Asymptotic Relative Equilibrium in the n-Body Problem: Relativistic Application in the Poincare Upper Half-Plane

Abstract

In this paper, we study the [Formula: see text]-body problem in the Poincaré upper half-plane [Formula: see text], where the radius [Formula: see text] of the Poincaré disk is fixed. We introduce a new potential to derive the condition for hyperbolic relative equilibria on [Formula: see text]. We analyze the relative equilibrium of positive masses moving along geodesics under the [Formula: see text] group. This result is utilized to establish the existence of relative equilibria for the [Formula: see text]-body problem on [Formula: see text] for [Formula: see text] and [Formula: see text]. We revisit previously known results and uncover new qualitative findings on relative equilibria that are not evident in an extrinsic context. Additionally, we provide a simple expression for the center of mass of a system of point particles on a two-dimensional surface with negative constant Gaussian curvature.