藉由符號運算研究六體問題中心構形的有限性

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研究生：	張格銘 Chang, Ke-Ming
論文名稱：	藉由符號運算研究六體問題中心構形的有限性 Toward finiteness of central configurations for the planar six-body problem by symbolic computations
指導教授：	陳國璋 Chen, Kuo-Chang
口試委員:	莊重 Juang, Jonq 蔡東和 Tsai, Dong-Ho 蔡亞倫 Tsai, Ya-Lun 黃信元 Huang, Hsin-Yuan
學位類別：	博士 Doctor
系所名稱：	理學院 - 數學系 Department of Mathematics
論文出版年：	2023
畢業學年度：	111
語文別：	英文
論文頁數：	170
中文關鍵詞：	中心構形、符號運算
外文關鍵詞：	central configurations, symbolic computations
相關次數：	點閱：3 下載：0
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在這篇論文中，我們探討中心構形的有限性問題。多體問題中心構形，隨著質點個數的增加，系統的複雜度會劇增。對於六體中心構形的有限性問題，由於可能涉及龐大的計算量，我們採用符號運算作為輔助。符號運算雖然是由電腦執行，不過只有牽涉到整數運算，因此計算的結果是準確的，不會有誤差的問題。

我們依據 Albouy 和 Kaloshin 在研究5體中心構形有限性的文章中所使用的方法，建立一些便於符號運算的定義及定理，接著生成演算法並不斷地改良使得他們的方法能夠部分自動化地應用在 6 體有限性問題，盡可能解決更多 6 體的部分情況。

在第一節，我們簡單介紹 n 體問題中的中心構形以及有限性問題；在第二節中，我們大略講述 Albouy 和 Kaloshin 的方法；在第三到第五節，我們根據他們的方法建立相關的理論；在第六到第八節，我們介紹演算法的內容；最後幾個小節中，我們展示了演算法應用在 n=4,5,6 的結果。

In this thesis, we study the finiteness problem of central configurations. In n-body problem, the complexity of system of central configurations increases drastically as n increases. To study the finiteness problem of six body central configurations, there are enormous amount of computations in the process, therefore we develop symbolic computations for this problem. Since all computations are exact and involve only integers, there is no round-off error.

Based on Albouy and Kaloshin's approach, we introduce some matrix algebra and establish several criteria. Then we implement our algorithms to solve as many cases as possible.

In section 1, we introduce central configurations and the finiteness problem. In section 2, we briefly present Albouy and Kaloshin's method. Based on their approach, we induce some matrix algebra and criteria in sections 3,4,5. In sections 6,7,8, we presents main steps of our algorithms. The last three sections contains applications to n-body problems.

1. Introduction ... 1
2. Singular sequences, zw-diagrams, and coloring rules ... 4
3. Matrix rules for singular sequences ... 14
4. Order matrices for singular sequences ... 26
5. Collect polynomial equations ... 45
6. Algorithm I - Determine zw-diagrams ... 49
7. Algorithm II - Determine orders of variables ... 53
8. Algorithm III - Eliminations and mass relations ... 56
9. The case n = 4 ... 60
10. The case n = 5 ... 65
11. The case n = 6 ... 78
Appendix I: Program for Algorithm I by Mathematica ... 108
Appendix II: Program for Algorithm II by Mathematica ... 116
Appendix III: Program for Algorithm III by Mathematica ... 139
Appendix IV: Some frequently appeared factors in mass relations ... 160
Bibliography ... 169
                                

1. Albouy, A., On a paper of Moeckel on central configurations. Regul. Chaotic Dyn. 8, 133–142 (2003).

2. Albouy, A., Cabral, H. E., Santos, A.A., Some problems on the classical n-body problem. Celest. Mech. Dyn. Astron. 113, 369–375 (2012)

3. Albouy, A., Chenciner, A.: Le probl`eme des n corps et les distances mutuelles. Invent. Math. 131, 151–184 (1998).

4. Albouy, A., Kaloshin, V., Finiteness of central configurations for five bodies in the plane. Ann. Math. 176, 535–588 (2012).

5. Chazy, J., Sur certaines trajectoires du probl`eme des n corps, Bull. Astronom. 35, 321–389 (1918).

6. Dias, T., Pan, BY., Generic finiteness for a class of symmetric planar central configurations of the six-body problem and the six-vortex problem. J. Dyn. Diff. Equations 32, 1579–1602 (2020).

7. Dziobek, O., ¨Uber einen merkw¨urdigen Fall des ielk¨orperproblems, Astron. Nach. 152, 33–46 (1900).

8. Euler. L., Considerationes de motu corporum coelestium. Novi commentarii academiae scientiarum Petropolitanae. V. 10. 1764. P. 544–558. Berlin acad.. april 1762. Also in Opera Omnia. V. 25, S. 2. P. 246–257. with corrections and comments by M. Sch¨urer.

9. Hampton, M., Finiteness of kite relative equilibria in the five-vortex and five- body problems, Qual. Theory Dyn. Syst. 8 349–356 (2009).

10. Hampton, M., Jensen, A., Finiteness of spatial central configurations in the five-body problem, Celest. Mech. Dyn. Astr. 109:321–332 (2011)

11. Hampton, M., Moeckel, R., Finiteness of relative equilibria of the four-body problem, Invent. Math. 163, 289–312 (2006).

12. Jensen, A., Leykin, A., Smale’s 6th problem for generic masses. Preprint 2023, arXiv:2301.02305.

13. Lagrange, J. L., Essai sur le probl´eme des trois corps. OEuvres. V. 6. 1772. pp.229–324.

14. Leandro, E., Finiteness and bifurcations of some symmetrical classes of central configurations. Arch. Rational Mech. Anal. 167, 147–177 (2003).

15. Moczurad, M., Zgliczy´nski, P., Central configurations in planar n-body problem with equal masses for n = 5, 6, 7. Celest. Mech. Dyn. Astr. 131, 46 (2019).

16. Moczurad, M., Zgliczy´nski, P., Central configurations in the spatial n-body problem for n = 5, 6 with equal masses Celest. Mech. Dyn. Astr. 132, 56 (2020).

17. Moeckel, R., On central configurations. Math. Zeit., 205, 499–517. (1990).

18. Moeckel, R., Generic finiteness for Dziobek configurations. Trans. Amer. Math. Soc. 353, 4673–4686.

19. Moeckel, R., Lectures on central configurations, 2014.
http://www.math.umn.edu/~rmoeckel/notes/Notes.html

20. Moulton, F. R., The straight line solutions of the problem of N bodies. Ann. of Math. (2) 12, 1–17 (1910).

21. Roberts, G. E., A continuum of relative equilibria in the five-body problem Physica D 127, 141–145 (1999).

22. Smale, S., Mathematical problems for the next century, Math. Intelligencer 20, no. 2, 7–15 (1998).

23. Yu, X.; Zhu, S., Finiteness of central configurations of the Coulomb four-body problem. Preprint 2021.

24. Wintner, A., The Analytical Foundations of Celestial Mechanics, Princeton Math. Ser. 5, Princeton Univ. Press, Princeton, NJ, 1941.

25. Xia, Z., Central configurations with many small masses, J. Differential Equations 91, 168–179 (1991).

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