A note on the two nested regular polygonal central configurations

Abstract

This paper is concerned with nested polygonal central configurations for the Newtonian 2n-body problem. We show that when two nested regular n-polygons ( n ≥ 3 n\geq 3 ) with masses located at the vertices form a central configuration where the twisted angle θ \theta is zero, then the value of masses in each separate polygon must be equal.

Twisted stacked central configurations for the spatial nine-body problem

“…, q 2N form a spatial central configuration. For more details in these direction, one can refer to [11,14,15,17,19].…”

“…Note that in 2003, for the planar twisted central configurations (i.e. h = 0) formed by two regular N −polygons with any twist angle θ, Zhang and Zhou [19] arrived at the conclusion that the values of masses in each separate regular N -polygons must be equal (but without detailed proof); in 2015, based on the eigenvalues of circulant matrices, Wang and Li [11] investigated the masses of the 2N bodies for h 0 with twist angle θ = 0, and they also obtained the values of masses in each separate regular N -polygons must be equal. Moreover, we also note that Yu and Zhang [16] proved that if the central configuration is formed by two twisted regular N -polygons with distance h 0, then the twist angles must be θ = 0 or θ = π/N , so we want to study that for the spatial central configuration (h > 0) formed by two twisted regular N -polygons with twist angle θ = π/N , whether the sizes of each separate regular N -polygons must be equal or not ?…”

Planar N-body central configurations with a homogeneous potential

“…including the central mass), the central mass should become inconsequential for large N and A. Much is also already known about nested and 'twisted' regular polygon configurations [42,100,145,126,78,82,31,27,144,17,138,146] which would be another relatively easy extension. (2) Numerically complete an analysis of all bifurcations in the equal mass N -body problem as the potential exponent A is varied in [2, ∞) for N ∈ {6, .…”

Eulerian collinear configuration for 3-body problem

“…The study of central configurations is a very important subject in celestial mechanics with a long and varied history [18], and a well-known fact is that finding the relative equilibrium solutions of the classical N -body problem and the planar central configurations are equivalent. There are a lot of elegant works on central configurations [8,12,15,19,20,24,27,28,30], but very few works are concentrated on finding the concrete central configurations. In this paper, we are interested in searching a kind of important central configuration: Eulerian collinear configuration.…”