The spatial $N$-centre problem: scattering at positive energies

TL;DR

This paper proves the existence of entire solutions with prescribed scattering angles for the spatial generalized N-centre problem at positive energies, using variational methods and collision avoidance strategies.

Contribution

It introduces a variational approach to construct solutions with specific scattering behavior in the spatial N-centre problem at positive energies.

Findings

Existence of solutions with prescribed scattering angles.

Application of variational methods to celestial mechanics problems.

Development of a collision exclusion strategy.

Abstract

For the spatial generalized $N$-centre problem $$ \ddot{x} = -\sum_{i=1}^{N} \frac{m_i (x - c_i)}{\vert x - c_i \vert^{\alpha+2}},\qquad x \in \mathbb{R}^3 \setminus \{c_1,\dots,c_N \}, $$ where $m_i > 0$ and $\alpha \in [1,2)$, we prove the existence of positive energy entire solutions with prescribed scattering angle. The proof relies on variational arguments, within an approximation procedure via (free-time) boundary value problems. A self-contained appendix describing a general strategy to rule out the occurrence of collisions is also included.