Central Configurations for NewtonianN+2p+1-Body Problems

Furong Zhao,Jian Chen

doi:10.1155/2013/272791

Abstract

We show the existence of spatial central configurations for theN+2p+1-body problems. In theN+2p+1-body problems,Nbodies are at the vertices of a regularN-gonT;2pbodies are symmetric with respect to the center ofT, and located on the straight line which is perpendicular to the regularN-gonTand passes through the center ofT; theN+2p+1th is located at the the center ofT. The masses located on the vertices of the regularN-gon are assumed to be equal; the masses located on the same line and symmetric with respect to the center ofTare equal.

Highlights

Introduction and Main ResultsThe Newtonian n-body problems [1,2,3] concern with the motions of n particles with masses mj ∈ R+ and positions qj ∈ R3 (j = 1, 2, . . . , n), and the motion is governed by Newton’s second law and the Universal law: mj q.. j = ∂U (q), ∂qj (1)where q = (q1, q2, . . . , qn) and with Newtonian potential: U (q) ∑1⩽j
We assume the claim is true for p − 1 and we will prove it for p
Since Mp = Mp = 0, we have that the first p − 1 equation of (12) is satisfied when λ = λ, Mi = Mi for i = 1, 2, . . . , p − 1 and Mp = Mp = 0

Summary

Introduction

By the symmetries of the system, (7) is equivalent to the following equations: N+2p+1 J=1,j =/ k mjmk 󵄨󵄨󵄨󵄨󵄨qj − qk󵄨󵄨󵄨󵄨󵄨3 In order to simplify the equations, we defined a0,j = −2/|1+rj2|3/2, aj,j = −1/4rj3, where j = 1, . Equations (9)–(11) can be written as a linear system of the form AX = b given by a0,1 a1,1 a2,1

Results

Conclusion