An intrinsic approach in the curved n-body problem: the negative curvature case

Abstract

We consider the motion of n point particles of positive masses that interact gravitationally on the 2-dimensional hyperbolic sphere, which has negative constant Gaussian curvature. Using the stereographic projection, we derive the equations of motion of this curved n-body problem in the Poincaré disk, where we study the elliptic relative equilibria. Then we obtain the equations of motion in the Poincaré upper half plane in order to analyze the hyperbolic and parabolic relative equilibria. Using techniques of Riemannian geometry, we characterize each of the above classes of periodic orbits. For n=2 and n=3 we recover some previously known results and find new qualitative results about relative equilibria that were not apparent in an extrinsic setting.

44 pages

References in corpus (3)

• Relative equilibria in the 3-dimensional curved n-body problem
• Saari's Conjecture for the Collinear $n$-Body Problem
• Saari's Homographic Conjecture of the Three-Body Problem

Cited by in corpus (18)

• Relative equilibria in the 3-dimensional curved n-body problem
• On the stability of tetrahedral relative equilibria in the positively curved 4-body problem
• Classification and stability of relative equilibria for the two-body problem in the hyperbolic space of dimension 2
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• Reduction and relative equilibria for the 2-body problem in spaces of constant curvature
• All the Lagrangian relative equilibria of the curved 3-body problem have equal masses
• Bifurcations of the Lagrangian orbits from the classical to the curved 3-body problem
• Computing hyperbolic choreographies
• The continuous transition of Hamiltonian vector fields through manifolds of constant curvature
• Choreography solutions of the $n$-body problem on $S^2$
• The $N$-Body Problem in Spaces with Uniformly Varying Curvature
• Möbius solutions of the curved $n$--body problem for positive curvature
• Compactness and index of ordinary central configurations for the curved n-body problem
• On certain Möbius type solutions for the n-body problem in a positive space form
• The Vlasov-Poisson equation, the Moebius Geometry and then-body problem in a negative space form
• Attracting and repelling 2-body problems on a family of surfaces of constant curvature
• Conjugated equilibrium solutions for the $2$--body problem in the two dimensional sphere ${\mathbb{M}}_R^2$ for equal masses
• 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