# Hyperbolic Center of Mass for a System of Particles on a two-dimensional Space with Constant Negative Curvature: An Application to the Curved $2$- and $3$-Body Problems

## TL;DR

This paper derives a simple, intrinsic formula for the center of mass on hyperbolic surfaces, extending Euclidean concepts and applying them to curved two- and three-body problems.

## Contribution

It introduces a new intrinsic expression for the center of mass in hyperbolic geometry, applicable to systems on negatively curved surfaces and related to curved celestial mechanics.

## Key findings

- Derived an explicit formula for the hyperbolic center of mass
- Extended the concept to one-dimensional hyperbolic space
- Applied the formula to curved 2- and 3-body problems

## Abstract

In this article is given a simple expression for the \textit{ center of mass} for a system of material points in a two-dimensional surface of constant negative Gaussian curvature. Using basic techniques of Geometry, an expression in intrinsic coordinates is obtained, and it is showed how it extends the definition for the Euclidean case. The argument is constructive and also serves for defining center of mass of a system of particles on the one-dimensional hyperbolic space $\mathbb{L}^1_R$. Finally, is showed some applications to the curved $2$- and $3$-body problems.