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論文名稱 Title |
穩定虛擬質量方法計算平面中心構型運動 Stable Virtual Mass Method for Computing the Planar Central Configuration Motion |
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系所名稱 Department |
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畢業學年期 Year, semester |
語文別 Language |
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學位類別 Degree |
頁數 Number of pages |
40 |
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研究生 Author |
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指導教授 Advisor |
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召集委員 Convenor |
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口試委員 Advisory Committee |
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口試日期 Date of Exam |
2017-06-29 |
繳交日期 Date of Submission |
2017-07-28 |
關鍵字 Keywords |
克卜勒定律、克卜勒方程、質心虛擬質量、中心構型、N體問題 Kepler’s equation, Kepler’s laws of motion, Virtual mass, Central configuration, N-body problem |
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統計 Statistics |
本論文已被瀏覽 5732 次,被下載 0 次 The thesis/dissertation has been browsed 5732 times, has been downloaded 0 times. |
中文摘要 |
N 體中心構型運動被列為二十一世紀問題之一,其依據牛頓運動力學運作,每個粒子所受的合力皆會指向質心,並與粒子到質心的距離成正比。這個問題開啟了 N 體動力學方面的相關研究。 N 體問題沒有一般解析解,通常會使用傳統數值方法計算平面中心構型運動,除了二體問題外,方法一般來說會是不穩定的。為了克服這個困難,我們在質心處引入對應每個粒子的虛擬質量,使得 N 體中心構型運動可以化簡為 N 個二體運動問題。在引入虛擬質量後,使用諸如 Runge–Kutta methods 之類的傳統數值方法所顯示的數值結果相當穩定,幾乎可以精準模擬其運動軌跡。 除此之外,一旦中心構型問題簡化為若干二體問題,粒子的運動軌跡必定會是滿足克卜勒定律的橢圓。因此,透過克卜勒方程建構橢圓的外接圓,我們可以推導出 ``真近點角" 及 ``偏近點角" 之間的關係,進一步得到二體問題的解。 |
Abstract |
The central configuration motion, listed as one of problems for the twenty-first century, is the N-body Newtonian motion along which the net force on each body by all the other bodies always points to the center of mass and is proportional to the distance to the center of mass. The problem leads to the study of the behavior of solutions near collisions and the N-body dynamics. While computing the planar central configuration motion by classical numerical methods is always unstable except for two body system. To overcome this difficulty, for each mass we introduce its corresponding virtual mass so that the motion of N-body central configuration is reduced to N two-body motions, which is quite stable by using the classical numerical methods such as Runge-Kutta method. Once the two-body motion is confirmed, the trajectory of the motion must be an ellipse by Kepler's laws. Thus, we can construct the circumcircle of the ellipse to derive the relation of true anomaly and eccentric anomaly by Kepler's equation, from which the solution of the two-body problem can be achieved. |
目次 Table of Contents |
論文審定書i 摘要ii Abstract iii 1 Introduction 1 1.1 Newton’s laws of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Central configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Preliminaries 4 2.1 Kepler’s laws of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Second law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.2 First law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.3 Third law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Vis-viva equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Albouy-Chenciner equations . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Formula for the orbit of C.C. 12 3.1 Initial velocity of C.C. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Virtual mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4 Numerical results 18 Bibliography 23 Appendix A 25 Appendix B 26 Appendix C Kepler’s equation 28 |
參考文獻 References |
[1] A. Albouy and A. Chenciner. Le probl´eme des n corps et les distances mutuells. Inventiones mathematicae, 131:151–184, 1998. [2] Dziobek. U¨ ber einen merkwu¨rdigen Fall des Vielko¨rperproblems. Astronomische Nachrichten, 152:33, Mar. 1900. [3] L. Euler. De motu rectilineo trium corporum se mutuo attrahentium. Novi Commentarii academiae scientiarum Petropolitanae, 11:144–151, 1767. [4] M. Hampton and R. Moeckel. Finiteness of relative equilibria of the four-body problem. Inventiones mathematicae, 163:289–312, 2006. [5] H. Karttunen, P. Kr¨oger, H. Oja, M. Poutanen, and K. Donner. Fundamental Astronomy. Springer Berlin Heidelberg, 2007. [6] J.-L. Lagrange. Essai sur le probl`eme des trois corps. Prix de l’Acad´emie royale des sciences de Paris, 6:229–331, 1772. [7] T. Lee, T. Li, and C. Tsai. Hom4ps-2.0: A software package for solving polynomial systems by the polyhedral homotopy continuation method. Computing (Vienna/New York), 83(2-3):109–133, 11 2008. [8] T.-L. Lee and M. Santoprete. Central configurations of the five-body problem with equal masses. Celestial Mechanics and Dynamical Astronomy, 104:369–381, Aug. 2009. [9] R. Moeckel. Central configurations. In J. Llibre, R. Moeckel, and C. Sim´o, editors, Central Configurations, Periodic Orbits, and Hamiltonian Systems, chapter 2, pages 105–167. C.R.M., Barcelona, 2015. [10] J. L. Russell. Kepler’s laws of planetary motion: 1609–1666. The British Journal for the History of Science, 2(1):1–24, 1964. |
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