Hyperbolic motions in the $N$-body problem with homogeneous potentials

TL;DR

This paper provides a simplified proof demonstrating the existence of hyperbolic motions with prescribed velocities in the $N$-body problem with homogeneous potentials, extending previous results for Newtonian gravity.

Contribution

It offers a more accessible proof for hyperbolic motions in the $N$-body problem with homogeneous potentials, building on recent generalized results.

Findings

Existence of hyperbolic motions with prescribed velocities

Simplified proof approach for homogeneous potentials

Extension of previous results to broader class of potentials

Abstract

In the $N$-body problem, a motion is called hyperbolic, when the mutual distances between the bodies go to infinity with non-zero limiting velocities as time goes to infinity. For Newtonian potential, in \cite{MV20} Maderna and Venturelli proved that starting from any initial position there is a hyperbolic motion with any prescribed limiting velocities at infinity. Recently based on a different approach, Liu, Yan and Zhou \cite{LYZ21} generalized this result to a larger class of $N$-body problem. As the proof in \cite{LYZ21} is quite long and technical, we give a simplified proof for homogeneous potentials following the approach given in the latter paper.