Variational approach to second species periodic solutions of Poincar\'e of the 3 body problem

TL;DR

This paper develops a variational method to rigorously prove the existence of second species periodic solutions in the nonrestricted planar 3 body problem, extending prior results from the restricted case.

Contribution

It introduces a variational approach to establish the existence of second species solutions in the nonrestricted 3 body problem, filling a gap in the literature.

Findings

Proved existence of second species solutions in the nonrestricted 3 body problem.

Extended Poincaré's second species solutions from restricted to nonrestricted case.

Provided a rigorous mathematical framework for these solutions.

Abstract

We consider the plane 3 body problem with 2 of the masses small. Periodic solutions with near collisions of small bodies were named by Poincar\'e second species periodic solutions. Such solutions shadow chains of collision orbits of 2 uncoupled Kepler problems. Poincar\'e only sketched the proof of the existence of second species solutions. Rigorous proofs appeared much later and only for the restricted 3 body problem. We develop a variational approach to the existence of second species periodic solutions for the nonrestricted 3 body problem. As an application, we give a rigorous proof of the existence of a class of second species solutions.