Periodic Solutions for Circular Restricted 4-body Problems with Newtonian Potentials

TL;DR

This paper proves the existence of non-collision periodic solutions in a planar restricted 4-body problem with Newtonian potentials, using variational methods and symmetry considerations.

Contribution

It introduces a variational approach to establish non-collision periodic solutions in a restricted 4-body problem with specific symmetric configurations.

Findings

Existence of minimizers for the Lagrangian action on symmetric loop spaces.

Minimizers are proven to be non-collision periodic solutions.

Solutions have fixed winding numbers.

Abstract

We study the existence of non-collision periodic solutions with Newtonian potentials for the following planar restricted 4-body problems: Assume that the given positive masses $m_{1},m_{2},m_{3}$ in a Lagrange configuration move in circular obits around their center of masses, the sufficiently small mass moves around some body. Using variational minimizing methods, we prove the existence of minimizers for the Lagrangian action on anti-T/2 symmetric loop spaces. Moreover, we prove the minimizers are non-collision periodic solutions with some fixed wingding numbers.