On the existence of symmetric bicircular central configurations of the 3n-body problem
Corbera Subirana, Montserrat (Universitat de Vic - Universitat Central de Catalunya)
Valls, Clàudia 1973- (Universidade de Lisboa. Instituto Superior Técnico. Departamento de Matemàtica)

Data:	2021
Resum:	In this paper, we consider central configurations of the planar 3n-body problem consisting of n masses at the vertices of a regular n-gon inscribed in a circle of radius r and 2n masses at the vertices of a second (not necessarily regular) concentric 2n-gon inscribed in a circle of radius ar which are symmetric in the sense that the set of positions of the 3n masses and the set of the corresponding masses are invariant under the action of a finite subgroup of O(2). There are two different types of such configurations. In the first type, called regular bicircular central configurations of the 3n-body problem, the second 2n-gon is regular, n of the vertices of the second n-gon are aligned with the vertices of the first regular n-gon and the masses at the vertices of this 2n-gon alternate values. In the second type, called semiregular bicircular central configurations of the 3n-body problem, the second 2n-gon is semiregular and the masses at its vertices are all of them equal. A semiregular 2n-gon has n pair of vertices symmetric by a reflection of an angle β with respect to the axis of symmetry of the first regular n-gon. Our aim is to analyze the set of values of the parameter a for the regular 2n-gon and of the parameters (a, β) for the semiregular 2n-gon providing symmetric bicircular central configurations. In particular, for all n≥ 2 we prove analytically the existence of symmetric bicircular central configurations with a (respectively (a, β)) satisfying some particular conditions. Using either computer-assisted results or numerical results, we also describe the complete set of values of a (respectively (a, β)) providing symmetric bicircular central configurations for n= 2, 3, 4, 5 and we give numerical evidences that the pattern for n> 5 is the same as the one for n= 5.
Ajuts:	Ministerio de Economía y Competitividad MTM2016-77278-P Agencia Estatal de Investigación PID2019-104658GB-I00
Drets:	Tots els drets reservats.
Llengua:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Matèria:	Bicircular central configuration ; N-body problem ; Regular n-gon
Publicat a:	Journal of Nonlinear Science, Vol. 31, Issue 6 (December 2021) , art. 88, ISSN 1432-1467

DOI: 10.1007/s00332-021-09743-z

Postprint 36 p, 1.9 MB

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