Regions of Central Configurations in a Symmetric 4 + 1-Body Problem

The symmetric central configurations of the 4-body problem with masses

We characterize the planar central configurations of the 4body problem with masses m1 = m2 ̸= m3 = m4 which have an axis of symmetry. It is known that this problem has exactly two classes of convex central configurations, one with the shape of a rhombus and the other with the shape of an isosceles trapezoid. We show that this 4-body problem also has exactly two classes of concave central configu...

Existence of symmetric central configurations

Central configurations have been of great interest over many years, with the earliest examples due to Euler and Lagrange. There are numerous results in the literature demonstrating the existence of central configurations with specific symmetry properties, using slightly different techniques in each. The aim here is to describe a uniform approach by adapting to the symmetric case the well-known ...

Central configurations and relative equilibria in the n–body problem

Stacked Central Configurations for the Spatial Nine-Body Problem

We show the existence of the twisted stacked central configurations for the 9-body problem. More precisely, the position vectors x₁, x₂, x₃, x₄, and x₅ are at the vertices of a square pyramid Σ; the position vectors x₆, x₇, x₈, and x₉ are at the vertices of a square Π.

Central Configurations for Newtonian N + 2 p + 1 - Body Problems

and Applied Analysis 3 −λ (0, 0, rp) = − N 󵄨󵄨󵄨󵄨󵄨1 + r2 p󵄨󵄨󵄨󵄨󵄨3/2 (0, 0, rp) − (0, 0, 1 r2 p)M0 − ∑ 1≤i<p ( 1 󵄨󵄨󵄨󵄨󵄨ri − rp󵄨󵄨󵄨󵄨󵄨2 + 1 󵄨󵄨󵄨󵄨󵄨ri + rp󵄨󵄨󵄨󵄨󵄨2)Mi (0, 0, 1) − 1 󵄨󵄨󵄨󵄨󵄨2rp󵄨󵄨󵄨󵄨󵄨2Mp (0, 0, 1) . (11) In order to simplify the equations, we defined a0,j = −2/|1+r2 j |3/2, aj,j = −1/4r3 j , where j = 1, . . . , p; ai,j = (1/|ri − rj|2ri − 1/|ri + rj|2ri), when i < j; ai,j = −(1/|ri − rj|2ri + 1/...