The Axisymmetric Central Configurations of the Four-Body Problem with Three Equal Masses

All the Lagrangian relative equilibria of the curved 3-body problem have equal masses

We consider the 3-body problem in 3-dimensional spaces of nonzero constant Gaussian curvature and study the relationship between the masses of the Lagrangian relative equilibria, which are orbits that form a rigidly rotating equilateral triangle at all times. There are three classes of Lagrangian relative equilibria in 3-dimensional spaces of constant nonzero curvature: positive elliptic and positive elliptic-elliptic, on 3-spheres, and negative elliptic, on hyperbolic 3-spheres. We prove that all these Lagrangian relative equilibria exist only for equal values of the masses.

三质量相等的平面四体问题中的凹风筝中心构型

Concave Kite Central Configurations in the Planar Four-Body Problem with Three Equal Masses

本文对三质量相等的平面四体问题中凹风筝中心构型进行了完整分类，通过严格计算机辅助分析方法证明了每种情况下的单参数族，并展示了分岔点的存在与类型。

Conjugated equilibrium solutions for the $2$--body problem in the two dimensional sphere $\mathbb{M}^2_R$ for equal masses

We study here the behaviour of solutions for conjugated (antipodal) points in the $2$-body problem on the two-dimensional sphere $\mathbb{M}^2_R$. We use a slight modification of the classical potential used commonly in \cite{Borisov}, \cite{Diacu} and \cite{Perez}, which avoids the conjugated (antipodal) points as singularities and permit us obtain solutions through these points, as limit of relative equilibria. Such limit solutions behave as relative equilibria because are invariant under Killing vector fields in the Lie Algebra ${\rm su} (2)$ and are geodesic curves.

关于三个不等质量的中心共圆中心构型的不存在性

On the Nonexistence of Centered Co-Circular Central Configurations With Three Unequal masses

本文针对一般幂律势n体问题，证明了含n-3个等质量体与三个任意质量体的系统在特定质量条件下不存在中心共圆中心构型。

Four-body central configurations with one pair of opposite sides parallel

We study four-body central configurations with one pair of opposite sides parallel. We use a novel constraint to write the central configuration equations in this special case, using distances as variables. We prove that, for a given ordering of the mutual distances, a trapezoidal central configuration must have a certain partial ordering of the masses. We also show that if opposite masses of a four-body trapezoidal central configuration are equal, then the configuration has a line of symmetry and it must be a kite. In contrast to the general four-body case, we show that if the two adjacent ma…

两对等质量凹形四体问题中的对称中心构型

Symmetric Central Configurations in the Concave 4-Body Problem with Two Pairs of Equal Masses

研究两对等质量四体问题中凹形风筝中心构型的存在性，通过严格计算机辅助分析证明对任意非负质量比，此类构型数量为零、一或二，并给出完整分类和分岔图。

Action Minimizing Solutions of The One-Dimensional $N$-Body Problem With Equal Masses

When we use variational methods to study the Newtonian $N$-body problem, the main problem is how to avoid collisions. C.Marchal got a remarkable result, that is, a path minimizing the Lagrangian action functional between two given configurations is always a true (collision-free) solution, so long as the dimension $d$ of physical space $\mathbb{R}^d$ satisfies $d\geq2$. But Marchal's idea can't apply to the case of the one-dimensional physical space. In this paper, we will study the fixed-ends problem for the one-dimensional Newtonian $N$-body problem with equal masses to supplement Marchal's r…

Action minimizing orbits in the 2-center problems with simple choreography constraint

The aim of this paper is to study the motion of $2+n$-body problem where two equal masses are assumed to be fixed. We assume that the value of each fixed mass is equal to $M>0$ and the remaining $n$ moving particles have equal masses $m>0$. According to Newton's second law and the universal gravitation law, the $n$ particles move under the interaction of each other and the affection of the two fixed particles. Also, this motion has a natural variational structure. Under the simple choreography constraint, we show that the Lagrangian action functional attains its absolute minimum on a uniform c…

Post-Newtonian effects on Lagrange's equilateral triangular solution for the three-body problem

Continuing work initiated in earlier publications [Yamada, Asada, Phys. Rev. D 82, 104019 (2010), 83, 024040 (2011)], we investigate the post-Newtonian effects on Lagrange's equilateral triangular solution for the three-body problem. For three finite masses, it is found that the equilateral triangular configuration satisfies the post-Newtonian equation of motion in general relativity, if and only if all three masses are equal. When a test mass is included, the equilateral configuration is possible for two cases: (1) one mass is finite and the other two are zero, or (2) two of the masses are fi…