On the regularization of the collision solutions of the one-center problem with weak forces

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

“…Following [36], the proofs of theorems 1.2, 1.3 are based on a constrained minimisation argument for the Maupertuis functional (the obstacle problem). Similar techniques have been exploited in the literature to solve minimisation problems in presence of singular potential, for instance in [11,16,17,40]. The core of the method consists in the analysis, close to the singularities, of a particular minimising sequence {u n } n .…”

“…In the latter case the set C is split in two subsets C int and C ext of those centres that respectively lie in the bounded and unbounded regions of the plane separated byγ (α). Condition (15) assures that both C int and C ext contain at least two centres, thus condition A1 is still satisfied.…”

“…The question about regularisation of the singular potential problem and the possible extension for the collision solutions has been proposed and discussed by different authors. We mention [20,30,31] for α = 1, and [4,12,32] for α = 1, [13][14][15]37] in case of logarithmic potential (α = 0).…”

A study of the apsidal angle and a proof of monotonicity in the logarithmic potential case

“…In the singular case (α ≥ 0), the behaviour of the apsidal angle plays a fundamental role in dealing with the collision solutions of (1) and related systems: a possible regularisation of the singularities of the flow is possible only in case that twice the apsidal angle tends to a multiple of π as → 0, see for instance [6,7,8,9]. Also a variational approach to system (1) may lead to collision avoidance provided ∆ α θ(u) > π as |u| → 0, see [10].…”

“…The purpose of this paper is to review the notion of the apsidal angle and to derive a formula for its derivative with respect to the angular momentum; from where, the monotonicity will be evident for any α ∈ (−2, 1) and proved in the logarithmic potential case. The plan of the paper is the following: in the next section we recall the orbital structure of the solutions of (1) and we write a formula for the apsidal angle as a fixed-ends integral (7), where the parameter q plays the role of the angular momentum. Then, section 3 is dedicated to the analytical study of the integrand, in particular of the function E α (s, q).…”