On Geodesic Rays of Newtonian Gravitational Systems

TL;DR

This paper investigates the structure of geodesic rays in the Newtonian N-body problem, revealing their limits and implications for the existence of certain orbits, and applies weak KAM theory to demonstrate the existence of parabolic solutions.

Contribution

It proves that limits of geodesic rays are also geodesic rays, and uses weak KAM theory to establish the existence of complete parabolic orbits from any initial position.

Findings

Limits of geodesic rays are also geodesic rays.

Many motions with non-negative energy are not geodesic rays.

Existence of complete parabolic orbits from any initial position.

Abstract

In this paper, we focus on the set of geodesics rays of the Newtonian N-body problem. We find that the limits of geodesic rays are also geodesic rays, hence they are not dense in the space of initial conditions. As a result, there are many motions whose domain is the half real line has non-negative total energy, and they are not geodesic rays, the set of such motions has positive measure, we think this set gives a large space to accommodate many solutions with "bad" behaviors. As an application of weak KAM theory for N-body problem, we give a brief proof of the existence of complete parabolic orbit starting from any given initial position.