Number Of Central Configurations Research Articles

Central Configurations of the Curved N-Body Problem

We consider the $N$-body problem of celestial mechanics in spaces of nonzero constant curvature. Using the concept of locked inertia tensor, we compute the moment of inertia for systems moving on spheres and hyperbolic spheres and show that we can recover the classical definition in the Euclidean case. After proving some criteria for the existence of relative equilibria, we find a natural way to define the concept of central configuration in curved spaces using the moment of inertia, and show that our definition is formally similar to the one that governs the classical problem. The existence criteria we develop for central configurations help us provide several examples and prove that, for any given point masses on spheres and hyperbolic spheres, central configurations always exist. We end our paper with results concerning the number of central configurations that lie on the same geodesic, thus extending the celebrated theorem of Moulton to hyperbolic spheres and pointing out that it has no straightforward generalization to spheres, where the count gets complicated even in the case $N=2$.

Dziobek configurations of the restricted (N + 1)-body problem with equal masses

In this article, we count the number of (N − 1)-dimensional central configurations of the restricted (N + 1)-body problem with equal masses. For N = 4, 5, there are 25, 56 central configurations. For N = 3 or N ⩾ 6, the number of central configurations is 2N + N − 1.

Number of central configurations and singular surfaces in the mass space in the collinear four-body problem

Number of central configurations and singular surfaces in the mass space in the collinear four-body problem

Central configurations of the collinear three-body problem and singular surfaces in the mass space

This Letter is to provide a new approach to study the phenomena of degeneracy of the number of the collinear central configurations under geometric equivalence. A direct and simple explicit parametric expression of the singular surface H 3 is constructed in the mass space ( m 1 , m 2 , m 3 ) ∈ ( R + ) 3 . The construction of H 3 is from an inverse respective, that is, by specifying positions for the bodies and then determining the masses that are possible to yield a central configuration. It reveals the relationship between the phenomena of degeneracy and the inverse problem of central configurations. We prove that the number of central configurations is decreased to 3 ! / 2 − 1 = 2 , m 1 , m 2 , and m 3 are mutually distinct if m ∈ H 3 . Moreover, we know not only the number of central configurations but also what the nonequivalent central configurations are.

On the relative equilibrium configurations in the planar five-body problem

The number of central configurations in the Grebenicov-Elmabsout model of the planar five-body problem is estimated. An appropriate rational parameterization is used to reduce the equations defining such configurations to some polynomial ones. For the restricted five-body problem a sharp estimation is given by using the Sturm separation theorem.

On the Dziobek configurations of the restricted $(N+1)$-body problem with equal masses

In this article we show that the $(N-1)$-dimensional central configurations of the restricted $(N+1)$-body problem with equal masses are symmetrical. As a consequence, we are able to prove finiteness and find upper and lower bounds for the number of central configurations.

Planar central configurations as fixed points

Planar central configurations can be seen as critical points of the reduced potential or solutions of a system of equations. By the homogeneity of the potential and its O(2)-invariance it is possible to see that the SO(2)- orbits of central configurations are fixed points of a map f. The purpose of the paper is to define and study this map and to derive some properties using topological fixed point theory. The generalized Moulton–Smale theorem for collinear configurations is proved, together with some estimates on the number of central configurations in the case of three bodies, using fixed point indices. Well-known results such as the compactness of the set of central configurations follow easily in this topological framework.

Bifurcation of central configuration in the 2N+1 body problem

The bifurcation of central configuration in the Newtonian N-body problem for any odd number N ≥ 7 is shown. We study a special case where 2n particles of mass m on the vertices of two different coplanar and concentric regular n-gons (rosette configuration) and an additional particle of mass m0 at the center are governed by the gravitational law he 2n+1 body problem. This system is of two degrees of freedom and permits only one mass parameter μ =m 0/m. This parameter μ controls the bifurcation. If n≥ 3, namely any odd N ≥ 7, then the number of central configurations is three when μ ≥ μ c , and one when μ ≥ μ c . By combining the results of the preceding studies and our main theorem, explicit examples of bifurcating central configuration are obtained for N ≤ 13, for any odd N ∈ [15,943], and for any N ≥ 945.

On the number of central configurations in the N-body problem

Central configurations are critical points of the potential function of the n-body problem restricted to the topological sphere where the moment of inertia is equal to constant. For a given set of positive masses m 1,..., m n we denote by N(m 1, ..., m n, k) the number of central configurations' of the n-body problem in ℝk modulus dilatations and rotations. If m n 1,..., m n, k) is finite, then we give a bound of N(m 1,..., m n, k) which only depends of n and k.