Globally minimizing parabolic motions in the Newtonian N-body problem

TL;DR

This paper proves the existence of collision-free parabolic solutions in the Newtonian N-body problem that are asymptotic to given central configurations, using variational methods to analyze minimizing trajectories.

Contribution

It establishes the existence of globally minimizing parabolic motions asymptotic to any normalized central configuration in the Newtonian N-body problem.

Findings

Existence of collision-free parabolic solutions for any initial configuration.

Solutions are minimizers in every time interval.

Solutions are asymptotic to given central configurations.

Abstract

We consider the $N$-body problem in $\mathbb{R}^d$ with the newtonian potential $1/r$. We prove that for every initial configuration $x_i$ and for every minimizing normalized central configuration $x_0$, there exists a collision-free parabolic solution starting from $x_i$ and asymptotic to $x_0$. This solution is a minimizer in every time interval. The proof exploits the variational structure of the problem, and it consists in finding a convergent subsequence in a family of minimizing trajectories. The hardest part is to show that this solution is parabolic and asymptotic to $x_0$.