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Convexity and symmetry of central configurations in the five-body problem: Lagrange plus two

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Abstract

We study convexity and symmetry of central configurations in the five body problem when three of the masses ara located at the vertices of an equilateral triangle, that we call Lagrange plus two central configurations. First, we prove that the two bodies out of the vertices of the triangle cannot be placed on certain lines. Next, we give a geometrical characterization of such configurations in the sense as that of Dziobek, and we describe the admissible regions where the two remaining bodies can be placed. Furthermore, we prove that any Lagrange plus two central configuration is concave. Finally, we show numerically the existence of non-symmetric central configurations of the five body problem.

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Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

First and second authors are supported by MINECO grants MTM2016-80117-P and MTM2016-77278-P (FEDER) and Catalan (AGAUR) grants 2017 SGR 1374 and SGR 1617. The third author is partially supported Convenio Marco UBB1755/2016-2020 \(N^o\) 84, FAPEMIG APQ-03149-18 and CNPq 433285/2018-4. The fourth author is partially supported by Math Amsud-Conicyt 17-Math-07 and Fondecyt 1180288.

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Authors and Affiliations

  1. Departament d’Informàtica, Matemàtica Aplicada i Estadística, Universitat de Girona, Girona, Spain

    E. Barrabés

  2. Departament de Matemàtiques, Universitat Politècnica de Catalunya, Manresa, Spain

    J. M. Cors

  3. Instituto de Matemática e Computação, Universidade Federal de Itajubá, Itajubá, MG, Brazil

    A. C. Fernandes

  4. Grupo de Investigación en Sistemas Dinámicos y Aplicaciones-GISDA, Departamento de Matemática, Universidad del Bío-Bío, Concepción, Chile

    C. Vidal

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  1. E. Barrabés
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  2. J. M. Cors
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  3. A. C. Fernandes
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  4. C. Vidal
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Correspondence to A. C. Fernandes.

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Appendix

Appendix

The expressions \(N_i\), \(i=1,2,3\) in Eq. (12) are the following:

$$\begin{aligned} N_1= & {} \Delta _{235}\Delta _{123} \Delta _{145} \Big ( (R_{35}-R_{45}) (R_{25}R_{34}+R_{12}R_{24}) \Big . \\&\Big . + (R_{45}-R_{34}) (R_{24}R_{35}+R_{12}R_{25}) +(R_{34}-R_{35}) (R_{24}R_{25}+R_{12}R_{45})\Big ) \\ N_2= & {} \Delta _{125} \Delta _{134}^2 (\Delta _{123}-\Delta _{125}) R_{34} (R_{25}-R_{35}) (R_{34}-R_{45}) \\&+ \Delta _{124} \Delta _{135}^2 (\Delta _{123}-\Delta _{124}) R_{35} (R_{24}-R_{34}) (R_{35}-R_{45}) \\&+ \Delta _{124} \Delta _{135}^2 \Delta _{134} \Big (R_{24} R_{35} \left( R_{34}-R_{45}\right) - R_{24} R_{25} \left( R_{34}-R_{35}\right) \\&- R_{25} R_{34} \left( R_{35}-R_{45}\right) \Big ) +\Delta _{123} \Delta _{145} \Delta _{134}(\Delta _{123}-\Delta _{125}) R_{12} (R_{25}-R_{35}) (R_{34}-R_{45}) \\&- \Delta _{123} \Delta _{145} \Delta _{135}(\Delta _{123}-\Delta _{124}) R_{12} (R_{24}-R_{34}) (R_{35}-R_{45}) \\&- \Delta _{123} \Delta _{145}\Delta _{134} \Delta _{135} R_{12} \Big ( R_{24}(R_{35}-R_{45})-R_{25}(R_{34}-R_{45})\\&+R_{45}(R_{34}-R_{35}) \Big ) + \Delta _{134} \Delta _{135}R_{34}R_{35} \Big ( \Delta _{123} \Delta _{124} \left( R_{35}-R_{45}\right) \\&+\Delta _{125} \left( \Delta _{123} \left( R_{34}-R_{45}\right) +\Delta _{124}\left( R_{34}+R_{35}-2 R_{45}\right) \right) \Big ) \\&+ \Delta _{134} \Delta _{135}R_{34}R_{25} \Big ( \Delta _{125}\left( \Delta _{124}-\Delta _{123}\right) R_{34}\\&+ \left( \Delta _{123} \left( \Delta _{125}-\Delta _{124}\right) +\Delta _{125} \Delta _{134}\right) R_{35} \Big .\\&\Big . +\left( \Delta _{123} \Delta _{124}-\Delta _{125} \left( \Delta _{124}+\Delta _{134}\right) \right) R_{45}\Big ) \\&- \Delta _{134} \Delta _{135}R_{24}R_{25} \left( \Delta _{123} \left( \Delta _{124}-\Delta _{125}\right) -\Delta _{125} \Delta _{134}\right) \left( R_{34}-R_{35}\right) \\&- \Delta _{134} \Delta _{135}R_{24}R_{35} \Big (\Delta _{124} \left( \Delta _{125}-\Delta _{123}\right) R_{35}+\left( \Delta _{123} \left( \Delta _{124}-\Delta _{125}\right) \right. \\&\left. -\Delta _{125} \Delta _{134}\right) R_{34}+\Delta _{125} \Delta _{234} R_{45}\Big ) \\ N_3= & {} \Delta _{123}\Delta _{124} \Delta _{125} \Delta _{345} R_{24} ( R_{23}R_{34}-R_{25}R_{35}-R_{35}^2) \\&+ \Delta _{123}\Delta _{124} \Delta _{145} \Delta _{235} R_{24} R_{35}R_{45}\\&- \Delta _{123} \Delta _{124} \Delta _{135} \Delta _{245} R_{24}^2 \left( R_{25}-R_{35}\right) \\&+ \Delta _{123} \Delta _{125} R_{25} R_{34} \left( \Delta _{124} \Delta _{345} R_{35} -\Delta _{134} \Delta _{245} R_{25}+\Delta _{145} \Delta _{234} R_{45}\right) \\&- \Delta _{123} \Delta _{145} R_{12} R_{45} \left( \Delta _{125} \Delta _{234} R_{34}-\Delta _{124} \Delta _{235} R_{35}\right) \\&-\Delta _{123} \Delta _{145} R_{12} R_{24} \left( -\Delta _{123}\Delta _{245} R_{25}-\Delta _{125} \Delta _{234} R_{45}+\Delta _{124} \Delta _{235} R_{35}\right) \\&- \Delta _{123} \Delta _{145} R_{12} R_{25} \left( \Delta _{124} \Delta _{235} R_{45}-\Delta _{125} \Delta _{234} R_{34}\right) \\&+ \Delta _{123} \Delta _{245} R_{24} R_{25} \left( \Delta _{125} \Delta _{134} R_{25}+ \Delta _{123} \Delta _{145} R_{45} \right) \end{aligned}$$

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Barrabés, E., Cors, J.M., Fernandes, A.C. et al. Convexity and symmetry of central configurations in the five-body problem: Lagrange plus two. Qual. Theory Dyn. Syst. 20, 63 (2021). https://doi.org/10.1007/s12346-021-00504-7

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