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

Abstract

By using, the Vlasov-Poisson equation defined in either a Riemannian or a semi-Riemannian space ${\mathbb{R}}_g^k$, 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 Möbius geometry and using the Iwasawa decomposition of the Möbius isometric group $SL(2,\mathbb{R})$ via its representation in one Clifford Algebra, we complete the study of the whole set of Möbius solutions (relative equilibria) of the problem begun by Diacu {\it et al.} in \cite{Diacu8}.

23 pages, 5 figures

References in corpus (2)

• An intrinsic approach in the curved n-body problem: the negative curvature case
• Möbius solutions of the curved $n$--body problem for positive curvature