Newton's method for computing periodic orbits of the planar three-body problem

TL;DR

This paper details a modified Newton's method utilizing continuous analogs and high-precision Taylor series to compute periodic orbits in the planar three-body problem, discovering new topological families.

Contribution

It introduces a novel modification of Newton's method using continuous analogs and high-precision ODE solutions for better orbit computation in the three-body problem.

Findings

Identified new topological families of periodic orbits.

Enhanced the accuracy of orbit computation with high-precision Taylor series.

Demonstrated the effectiveness of the modified Newton's method in complex orbital calculations.

Abstract

In this paper we present in detail Newton's method and its modification, based on the Continuous analog of Newton's method for computing periodic orbits of the planar three-body problem. The linear system at each step of the method is formed by solving a system of ODEs with the multiple precision Taylor series method. We consider zero angular momentum symmetric initial configuration with parallel velocities, bodies with equal masses and relatively short periods. Taking candidates for the correction method with greater return proximity as usual and correcting with the modified Newton's method, allows us to find some new topological families that are not included in the database in [SCIENCE CHINA Physics, Mechanics & Astronomy 60.12 (2017)]