New Periodic Solutions for Newtonian n-Body Problems with Dihedral Group Symmetry and Topological Constraints

Abstract

Abstract. In this paper, we prove the existence of a family of new noncollision periodic solutions for the classical Newtonian n-body problems.In our assumption, the n = 2l ≥ 4 particles are invariant under the dihedral rotation group D l in R 3 such that, at each instant, the n particles form two twisted l-regular polygons. Our approach is variational minimizing method and we show that the minimizers are collision-free by level estimates and local deformations.

Spatial Double Choreographies of the Newtonian 2n-Body Problem

“…It will be interesting to compare our result with those obtained in [18] and [30], where the authors also considered minimization problems with mixed symmetric and topological constraints, and proved the existence of certain spatial double choreographies as collision-free minimizers similar to ours. When n = 2, the solution obtained in Theorem 1.1 are likely to be the same as those obtained in [18, Theorem 2.1] and [30,Theorem 3] with the proper topological constraints.…”

“…While the strong symmetric constraints helped simplify the problem, it also restricted the possible choices of topological constraints. For example in our terminology, roughly speaking the solutions found in [30] all correspond to ω-topological constraints with ω i = ω i+1 , for each i.…”

“…For one reason, Marchal's average method or the rotating circle property does not work, and in some cases it has been proven that an action minimizer does contain collision, see [19], [29] and [22]. Because of this, fewer results are available along this line, see [4], [5], [7], [18], [26], [30] and [33].…”

Torus knot choreographies in the $n$-body problem

“…In summary, we have that the map G has 3-dimensional families of zeros and also 3restrictions given by (12). To prove the existence of solutions, we could take 3-restrictions in the domain and range of G. But given that the range is a non-flat manifold, it is simpler to augment the delay differential equation G = 0 with the three Lagrangian multipliers λ j for j = 1, 2, 3,…”

“…Notable exceptions include works on: the figure-eight of three bodies [2], the rotating n-gon [5], the figure-eight type for odd bodies [4] and the super-eight of four bodies [6]. Other variational approaches related to existence of planar choreographies can be found in [7,8,9,10,11,12] and the references therein.…”