Regular Polygonal Equilibria on $$\mathbb {S}^1$$ and Stability of the Associated Relative Equilibria

Abstract

For the curved n-body problem in S 3 , we show that a regular polygonal configuration for n masses on a geodesic is an equilibrium configuration if and only if n is odd and the masses are equal. The equilibrium configuration is associated with a one-parameter family (depending on the angular velocity) of relative equilibria, which take place on S 1 embedded in S 2 . We then study the stability of the associated relative equilibria on two invariant manifolds, T * ((S 1 ) n \ ∆) and T * ((S 2 ) n \ ∆). We show that they are Lyapunov stable on S 1 , they are Lyapunov stable on S 2 if the absolute value of angular velocity is larger than a certain value, and that they are linearly unstable on S 2 if the absolute value of angular velocity is smaller than that certain value.

Classification of positive elliptic-elliptic rotopulsators on Clifford tori

“…However, the first paper giving an explicit n-body problem in spaces of constant Gaussian curvature for general n ≥ 2 was published in 2008 by Diacu, Pérez-Chavela and Santoprete (see [15][16][17]). This breakthrough then gave rise to further results for the n ≥ 2 case in [6], [7][8][9][10][11][12][13][14][18][19][20][21][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] and the references therein. Rotopulsators are solutions to (1.1) consisting of any orbit induced by a (possibly hyperbolic) rotation, but otherwise impose few restrictions on the position vectors of the point masses.…”