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On the relative equilibria of the (rhombus+1)-body problem

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Abstract

We consider some basic issues on the relative equilibria of the restricted five-body problem in which the primaries form a rhombus relative equilibrium. Firstly, we elicit a basic structure underlying the relative equilibrium equations, an involutive diffeomorphism which has proved very useful as a simplifying tool throughout our work. Next, we exhibit a bounded neighborhood of the rhombus family containing all relative equilibria, determine all the solutions for the values of the mass ratio m of the primaries either sufficiently large or close to zero, and enumerate the symmetric classes of relative equilibria for any allowable choice of values for m and for the rhombus semi-diagonal length d. Degeneracy of symmetric classes is also examined, and the related bifurcations are discussed. The presence of a nonlinear constraint to be satisfied by the parameters m and d represents a significant challenge to a rigorous analysis and, to the best of our knowledge, a novelty in analytical investigations of bifurcations in the N-body problem.

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Authors and Affiliations

  1. Departamento de Ciências Exatas, Universidade de Pernambuco, Rodovia BR 203, Km 2, S/N, Petrolina, PE, 56328-900, Brazil

    Juscelino G. Lopes

  2. Departamento de Matemática, Universidade Federal de Pernambuco, Av. Jornalista Aníbal Fernandes S/N, Recife, PE, 50740-560, Brazil

    Eduardo S. G. Leandro

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  1. Juscelino G. Lopes
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  2. Eduardo S. G. Leandro
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Correspondence to Eduardo S. G. Leandro.

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This article is part of the topical collection on Theory and applications of resonances and central configurations.

Guest Editors: Antonio Bertachini Prado, Eduardo S.G. Leandro, Nelson Callegari Jr., and Alessandra Celletti.

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Lopes, J.G., Leandro, E.S.G. On the relative equilibria of the (rhombus+1)-body problem. Celest Mech Dyn Astron 134, 47 (2022). https://doi.org/10.1007/s10569-022-10100-9

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  • DOI: https://doi.org/10.1007/s10569-022-10100-9

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