Some problems on the classical n-body problem

Abstract

Our idea is to imitate Smale's list of problems, in a restricted domain of mathematical aspects of Celestial Mechanics. All the problems are on the n-body problem, some with different homogeneity of the potential, addressing many aspects such as central configurations, stability of relative equilibrium, singularities, integral manifolds, etc. Following Steve Smale in his list, the criteria for our selection are: (1) Simple statement. Also preferably mathematically precise, and best even with a yes or no answer. (2) Personal acquaintance with the problem, having found it not easy. (3) A belief that the question, its solution, partial results or even attempts at its solution are likely to have great importance for the development of the mathematical aspects of Celestial Mechanics.

Uniqueness results for co-circular central configurations for power-law potentials

“…If equation (9) holds for some i, then the magnitude of the force vector acting on the i-th body is the correct length to satisfy equation (1). Equations (7) and (9) define a system of 2n equations that are both necessary and sufficient for a central configuration on the unit circle to have its center of mass at the origin.…”

“…Llibre and Valls have announced that the regular pentagon (again with equal masses) is the only co-circular central configuration with this special property for n = 5 [10]. This question is listed as Problem 12 in a collection of important open problems in celestial mechanics compiled by Albouy, Cabral and Santos [1].…”

Linear stability of the elliptic relative equilibrium with (1 + n)-gon central configurations in planar n-body problem

“…Another question is whether the linearly stable relative equilibrium is always a non-degenerate minimum of the U| E (cf. [1], Problem 15, 16).…”

“…Choose α 0 = 1 2 , (β 0 , e 0 ) ∈ U 1 . Then Remark 3.9 and Theorem 3.7 imply that if (α, β, e) satisfies e 0 ≤ e, β, e) > 0 inD1 (ω, 2π) ∀ω ∈ U. where U 1 = {(β 0 , e 0 )| 0 < β 0 < min{ √ 0 −2y 0 ) , e 0 ∈ [0, 1)}, (x 0 ,y 0 ) = (1.5, 0.108). Simple computations show that (0, 0.7237] × [0, 1) ⊂ (0, 0 ) × [0, 1) ⊂ U 1 .…”

“…For 9 ≤ n ≤ 27, the equality A(R 1 , e) > 0 inD 2 (1, 2π) holds for all (m, e) ∈ [0, +∞) × [0, 1).Moreover, ifn < σ n , n ≤ 4ď n , 1 , β 1 , e) > 0 inD1 (1, 2π), for ∀(m, e) ∈ (0, +∞) × [0, 1), hence A(R 1 , e) > 0 inD2 (1, 2π), for ∀(m, e) ∈ (0, +∞) × [0, 1).…”