Central configurations in the five-body problem: rhombus plus one
The paper demonstrates the existence of a particular central configuration of five bodies. Specifically, four masses lie at the vertices of a rhombus with the remaining fifth body lying on one of the diagonals of the rhombus. The authors consider both the case where the fifth body lies inside and outside the rhombus. Positions and masses of the five bodies are determined by examination of the Dziobek/Laura/Andoyer equations. These give a system of equations in terms of the positions and masses of the bodies and the signed areas of the triangles formed by sets of three bodies. Under these conditions, it can be shown that on the diagonal containing two bodies, both must have equal mass. With suitable rescaling, these are given unit mass, and the length of the diagonal is set to one unit as well. A numerical plot shows a 2-dimensional region of possible values for the positions of the bodies in terms of the position of the non-vertex body and the length of the non-unit diagonal. The remainder of the paper is dedicated to showing that for a given set of positions, the mass values are unique. This is done by writing the remaining unsolved Dziobek/Laura/Andoyer equations in terms of a matrix equation with the non-unit masses as variables. The proof is given by establishing the invertibility of the coefficient matrix. This is done by reducing the determinant of the matrix to a polynomial equation and proving that there are no zeros within particular suitable intervals. For those who research central configurations, the results are worth noting.
scientific article; zbMATH DE number 7383378
scientific article