The Vlasov-Poisson equation, the Moebius Geometry and then-body problem in a negative space form

By using, the Vlasov-Poisson equation defined in either a Riemannian or a semi-Riemannian space $\mathbb{R}^k_g$, and a Dirac distribution function, we re-obtain the well known and classical equations of motion of a mechanical system with a pairwise acting potential function. We apply this result to the study of an $n$--body problem in a two dimensional negative space form with the hyperbolic cotangent potential. Following the Klein's geometric Erlangen program, with methods of M\"{o}bius geometry and using the Iwasawa decomposition of the M\"{o}bius isometric group $SL(2,\mathbb{R})$ via its representation in one Clifford Algebra, we complete the study of the whole set of M\"{o}bius solutions (relative equilibria) of the problem begun by Diacu {\it et al.} in \cite{Diacu8}.