Abstract
For the 4-body problem there is the following conjecture: Given arbitrary positive masses, the planar 4-body problem has a unique convex central configuration for each ordering of the masses on its convex hull. Until now this conjecture has remained open. Our aim is to prove that this conjecture cannot be extended to the (ℓ + 2)-body problem with ℓ ⩾ 3. In particular, we prove that the symmetric (2n + 1)-body problem with masses m1 = … = m2n−1 = 1 and m2n = m2n+1 = m sufficiently small has at least two classes of convex central configuration when n = 2, five when n = 3, and four when n = 4. We conjecture that the (2n + 1)-body problem has at least n classes of convex central configurations for n > 4 and we give some numerical evidence that the conjecture can be true. We also prove that the symmetric (2n + 2)-body problem with masses m1 = … = m2n = 1 and m2n+1 = m2n+2 = m sufficiently small has at least three classes of convex central configuration when n = 3, two when n = 4, and three when n = 5. We also conjecture that the (2n + 2)-body problem has at least [(n +1)/2] classes of convex central configurations for n > 5 and we give some numerical evidences that the conjecture can be true.
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Funding
The first two authors are partially supported by the Ministerio de Economía, Industria y Competitividad, Agencia Estatal de Investigatión grants MTM2016-77278-P (FEDER). The second author is also partially supported by the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The third author is partially supported by Fundamental Research Funds for the Central Universities (NO.XDJK2015C139), China Scholarship Council (No. 201708505030), the National Natural Science Foundation of China (grant No. 11626193).
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The authors declare that they have no conflicts of interest.
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Corbera, M., Llibre, J. & Yuan, P. On the Convex Central Configurations of the Symmetric (ℓ + 2)-body Problem. Regul. Chaot. Dyn. 25, 250–272 (2020). https://doi.org/10.1134/S1560354720030028
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DOI: https://doi.org/10.1134/S1560354720030028