Scattering Parabolic Solutions for the Spatial N-Centre Problem

Abstract

For the N -centre problem in the three dimensional space,where N ≥ 2, mi > 0 and α ∈ [1, 2), we prove the existence of entire parabolic trajectories having prescribed asymptotic directions. The proof relies on a variational argument of min-max type. Morse index estimates and regularization techniques are used in order to rule out the possible occurrence of collisions.

The spatial N-centre problem: scattering at positive energies

“…Blow-up arguments, Morse index estimates and regularization techniques play a crucial role in making our procedure effective. More precisely, the blow analysis takes advantage of arguments previously developed both in [11] (dealing with a onecenter like potential, under strong force type assumptions both at the singularity and at infinity) and in [7,8] (dealing with the generalized N -centre problem, at the zero-energy level). On the other hand, the strategy to rule out the occurrence of collisions is inspired by the one in [19,20] but is here sharpened by the use of the classical estimates at collisions by Sperling [18]: all this is carefully presented in a final Appendix, hopefully of independent interest.…”

“…The same arguments of [8] can then be used to show that the above convergence forces the Morse index of x R to be greater than a quantity i(α) such that i(α) ≥ 2 when α > 1. Therefore, a contradiction with (28) is obtained.…”

“…for every t ∈ [−1, 1], a simple argument (see [8,Remark 4.7]) shows that M 0 (u R ) = c 0 , thus implying…”

Euler integral and perihelion librations

“…Using the formulae in the previous section, we provide the expressions of J in (4) and and E in (8) in terms of K:…”

“…Such argument, however, does not provide a complete outline of the problem, since, as a matter of fact, leaves important questions unanswered, like, as an example, the existence of action-angle coordinates, of periodic orbits, the complete picture of the bifurcation diagram. Because of this, the problem has received, in the last decades, a renewed interest and noticeable papers appeared [16,3,8,4]. A common ingredient of the mentioned literature is a separation-like change of coordinates, possibly combined with a "regularising" change of time, which allows, following Euler's ideas, to decouple the Hamiltonian.…”