Exponential stability of Euler integral in the three--body problem

TL;DR

This paper demonstrates that the Euler integral acts as an approximate integral in the three-body problem under specific conditions, using novel normal form techniques and phase portrait analysis, with applications to collision prediction.

Contribution

It introduces a new normal form approach and phase portrait analysis to establish the approximate stability of the Euler integral in the three-body problem.

Findings

Euler integral is an approximate integral for the three-body problem with very different masses.

The method applies to particles constrained on the same plane.

Applications include predicting collisions between minor bodies.

Abstract

The first integral characteristic of the two--centres problem is proven to be an approximate integral (in the sense of N.N.Nekhorossev) to the three--body problem, at least if the masses are very different and the particles are constrained on the same plane. The proof uses a new normal form result, carefully designed around the degeneracies of the problem, and a new study of the phase portrait of the unperturbed problem. Applications to the prediction of collisions between the two minor bodies are shown.