New classes of spatial central configurations for the 7-body problem

Abstract

In this paper we show the existence of new families of spatial central configurations for the 7-body problem. In the studied spatial central configurations, six bodies are at the vertices of two equilateral triangles T 1 , T 2 and one body is located out of the parallel distinct planes containing T 1 and T 2 . The results have simple and analytic proofs.

Central configurations of the spatial seven-body problem: two twisted triangles plus one

“…Our result is in accordance with the provisions by Fernandes and Mello (2014). In fact, we obtain basically the same results but the calculation here were a little easier.…”

“…In fact, we obtain basically the same results but the calculation here were a little easier. If we take a system of coordinates in which Again, as well as the results and provisions by Fernandes and Mello (2014), there is not restriction on the choice of m 7 . Equation (2) says that the masses m 1 , m 2 , m 3 , m 4 , m 5 and m 6 are at the vertices of a regular octahedron, a well known central configuration.…”

“…The seventh body is located between these planes on the line perpendicular passing through the center of the triangles and equidistant to them. Fernandes and Mello (2014) studied the case where the triangles are not rotated and they proof the existence of one and only one class of central configuration of that problem. Our results are similar to them as they had predicted.…”