On the convex central configurations of the symmetric (ℓ + 2)-body problem
Corbera Subirana, Montserrat (Universitat de Vic. Departament d'Enginyeries)
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Yuan, Pengfei (Southwest University. School of Mathematics and Statistics (China))

Date:	2020
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.
Grants:	Ministerio de Economía y Competitividad MTM2016-77278-P Agència de Gestió d'Ajuts Universitaris i de Recerca 2017/SGR-1617 European Commission 777911
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Convex central configurations ; (ℓ + 2)-body problem
Published in:	Regular and Chaotic Dynamics, Vol. 25, Issue 3 (May 2020) , p. 250-272, ISSN 1468-4845

DOI: 10.1134/S1560354720030028

Postprint 29 p, 827.1 KB

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