bodies, the sun- earth-moon are the standard three-body problem. However, the Jupiter, the Trojan and the other planets site forms an equilateral triangle with the sun. These are also very important studies. According to Hamilton＇s theory- form N-body problem to three-body problem, the analysis qualitative of central configuration problem and collinear, We build up the math model and discuss the stability motion of three-body. We also use computer programming to simulate the movement of three-body in the original condition in order to make sure the resemblance of the math model.
 第一章 緒論............................................ 5 第一節 研究動機與問題之源起...........................................................6 第二節 研究方法...................................................................................7 第二章 問題的介紹與定性的探討........................... 8 第一節 概說...........................................................................................8 第二節 多體問題...................................................................................8 第三節 漢密頓形與不動點的解...........................................................9 第四節 向心型問題.............................................................................10 第五節 拉格朗日等邊三角形.............................................................11 第六節 太陽系裡的拉格朗日點.........................................................12 第三章 限制性三體問題的計算............................. 14 第一節 限制性三體問題的數學模型.................................................14 第二節 找出平衡點附近的解.............................................................15 第四章 電腦模擬三體運動................................. 25 第一節 簡介.........................................................................................25 第二節‧演算法...................................................................................25 第三節‧主程式...................................................................................26 第四節‧執行結果與研討...................................................................47 第五章 討論............................................. 55 第一節 模擬結果分析.........................................................................55 第二節 結論.......................................................................................57 參考文獻................................................ 55
 1. Abraham, R. and Marsden, J. 1978, Foundations of Mechanics. 2. Alfriend,J. 1970, The stability of the triangular Lagrangian points for commensurability of order 2. 3. Angenent, S.B. 1990, Monotone recurrence relations. 4. Arnold,V.I. 1968, New periodic solution of the plane three-body problem corresponding to elliptic notion in the lunar theory. 5. Arnold, V.I. 1978, Mathematical Methods of Classical Mechanics. 6. Besicovitch,A.S.1932,Almost Periodic Functions 7. Bowen,R.1975, Equilibrium States And the Ergodic theory of Anosov Diffeomorphism. 8. Buchanan, D. 1941, Trojan Satellites─limiting case. 9. Chrikov, B.V.1979.A universal instability of many dimensional oscillator systems. 10. Coddington, E. and Levinson, N. 1955, Theory of ordinary Differential Equations. 11. Contopoulos,G..1981,The 4：1 resonance. 12. Danby, J.M.A.1962, Fundamentals of Celestial Mechanics. 13. Devaney, R. 1986, An Introduction to Chaotic Dynamical Systems. 14. Dieudonne, J.A. 1960, Foundations of Modern Analysis. 15. Diliberto, J.A.1961, Perturbation theorems for periodic systems. 16. Favar, J. 1933 Lecons sur les fonctions presque periodiques. 17. Fink, A.M. 1975, Almost periodic Differential Equation. 18. Flanders, H.1963, Differential Forms with Application to the Physical Sciences. 19. Floer, A. and Zehnder,E.1985.Fixed points Results for symplectic maps related to the Arnold conjecture. 20. Guckenheimer, J. and Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcation Theory. 21. Hale, J.k. 1972, Ordinary Differential Equations. 22. Hale, J.k. and Chow, S-N, 1982 , Methods of Bifurcation theory. 23. Halmors, P. 1958, Finite Dimensional Vector Spaces. 24. Hill, G.W. 1878, Researches in the lunar theory. 25. Hirsch,M. ,Pugh,C., and Shub,M. 1977,Invariant Manifolds. 26. Kelley, A.1967, On the Liapunov subcenter theorem. 27. Kelley, A.1955, General Topology. 28. Kobayashi, S. and Nomizu,K. 1969, Foundations of Differential Geometry. 29. Kenneth, R. and Glen, R. 1937, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem. 30. Lefshetz, S. 1963, Differential Equations. 31. Marsden, J. and Weinstein, A. 1974,Reduction of symplectic manifold with symmetrices . 32. Markeev, A.P. 1966,On the stability of the triangular libration points in the circular bounded three-body problem. 33. Meyer, K.R. 1967, On contact transformations. 34. Meyer, K.R. 1970, Generic bifurcation of periodic points. 35. Meyer, K.R. 1970, Generic stability properties of periodic points. 36. Meyer, K.R. 1981, Periodic solutions of the N-body problem. 37. Meyer, K.R. 1987, Bifurcation of the central configuration. 38. Moser, J.K. 1978, A fixed point theorem in symplectic geometry. 39. Moser, J.K. 1986, Minimal solutions of variational problem on a tours. 40. Moulton, F.R. 1914, An introduction to Celestial Mechanics. 41. Nitecki, Z. 1971, Differentiable Dynamics. 42. Palmore, J.I. 1969, Bridges and Natural Centers in the restricted Three-Body Problem. 43. Siegel, C.L. and Moser, J.K. 1971, Lectures on Celestial Mechanics.
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 1 三體問題極小反轉解的距離估計

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