On the stability of tetrahedral relative equilibria in the positively curved 4-body problem

Abstract

We consider the motion of point masses given by a natural extension of Newtonian gravitation to spaces of constant positive curvature. Our goal is to explore the spectral stability of tetrahedral orbits of the corresponding 4-body problem in the 2-dimensional case, a situation that can be reduced to studying the motion of the bodies on the unit sphere. We first perform some extensive and highly precise numerical experiments to find the likely regions of stability and instability, relative to the values of the masses and to the latitude of the position of three equal masses. Then we support the numerical evidence with rigorous analytic proofs in the vicinity of some limit cases in which certain masses are either very large or negligible, or the latitude is close to zero.

32 pages, 6 figures

References in corpus (4)

• An intrinsic approach in the curved n-body problem: the negative curvature case
• 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 (11)

• Relative equilibria in the 3-dimensional curved n-body problem
• The classical N-body problem in the context of curved space
• 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
• Rotopulsators of the curved N-body problem
• Central configurations of the curved $N$-body problem
• Choreography solutions of the $n$-body problem on $S^2$
• Stability of fixed points and associated relative equilibria of the $3$-body problem on ${\mathbb{S}}^1$ and ${\mathbb{S}}^2$
• Existence of a lower bound for the distance between point masses of relative equilibria in ${\mathbb{S}}^{k-1}$, $k\ge 3$
• Computing planar and spherical choreographies