Research article Special Issues

Remarks on the inverse problem of the collinear central configurations in the $ N $-body problem

  • Received: 13 October 2021 Revised: 17 January 2022 Accepted: 04 February 2022 Published: 11 May 2022
  • For a fixed configuration in the collinear $ N $-body problem, the existence of the central configurations is determined by a system of linear equations, which in turn is determined by certain Pfaffians for even or odd $ N $ in literatures. In this short note, we prove that the Pfaffians of the associate matrices for all even number collinear configurations are nonzero if and only if the extended Pfaffians of the associate matrices for all odd number collinear configurations are nonzero. Therefore, the inverse problem of the collinear central configurations can be answered and each collinear configuration determines a one-parameter family of masses with a fixed total mass if the Pfaffians of the associate matrix for all collinear even number bodies are nonzero. We also make some remarks on the super central configurations and the number of collinear central configurations under different equivalences, especially a lower bound for the number of collinear central configurations under the geometric equivalence.

    Citation: Zhifu Xie. Remarks on the inverse problem of the collinear central configurations in the $ N $-body problem[J]. Electronic Research Archive, 2022, 30(7): 2540-2549. doi: 10.3934/era.2022130

    Related Papers:

  • For a fixed configuration in the collinear $ N $-body problem, the existence of the central configurations is determined by a system of linear equations, which in turn is determined by certain Pfaffians for even or odd $ N $ in literatures. In this short note, we prove that the Pfaffians of the associate matrices for all even number collinear configurations are nonzero if and only if the extended Pfaffians of the associate matrices for all odd number collinear configurations are nonzero. Therefore, the inverse problem of the collinear central configurations can be answered and each collinear configuration determines a one-parameter family of masses with a fixed total mass if the Pfaffians of the associate matrix for all collinear even number bodies are nonzero. We also make some remarks on the super central configurations and the number of collinear central configurations under different equivalences, especially a lower bound for the number of collinear central configurations under the geometric equivalence.



    加载中


    [1] S. Smale, Mathematical problems for the next century, Math. Intell., 20 (1998), 7–15. https://doi.org/10.1007/BF03025291 doi: 10.1007/BF03025291
    [2] F. R. Moulton, The straight line solutions of the problem of $N$-bodies, Ann. Math., 2 (1910), 1–17. https://doi.org/10.2307/2007159 doi: 10.2307/2007159
    [3] L. Euler, De motu rectilineo trium corporum se mutuo attahentium, Novi Comm. Acad. Sci. Imp. Petrop., 11 (1767), 144–151.
    [4] J. Lagrange, Essai sur le probleme des trois corps, Euvres, Gauthier-Villars, Paris, 6 (1772), 2721–7292.
    [5] M. Hampton, R. Moeckel, Finiteness of relative equilibria of the four body problem, Invent. Math., 163 (2006), 289–312. https://doi.org/10.1007/s00222-005-0461-0 doi: 10.1007/s00222-005-0461-0
    [6] A. Albouy, V. Kaloshin, Finiteness of central configurations of five bodies in the plane, Ann. Math., 176 (2012), 1–54. https://doi.org/10.4007/annals.2012.176.1.10 doi: 10.4007/annals.2012.176.1.10
    [7] A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton Math., Series 5,215. Princeton Univ. Press, Princeton, NJ. 1941. 3rd printing 1952.
    [8] H. E. Buchanan, On certain determinants connected with a problem in Celestial Mechanics, Bull. Amer. Math. Soc., 5 (1909), 227–231. https://doi.org/10.1090/S0002-9904-1909-01744-6 doi: 10.1090/S0002-9904-1909-01744-6
    [9] A. Albouy, R. Moeckel, The inverse problem for collinear central configuration, Celest. Mech. Dyn. Astron., 77 (2000), 77–91. https://doi.org/10.1023/A:1008345830461 doi: 10.1023/A:1008345830461
    [10] Z. Xie, An analytical proof on certain determinants connected with the collinear central configurations in the $n$-body problem, Celest. Mech. Dyn. Astron., 118 (2014), 89–97. https://doi.org/10.1007/s10569-013-9525-4 doi: 10.1007/s10569-013-9525-4
    [11] D. Ferrario, Pfaffians and the inverse problem for collinear central configurations. Celest. Mech. Dyn. Astron., 132 (2020), 1–16. https://doi.org/10.1007/s10569-020-09975-3
    [12] T. Ouyang, Z. Xie, Collinear Central Configuration in Four-body Problem, Celest. Mech. Dyn. Astron., 93 (2005), 147–166. https://doi.org/10.1007/s10569-005-6596-x doi: 10.1007/s10569-005-6596-x
    [13] Y. Long, S. Sun, Collinear Central Configurations and Singular Surfaces in the Mass Space, Arch. Rational Mech. Anal., 173 (2004), 151–167. https://doi.org/10.1007/s00205-004-0314-9 doi: 10.1007/s00205-004-0314-9
    [14] Y. Long, S. Sun, Collinear central configurations in celestial mechanics, In Topological Methods, Variational Methods and Their Applications: ICM 2002 Satellite Conference on Nonlinear Functional Analysis, World Scientifc, 2003. p159–165.
    [15] T. Ouyang, Z. Xie, Number of Central Configurations and Singular Surfaces in Mass Space in the Collinear Four-body Problem, Trans. Am. Math. Soc., 364 (2012), 2909–2932. https://doi.org/10.1090/S0002-9947-2012-05426-2 doi: 10.1090/S0002-9947-2012-05426-2
    [16] Z. Xie, Central Configurations of the Collinear Three-body Problem and Singular Surfaces in the Mass Space, Phys. Lett. A, 375 (2011), 3392–3398. https://doi.org/10.1016/j.physleta.2011.07.047 doi: 10.1016/j.physleta.2011.07.047
    [17] Z. Xie, Super Central Configurations of the $n$-body Problem, J. Math. Phys., 51 (2010), 042902. https://doi.org/10.1063/1.3345125 doi: 10.1063/1.3345125
    [18] Z. Xie, W. Johnson, Super Central configurations in the Collinear 5-body Problem, Appl. Math. Comput., 381 (2020), 125194. https://doi.org/10.1016/j.amc.2020.125194 doi: 10.1016/j.amc.2020.125194
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(827) PDF downloads(48) Cited by(0)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog