Scattering parabolic solutions for the spatial N-centre problem

TL;DR

This paper proves the existence of special scattering solutions in the spatial N-centre problem with prescribed directions, using variational methods, Morse index estimates, and regularization to avoid collisions.

Contribution

It introduces a variational min-max approach to construct parabolic trajectories with specified asymptotic directions in the 3D N-centre problem, extending previous results.

Findings

Existence of entire parabolic trajectories with prescribed asymptotic directions.

Application of Morse index estimates to exclude collisions.

Use of regularization techniques to ensure collision-free solutions.

Abstract

For the $N$-centre problem in the three dimensional space, $$ \ddot x = -\sum_{i=1}^{N} \frac{m_i \,(x-c_i)}{\vert x - c_i \vert^{\alpha+2}}, \qquad x \in \mathbb{R}^3 \setminus \{c_1,\ldots,c_N\}, $$ where $N \geq 2$, $m_i > 0$ and $\alpha \in [1,2)$, 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.