Article Contents
Article Contents

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

• * 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)

• 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.

 Citation:

• 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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