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On the Uniqueness of Convex Central Configurations in the Planar \(4\)-Body Problem

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Abstract

In this paper, we provide a rigorous computer-assisted proof (CAP) of the conjecture that in the planar four-body problem there exists a unique convex central configuration for any four fixed positive masses in a given order belonging to a closed domain in the mass space. The proof employs the Krawczyk operator and the implicit function theorem (IFT). Notably, we demonstrate that the implicit function theorem can be combined with interval analysis, enabling us to estimate the size of the region where the implicit function exists and extend our findings from one mass point to its neighborhood.

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  • 29 October 2023

    The numbering issue has been changed to 4-5.

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ACKNOWLEDGMENTS

We would like to acknowledge the valuable suggestions made by Professor Yiming Long, Professor Piotr Zgliczynski, and the anonymous referee.

Funding

Shanzhong Sun is partially supported by the National Key R&D Program of China (2020YFA 0713300), NSFC (Nos. 11771303, 12171327, 11911530092, 12261131498, 11871045). Zhifu Xie is partially supported by Wright W. and Annie Rea Cross Endowment Funds at the University of Southern Mississippi.

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Authors and Affiliations

  1. Department of Mathematics and Academy for Multidisciplinary Studies, Capital Normal University, 100048, Beijing, P. R. China

    Shanzhong Sun

  2. School of Mathematics and Natural Science, The University of Southern Mississippi, MS 39406, Hattiesburg, USA

    Zhifu Xie

  3. School of Mathematics and Statistics, Hebei University of Economics and Business, 050061, Shijiazhuang Hebei, P. R. China

    Peng You

Authors
  1. Shanzhong Sun
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  2. Zhifu Xie
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  3. Peng You
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Corresponding authors

Correspondence to Shanzhong Sun, Zhifu Xie or Peng You.

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The authors declare that they have no conflicts of interest.

Additional information

Dédié à Alain Chenciner avec admiration et amitié

MSC2010

70F10, 70F15

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Sun, S., Xie, Z. & You, P. On the Uniqueness of Convex Central Configurations in the Planar \(4\)-Body Problem. Regul. Chaot. Dyn. 28, 512–532 (2023). https://doi.org/10.1134/S1560354723520076

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  • DOI: https://doi.org/10.1134/S1560354723520076

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