Abstract view
Eulertype relative equilibria in spaces of constant curvature and their stability


Published:20170525
Printed: Apr 2018
Ernesto PérezChavela,
Departamento de Matemáticas, Instituto Tecnológico Autónomo de México, México D.F., México
Juan Manuel SánchezCerritos,
Departamento de Matemáticas, Universidad Autónoma Metropolitana  Iztapalapa, México D.F., México
Abstract
We consider three point positive masses moving on $S^2$ and $H^2$.
An Eulerianrelative 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
Eulerianrelative 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 Eulerianrelative equilibria
are unstable.