Continuations and bifurcations of relative equilibria for the positive curved three body problem

2023/06/24 by Toshiaki Fujiwara, Fujiwara, Toshiaki, Ernesto Pérez-Chavela +1
Engineering · Physics and Astronomy · #70F07 #70F10 #70F15 #Astro and Planetary Science #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Spacecraft Dynamics and Control #Stellar, planetary, and galactic studies

paper · pdf · doi:10.48550/arxiv.2306.13838

openalex publication_date 2023/06/24 · openalex created_date 2023/06/28 · openalex updated_date 2026/07/28

Abstract

The positive curved three body problem is a natural extension of the planar Newtonian three body problem to the sphere \mathbbS². In this paper we study the extensions of the Euler and Lagrange Relative equilibria (RE in short) on the plane to the sphere. The RE on \mathbbS² are not isolated in general. They usually have one-dimensional continuation in the three-dimensional shape space. We show that there are two types of bifurcations. One is the bifurcations between Lagrange RE and Euler RE. Another one is between the different types of the shapes of Lagrange RE. We prove that bifurcations between equilateral and isosceles Lagrange RE exist for equal masses case, and that bifurcations between isosceles and scalene Lagrange RE exist for partial equal masses case.

Related

• Continuation of relative equilibria in the n--body problem to spaces\n of constant curvature
• Mass independent shapes for relative equilibria in the two dimensional constant positive curved three body problem
• A new method to study relative equilibria on \mathbbS²
• Three-body relative equilibria on \mathbbS² I: Euler configurations
• Three-body relative equilibria on S² II: Extended Lagrangian configurations