Topologically Distinct Collision-Free Periodic Solutions for the $${N}$$ N -Center Problem

Abstract

This work concerns the planar N -center problem with homogeneous potential of degree −α (α ∈ [1, 2)). The existence of infinitely many, topologically distinct, non-collision periodic solutions with a prescribed energy is proved. A notion of admissibility in the space of loops on the punctured plane is introduced so that in any admissible class and for any positive h the existence of a classical periodic solution with energy h for the N -center problem with α ∈ (1, 2) is proven. In case α = 1 a slightly different result is shown: it is the case that there is either a non-collision periodic solution or a collision-reflection solution. The results hold for any position of the centres and it is possible to prescribe in advance the shape of the periodic solutions. The proof combines the topological properties of the space of loops in the punctured plane with variational and geometrical arguments.

Topological approach to the generalized $ n$-centre problem

“…For strong singularities (α j > 2) the n center problem has a chaotic invariant set on the level H = h > sup V for n ≥ 2, again for purely topological reasons. In the present paper we prove topological sufficient conditions for chaotic behavior of the n center problem for arbitrary α j > 0, generalizing some results of [14,30,31]. For recent references see [14].…”

“…We prove a slightly more general proposition in section 5. The result of [14] implies chaotic behavior for the n center problem with n ≥ 4 moderate singularities. The assumption of Theorem 3.1 is considerably weaker.…”

“…Let Σ = ∆ newt ∪∆ mod . Following [14], we call a homotopy class Γ of closed curves in C 0 (S 1 , D \ Σ) admissible if its representative with a minimal number of self intersections has no simple subloops which bound a disk containing a single singularity a j ∈ Σ.…”

Periodic solutions to a forced Kepler problem in the plane

“…therefore, taking again into account (14), we deduce that there exist λ ′ ϕ ≤ λ ϕ , M ∈ (0, 1) and M ′ > 0 such that…”

Nonconstant periodic solutions with any fixed energy for singular Hamiltonian systems

“…In 1978, by imposing some constraints on the energy sphere, Rabinowitz [19] used variational methods to prove the existence of periodic solutions for a class of firstorder Hamiltonian systems with fixed energy. Since then, the existence of periodic solutions of Hamiltonian system with fixed energy is extensively investigated [1,2,5,7,9,10,18,21,22,23,24,25] etc. For nonsingular second-order Hamiltonian 1618 LIANG DING, RONGRONG TIAN AND JINLONG WEI system with fixed energy, there were some references: Benci [5] obtained one periodic solution for C 2 nonsingular potential; for nonconstant periodic solution with C 1 nonsingular potential, Zhang [25] gained one nonconstant periodic solution with positive fixed energy.…”