Spatial Central Configurations Research Articles

Finiteness of Spatial Central Configurations with Fixed Subconfigurations

Finiteness of Spatial Central Configurations with Fixed Subconfigurations

Notes on spatial twisted central configurations for $2N$-body problem

We are concerned with the spatial twisted central configuration formed by two paralleled regular \(N\)-polygons (\(N\geqslant 3\)) with twist angle \(\theta \), and we prove that the values of the masses in each separate regular \(N\)-polygon are equal if and only if \(\theta =0\) or \(\theta =\pi /N\). Moreover, suppose that all of the sides in one regular \(N\)-polygon have length \(a\) and all of the sides in the other regular \(N\)-polygon have length 1, then for the case of the twist angle is \(\theta =\pi /N\), if \(3\leqslant N\leqslant 8\), for any \(a>0\), the spatial twisted central configuration for \(2N\)-body problem exists. Furthermore, whether the twist angle is \(\theta =0\) or \(\theta =\pi /N\): for every \(N\geqslant 3\), if all the masses of the \(2N\) bodies or the lengths of all the sides of the two regular \(N\)-polygons are equal to each other, then there is only one spatial central configuration, and the distance between the two paralleled regular \(N\)-polygons with \(\theta =0\), must be larger than the case of \(\theta =\pi /N\).

New central configurations of the (n+1)-body problem

Abstract In this article we study central configurations of the ( n + 1 ) -body problem. For the planar ( n + 1 ) -body problem we study central configurations performed by n ≥ 2 bodies with equal masses at the vertices of a regular n -gon and one body with null mass. We also study spatial central configurations considering n bodies with equal masses at the vertices of a regular polyhedron and one body with null mass.

Topological bifurcations of spatial central configurations in the N-body problem

We study topological bifurcations of classes of spatial central configurations (s.c.c.) from the following highly symmetrical families: two nested regular tetrahedra, octahedra and cubes, two nested rotated regular tetrahedra and two dual regular polyhedra for 14 bodies. We prove the existence of local and global topological bifurcations of s.c.c. from these families. We seek new classes of s.c.c. by using equivariant bifurcation theory. It is worth pointing out that the shapes of the bifurcating families are less symmetrical than the shapes of the considered families of s.c.c.

New classes of spatial central configurations for the 7-body problem

In this paper we show the existence of new families of spatial central configurations for the 7-body problem. In the studied spatial central configurations, six bodies are at the vertices of two equilateral triangles , and one body is located out of the parallel distinct planes containing and . The results have simple and analytic proofs.

Spatial bi-stacked central configurations formed by two dual regular polyhedra

In this paper we prove the existence of two new families of spatial stacked central configurations, one consisting of eight equal masses on the vertices of a cube and six equal masses on the vertices of a regular octahedron, and the other one consisting of twenty masses at the vertices of a regular dodecahedron and twelve masses at the vertices of a regular icosahedron. The masses on the two different polyhedra are in general different. We note that the cube and the octahedron, the dodecahedron and the icosahedron are dual regular polyhedra. The tetrahedron is itself dual. There are also spatial stacked central configurations formed by two tetrahedra, one and its dual.

New Stacked Central Configurations for the Spatial Seven-Body Problem

In this paper we show the existence of three new families of spatial central configurations for the 7-body problem. More precisely, the position vectors x 1 , x 2 , x 3 , x 4 and x 5 are at the vertices of a square pyramid, the position vectors x 6 and x 7 are on a line connecting x 5 with the center of the opposite square.

Stacked central configurations for Newtonian [formula omitted]-body problems

We show the existence of spatial central configurations for the N+2p-body problems. In such a configuration, N bodies are at the vertices of a regular N-gon T; 2p bodies are symmetric with respect to the center of T, and located on the straight line which is perpendicular to the regular N-gon T and passes through the center of T. The masses located on the vertices of the regular N-gon are assumed to be equal; the masses located on the same line and symmetric with respect to the center of T are equal.

Spatial Central Configurations with Two Twisted Regular 4-Gons

We study the configuration formed by two squares in two parallel layers separated by a distance. We picture the two layers horizontally with thez-axis passing through the centers of the two squares. The masses located on the vertices of each square are equal, but we do not assume that the masses of the top square are equal to the masses of the bottom square. We prove that the above configuration of two squares forms a central configuration if and only if the twist angle is equal tokπ/2or (π/4+kπ/2)(k=1,2,3,4).

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

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.

Stacked Central Configurations for the Spatial Seven–Body Problem

In this article we study the existence of new families of spatial stacked central configurations for the 7–body problem. We study spatial central configurations where four bodies are at the vertices of a regular tetrahedron $${\mathcal T}$$ and the other three bodies are symmetrically located at the vertices of an equilateral triangle $${\mathcal{E}}$$ in the exterior of $${\mathcal{T}}$$ .

