All the Lagrangian relative equilibria of the curved 3-body problem have equal masses

Abstract

Abstract. We consider the 3-body problem in 3-dimensional spaces of nonzero constant Gaussian curvature and study the relationship between the masses of the Lagrangian relative equilibria, which are orbits that form a rigidly rotating equilateral triangle at all times. There are three classes of Lagrangian relative equilibria in 3-dimensional spaces of constant nonzero curvature: positive elliptic and positive elliptic-elliptic, on 3-spheres, and negative elliptic, on hyperbolic 3-spheres. We prove that all these Lagrangian relative equilibria exist only for equal values of the masses.

Bifurcations of the Lagrangian orbits from the classical to the curved 3-body problem

“…We can now completely eliminate the three equations involving ω 1 , ω 2 , ω 3 , but these variables still occur in the terms r 2 ij , which show up in the other equations. Actually these variables appear in the particular form σ(ω i − ω j ) 2 , which using (15) can be written as (16) σ…”

Polygonal rotopulsators of the curved n-body problem

“…However, a working model for the n ≥ 2 case was not found until 2008 by Diacu, Pérez-Chavela and Santoprete (see [9], [10] and [11]). This breakthrough then gave rise to further results for the n ≥ 2 case in [1]- [8] and [12], [14] and [16]- [27] and the references therein. Rotopulsators are solutions to (1.1) for which the configuration of the point massses may only rotate or change size, but retains its shape over time.…”

Circular Non-collision Orbits for a Large Class of n-Body Problems

“…The curved n-body problem for n = 2 goes back as far as the 1830s, but a working model for the n ≥ 2 case was not found until 2008 by Diacu, Pérez-Chavela and Santoprete (see [16], [17] and [18]). This breakthrough then gave rise to further results for the n ≥ 2 case in [9]- [15] and [20], [21] and [30]- [38]. See [15], [16], [17] and [18] for a historical overview.…”