Continuations and bifurcations of relative equilibria for the positive curved three body problem
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917629734682624 |
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| author | Fujiwara, Toshiaki Pérez-Chavela, Ernesto |
| author_facet | Fujiwara, Toshiaki Pérez-Chavela, Ernesto |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_13838 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Continuations and bifurcations of relative equilibria for the positive curved three body problem Fujiwara, Toshiaki Pérez-Chavela, Ernesto Classical Analysis and ODEs 70F07, 70F10, 70F15 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. |
| title | Continuations and bifurcations of relative equilibria for the positive curved three body problem |
| topic | Classical Analysis and ODEs 70F07, 70F10, 70F15 |
| url | https://arxiv.org/abs/2306.13838 |