Talk 8

Title: On the Relative Equilibria of the (Rhombus+1)-Body Problem 

Speaker:  Juscelino Grigório Lopes

Affiliation: Professor,  Universidade de Pernambuco - Campus Petrolina

Abstract: In this talk, 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. Our results were published in the journal  Celestial Mechanics and Dynamical Astronomy, Vol. 134:47, in October, 2022.