New Classes of Spatial Central Configurations forN+N+ 2-Body Problem

Under arbitrary masses, in this paper, we discuss the existence of new families of spatial central configurations for the <i >N</i> + <i >N</i> + 2-body problem, <svg style="vertical-align:-1.29163pt;width:40.625px;" id="M1" height="12.3" version="1.1" viewBox="0 0 40.625 12.3" width="40.625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.3)"> <g transform="translate(72,-62.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑁</tspan> <tspan style="font-size: 12.50px; " x="14.103384" y="0">≥</tspan> <tspan style="font-size: 12.50px; " x="26.143772" y="0">2</tspan> </text> </g> </g> </svg>. We study some necessary conditions and sufficient conditions for a families of spatial double pyramidical central configurations (d.p.c.c.), where 2<i >N</i> bodies are at the vertices of a nested regular <i >N</i>-gons <svg style="vertical-align:-3.13504pt;width:47.299999px;" id="M2" height="14.375" version="1.1" viewBox="0 0 47.299999 14.375" width="47.299999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.375)"> <g transform="translate(72,-60.5)"> <text transform="matrix(1,0,0,-1,-71.95,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0">Γ</tspan> </text> <text transform="matrix(1,0,0,-1,-64.61,60.55)"> <tspan style="font-size: 8.75px; " x="0" y="0">1</tspan> </text> <text transform="matrix(1,0,0,-1,-56.96,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0">∪</tspan> <tspan style="font-size: 12.50px; " x="10.527526" y="0">Γ</tspan> </text> <text transform="matrix(1,0,0,-1,-39.09,60.55)"> <tspan style="font-size: 8.75px; " x="0" y="0">2</tspan> </text> </g> </g> </svg>, and the other two bodies are symmetrically located on the straight line that is perpendicular to the plane that contains <svg style="vertical-align:-3.13504pt;width:47.299999px;" id="M3" height="14.375" version="1.1" viewBox="0 0 47.299999 14.375" width="47.299999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.375)"> <g transform="translate(72,-60.5)"> <text transform="matrix(1,0,0,-1,-71.95,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0">Γ</tspan> </text> <text transform="matrix(1,0,0,-1,-64.61,60.55)"> <tspan style="font-size: 8.75px; " x="0" y="0">1</tspan> </text> <text transform="matrix(1,0,0,-1,-56.96,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0">∪</tspan> <tspan style="font-size: 12.50px; " x="10.527526" y="0">Γ</tspan> </text> <text transform="matrix(1,0,0,-1,-39.09,60.55)"> <tspan style="font-size: 8.75px; " x="0" y="0">2</tspan> </text> </g> </g> </svg> and passes through the geometric center of <svg style="vertical-align:-3.13504pt;width:47.299999px;" id="M4" height="14.375" version="1.1" viewBox="0 0 47.299999 14.375" width="47.299999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.375)"> <g transform="translate(72,-60.5)"> <text transform="matrix(1,0,0,-1,-71.95,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0">Γ</tspan> </text> <text transform="matrix(1,0,0,-1,-64.61,60.55)"> <tspan style="font-size: 8.75px; " x="0" y="0">1</tspan> </text> <text transform="matrix(1,0,0,-1,-56.96,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0">∪</tspan> <tspan style="font-size: 12.50px; " x="10.527526" y="0">Γ</tspan> </text> <text transform="matrix(1,0,0,-1,-39.09,60.55)"> <tspan style="font-size: 8.75px; " x="0" y="0">2</tspan> </text> </g> </g> </svg>. We prove that if the bodies are in a d.p.c.c., then the masses on each <i >N</i>-gon are equal, and the other two are also equal. And also we prove the existence and uniqueness of the central configurations for any given ratios of masses.

Convex Central Configurations of the n-Body Problem Which are not Strictly Convex

It is well-known that if a planar central configuration for the Newtonian 4-body problem is convex, then it must be strictly convex. In some literature, same conclusion was believed to hold for the case of five or even more bodies but rigorous treatments are absent. With the help of some numerical calculations, in this paper we provide concrete examples of central configurations which are convex but not strictly convex. Our examples include planar central configurations with five bodies and spatial central configurations with seven bodies.

New spatial central configurations in the 5-body problem

In this paper we show the existence of new families of convex and concave spatial central configurations for the 5-body problem. The bodies studied here are arranged as follows: three bodies are at the vertices of an equilateral triangle T, and the other two bodies are on the line passing through the barycenter of T that is perpendicular to the plane that contains T.

Finiteness of spatial central configurations in the five-body problem

We strengthen a generic finiteness result due to Moeckel by showing that the number of spatial central configurations of the Newtonian five-body problem with positive masses is finite, apart from some explicitly given special cases of mass values.

New classes of spatial central configurations for the [formula omitted]-body problem

In this paper, we show the existence of new families of spatial central configurations for the n + 3 -body problem, n ≥ 3 . We study spatial central configurations where n bodies are at the vertices of a regular n -gon T and the other three bodies are symmetrically located on the straight line that is perpendicular to the plane that contains T and passes through the center of T . The results have simple and analytic proofs.

Stacked central configurations for the spatial six-body problem

In this paper we show the existence of three new families of stacked spatial central configurations for the six-body problem with the following properties: four bodies are at the vertices of a regular tetrahedron and the other two bodies are on a line connecting one vertex of the tetrahedron with the center of the opposite face.

On the spatial central configurations of the 5--body problem and their bifurcations

Central configurations provide special solutions of the general $n$--body problem. Using the mutual distances between the $n$ bodies as coordinates we study the bifurcations of the spatial central configurations of the $5$--body problem going from the problem with equals masses to the $1+4$-- body problem which has one large mass and four infinitesimal equal masses. This study is made by giving a computer--aided proof.

Bifurcation of Spatial Central Configurations from Planar Ones

Central configurations are important special solutions of the Newtonian N-body problem of celestial mechanics. In this paper a highly symmetrical case is studied. As the masses are varied, spatial central configurations appear through bifurcation from planar ones. In particular, spatial configurations can be found which are arbitrarily close to being planar.