Finiteness of polygonal relative equilibria for generalised quasi-homogeneous n-body problems and n-body problems in spaces of constant curvature

The n-Body Problem in Spaces of Constant Curvature. Part I: Relative Equilibria

We extend the Newtonian n-body problem of celestial mechanics to spaces of curvature κ = constant and provide a unified framework for studying the motion. In the 2-dimensional case, we prove the existence of several classes of relative equilibria, including the Lagrangian and Eulerian solutions for any κ 6= 0 and the hyperbolic rotations for κ < 0. These results lead to a new way of understandi...

THE n-BODY PROBLEM IN SPACES OF CONSTANT CURVATURE

We generalize the Newtonian n-body problem to spaces of curvature κ = constant, and study the motion in the 2-dimensional case. For κ > 0, the equations of motion encounter non-collision singularities, which occur when two bodies are antipodal. These singularities of the equations are responsible for the existence of some hybrid solution singularities that end up in finite time in a collision-a...

Finiteness of Relative Equilibria in the Planar Generalized N-body Problem with Fixed Subconfigurations

We prove that a fixed configuration of N − 1 masses in the plane can be extended to a central configuration of N masses by adding a specified additional mass only in finitely many ways. This holds for a family of potential functions including the Newtonian gravitational case and the classical planar point vortex model.

Central configurations and relative equilibria in the n–body problem

Finiteness of Relative Equilibria of the Four-body Problem

We show that the number of relative equilibria of the Newtonian four-body problem is finite, up to symmetry. In fact, we show that this number is always between 32 and 8472. The proof is based on symbolic and exact integer computations which are carried out by computer.