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

In this article is given a simple expression for the $\mathit{\text{center}}\mathit{\text{of}}\mathit{\text{mass}}$ 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 ${\mathbb{L}}_R^1$. Finally, is showed some applications to the curved $2$- and $3$-body problems.