Wednesday, April 13, 2022
Room 310, Online seminar via Webex
I. Hristov(1), R. Hristova(1), I. Puzynin(2), T. Puzynina(2), Z. Sharipov(2), Z. Tukhliev(2)
1 - Faculty of Mathematics and Informatics, Sofia University, Sofia, Bulgaria, 2 - Meshcheryakov Laboratory of Information Technologies, JINR, Dubna

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

A breakthrough in the numerical search for periodic orbits of the planar three-body problem has been made in recent years. In 2013, M. Shuvakov and V. Dmitrashinovich found 13 new topological families applying a clever numerical algorithm in the standard double-precision arithmetic [1]. Since the three-body problem is very sensitive to the initial conditions, working with double precision strongly limits the number of solutions that can be found. This limitation was recognized by S. Li and Sh. Liao; in 2017, they applied Newton's method to find more than 600 new families of periodic orbits [2]. They formed a linear system at each step of Newton's method by solving a system of ODEs with the high-order multiple-precision Taylor series method. However, no details of the numerical procedure are given in [2]. This numerical procedure is rather technical and deserves its own attention. In this work, we present 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. Our programs are first tested with a general search for relatively short periods and with a relatively coarse search grid. As a result, we found 105 new topological families that are not included in the database in [2]. A purposeful search for the so-called figure-eight satellites was also made. As a result, about 400 new satellites, including 7 new stable “choreographies”(trajectory families), were found. Until now there have been known only two stable “choreographies”, namely, the famous Moore’s figure-eight orbit and one “choreography” found by M. Shuvakov. The computations were performed on the "Nestum" cluster, Sofia, Bulgaria and on the "Govorun" supercomputer, JINR, Dubna, Russia.

[1] Shuvakov, M., et al. "Three classes of Newtonian three-body planar periodic orbits." Physical Review Letters 110.11 (2013): 114301.
[2] Li, et al. "More than six hundred new families of Newtonian periodic planar collisionless three-body orbits." SCIENCE CHINA Physics, Mechanics and Astronomy 60.12 (2017): 1-7.

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