Almost all 3-body relative equilibria on $ \mathbb S^2 $ and $ \mathbb H^2 $ are inclined

Florin Diacu; Shuqiang Zhu

doi:10.3934/dcdss.2020067

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2020, Volume 13, Issue 4: 1131-1143. Doi: 10.3934/dcdss.2020067

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Almost all 3-body relative equilibria on $ \mathbb S^2 $ and $ \mathbb H^2 $ are inclined

Florin Diacu^1,†, and
Shuqiang Zhu^2, ,

1.
Yale-NUS College, National University of Singapore, Republic of Singapore
2.
School of Mathematical Sciences, University of Science and Technology of China, Hefei, China

^* Corresponding author: Shuqiang Zhu
^* Corresponding author: Shuqiang Zhu

Dedicated to Jürgen Scheurle on the occasion of his 65th birthday
^† Editors' Note: Florin Diacu passed away on February 13, 2018 before this manuscript could be published. He will be missed by his colleagues, as a mathematician and as a person.

Received: October 31, 2017

Revised: February 28, 2018

Published: April 2020

Florin Diacu is supported by Yale-NUS startup grant, and Shuqiang Zhu is supported by NSFC(No.11801537, No.11721101) and the Fundamental Research Funds for the Central Universities (No.WK0010450010)

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Abstract

We answer here a question posed by F. Diacu in 2012 that asked whether there exist relative equilibria on $ \mathbb S^2 $ and $ \mathbb H^2 $ that move in a plane non-perpendicular to the rotation axis. For 3-body non-geodesic ordinary central configurations on $ \mathbb S^2 $ and $ \mathbb H^2 $, we find all relative equilibria that move in a plane perpendicular to the rotation axis. We also show that the set of shapes of 3-body non-geodesic ordinary central configurations on $ \mathbb S^2 $ and $ \mathbb H^2 $ is a 3-dimensional manifold. Then we conclude that almost all 3-body relative equilibria move in planes non-perpendicular to the rotation axis.

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Mathematics Subject Classification: Primary: 70F15; Secondary: 70F07.

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Figure 1. Lagrangian central configurations on $ \mathbb H^2 $

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Figure 2. An $ \mathbb S^2 $ central configuration on $ z = c $

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Figure 3. The projection of one shape

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