Polygonal rotopulsators of the curved n-body problem

Abstract

Let q 1 ,...,q n be the position vectors of the point masses of the curved n-body problem. Consider any positive elliptic-elliptic rotopulsator so-.., n}, where α 1 , ..., α n , β 1 , ..., β n ∈ [0, 2π) are constants, φ, θ, r and ρ are twice-differentiable, continuous, nonconstant functions, r 2 + ρ 2 = 1, r ≥ 0 and ρ ≥ 0. We prove that the configuration of the vectors (r cos (θ + α i ), r sin (θ + α i )) T is a regular polygon, as is the configuration of the vectors (ρ cos (φ + β i ), ρ sin (φ + β i )), i ∈ {1, ..., n}.

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.…”

“…Secondly, with continued investigations into whether solutions to the curved n-body problem can be related to the classical n-body problem (see for example [2,13]), rotopulsators seem to be good candidates for solutions to the curved n-body problem that are somehow related to homographic orbit solutions and relative equilibria solutions to the Newtonian n-body problem. It may be possible to use results for rotopulsators (see for example [28,29,33]) and using such a relationship to shed a new light on various open problems related to polygonal relative equilibria (see [1] for such open problems).…”

“…. , n} (see for example [14,21,28,29,33]). For general n it was proven in [33] for this case that hyperbolic rotopulsators, i.e.…”

Classification of positive elliptic-elliptic rotopulsators on Clifford tori

“…For general n it was proven in [30] that if σ = −1 and r i and ρ i are independent of i and not constant, then the configuration of the point masses has to be a regular polygon and all masses have to be equal. In [26] it was proven that if the β i are all equal, r i and ρ i are independent of i and not constant, then if σ = 1, the configuration of the point masses has to be a regular polygon as well and in [30] it was proven that for this case the masses have to be equal as well.…”

“…For general n it was proven in [30] that if σ = −1 and r i and ρ i are independent of i and not constant, then the configuration of the point masses has to be a regular polygon and all masses have to be equal. In [26] it was proven that if the β i are all equal, r i and ρ i are independent of i and not constant, then if σ = 1, the configuration of the point masses has to be a regular polygon as well and in [30] it was proven that for this case the masses have to be equal as well. If r i and ρ i are independent of i and not constant, the only remaining case to be conclusively investigated is the case that the solution to (1.1) is a positive elliptic-elliptic rotopulsator and r i =: r and ρ i =: ρ are independent of i and not constant, i.e.…”