An intrinsic approach in the curved n-body problem: the negative curvature case

TL;DR

This paper investigates the dynamics of n gravitationally interacting particles on a negatively curved 2D surface, deriving equations in different models and characterizing various types of relative equilibria using Riemannian geometry.

Contribution

It introduces an intrinsic geometric framework for the curved n-body problem on hyperbolic surfaces, providing new insights into relative equilibria in negative curvature.

Findings

Characterization of elliptic, hyperbolic, and parabolic relative equilibria.

Recovery of known results for n=2 and n=3.

Discovery of new qualitative behaviors of orbits in negative curvature.

Abstract

We consider the motion of n point particles of positive masses that interact gravitationally on the 2-dimensional hyperbolic sphere, which has negative constant Gaussian curvature. Using the stereographic projection, we derive the equations of motion of this curved n-body problem in the Poincar\'e disk, where we study the elliptic relative equilibria. Then we obtain the equations of motion in the Poincar\'e upper half plane in order to analyze the hyperbolic and parabolic relative equilibria. Using techniques of Riemannian geometry, we characterize each of the above classes of periodic orbits. For n=2 and n=3 we recover some previously known results and find new qualitative results about relative equilibria that were not apparent in an extrinsic setting.