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

  • * 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.

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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  • 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.

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


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

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

    Figure 3.  The projection of one shape

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