Convex symmetric rectangular pentagon central configurations (Q6591008)

The paper deals with central configurations, which are special solutions of the \(n\)-body problem, which take place when the acceleration vector for each particle lines up with its force vector, and the scalar difference is the same for all particles.\N\NA \textit{symmetric rectangular pentagon configuration} of five bodies is one in which a single body lies along the symmetry axis, while four bodies are positioned at the vertices of a rectangle, and the fifth body is placed on a line which is parallel to and equidistant from the two parallel sides of the rectangle. A convex rectangular pentagon, also called \textit{house-shaped}, is a pentagon with the added restriction that two non-adjacent sides have equal lengths, each of which forms a right angle with the intervening side.\N\NIn this paper, the authors establish the existence of central configurations of the five-body problem, where the five bodies lie at the vertices of a symmetric house-shaped configuration. They achieve this by using the characterization of planar central configurations given by the Laura/Andoyer/Dziobek algebraic equations, that make use of the mutual distances between bodies as coordinates, see [\textit{O. Dziobek}, Astron. Nachr. 152, No. 3, 33--46 (1900; \url{doi:10.1002/asna.19001520302})]. The main results are formulated and proven in Section 3. Then, numerical and theoretical conclusions are presented using concrete examples of convex and concave configurations. Unfortunately, the existence of concave configurations is supported only by numerical evidence. This article concludes with a discussion section.