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# Euler-type relative equilibria in spaces of constant curvature and their stability

Published:2017-05-25
Printed: Apr 2018
• Ernesto Pérez-Chavela,
Departamento de Matemáticas, Instituto Tecnológico Autónomo de México, México D.F., México
• Juan Manuel Sánchez-Cerritos,
Departamento de Matemáticas, Universidad Autónoma Metropolitana - Iztapalapa, México D.F., México
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## Abstract

We consider three point positive masses moving on $S^2$ and $H^2$. An Eulerian-relative equilibrium, is a relative equilibrium where the three masses are on the same geodesic, in this paper we analyze the spectral stability of these kind of orbits where the mass at the middle is arbitrary and the masses at the ends are equal and located at the same distance from the central mass. For the case of $S^2$, we found a positive measure set in the set of parameters where the relative equilibria are spectrally stable, and we give a complete classification of the spectral stability of these solutions, in the sense that, except on an algebraic curve in the space of parameters, we can determine if the corresponding relative equilibria is spectrally stable or unstable. On $H^2$, in the elliptic case, we prove that generically all Eulerian-relative equilibria are unstable; in the particular degenerate case when the two equal masses are negligible we get that the corresponding solutions are spectrally stable. For the hyperbolic case we consider the system where the mass in the middle is negligible, in this case the Eulerian-relative equilibria are unstable.
 Keywords: curved space, relative equilibrium, spectral stability
 MSC Classifications: 70F07 - Three-body problems 70G60 - Dynamical systems methods

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