On the Lagrange Stability of Motion and final Evolutions in the Three-Body Problem

TL;DR

This paper proves a theorem on the Lagrange stability in the three-body problem, providing insights into the nature of final evolutions such as hyperbolic-elliptic and parabolic-elliptic trajectories using new relations involving mutual distances.

Contribution

It introduces a Lagrange stability theorem for the three-body problem utilizing relations that connect squared mutual distances and the barycenter, enhancing understanding of final evolutions.

Findings

Proves the Lagrange stability theorem for three-body motions.

Defines the character of hyperbolic-elliptic and parabolic-elliptic evolutions.

Uses relations involving mutual distances and barycenter for analysis.

Abstract

For the three-body problem, we consider the Lagrange stability. To analyze the stability, along with integrals of energy and angular momentum, we use relations by the author from Sosnitskii (2005), which band together separately squared mutual distances between bodies (mass points) and squared mutual distances from bodies to the barycenter of the system. In this case, we prove the Lagrange stability theorem, which allows us to define more exactly the character of hyperbolic-elliptic and parabolic-elliptic final evolutions