Twisted stacked central configurations for the spatial nine-body problem

Abstract

We study the spatial central configuration formed by two twisted regular N -polygons with any twist angle θ, and prove that the sizes of the two regular N -polygons must be equal to each other.

Classification of stacked central configurations in R^3

“…Assume that all the particles are on the x-axis, and use x i to denote the position of m i . The equations become (7) Assume that z 1 , z 2 , z 3 are the roots of equation ( 8) and z 4 , z 5 , z 6 are the roots of equation (9). We are going to finish the proof by showing that the sum of the number of negative roots of equation ( 8) and the number of positive roots of equation ( 9) is not greater than 2.…”

“…Hampton introduced stacked central configurations in 2005 [17]. Many other examples of stacked central configurations were constructed, see [7,9,18,24].…”