Variational Construction of Homoclinic and Heteroclinic Orbits in the Planar Sitnikov Problem

TL;DR

This paper uses variational methods to prove the existence of infinitely many homoclinic and heteroclinic orbits in the planar Sitnikov problem as the eccentricity approaches 1, revealing complex chaotic dynamics.

Contribution

It demonstrates the existence of non-periodic homoclinic and heteroclinic solutions in the Sitnikov problem using variational techniques, extending previous results on periodic orbits.

Findings

Existence of infinitely many homoclinic and heteroclinic solutions.

Solutions connect some periodic orbits with non-periodic symbolic sequences.

Analysis in the limiting case as eccentricity tends to 1.

Abstract

The Sitnikov problem is a special case of the three-body problem. The system is known to be chaotic and has been studied by symbolic dynamics (J. Moser, Stable and random motions in dynamical systems, Princeton University Press, 1973). We study the limiting case of the Sitnikov problem as the eccentricity of the massive particles tends to 1. By variational method, we show the existence of infinitely many homoclinic and heteroclinic solutions in the planar Sitnikov problem. In a previous work, for certain periodic symbolic sequences, the second author showed the existence of periodic solutions realizing them. In this paper, we show the existence of homoclinic and heteroclinic solutions between some of these periodic orbits which realize certain non-periodic symbolic sequences.