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arXiv · 2306.13838

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

Toshiaki Fujiwara·Ernesto Pérez-Chavela
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Abstract

The positive curved three body problem is a natural extension of the planar Newtonian three body problem to the sphere $\mathbb{S}^2$. 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 $\mathbb{S}^2$ 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.

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Toshiaki Fujiwara, Ernesto Pérez-Chavela. 2023-06-24. Continuations and bifurcations of relative equilibria for the positive curved three body problem. https://arxiv.org/abs/2306.13838

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