Geometric interpretation for the spectral stability in the charged three-body problem

Abstract

We present a geometric interpretation of the spectral stability of the triangular libration points in the charged three-body problem. We obtain that the spectral stability varies with the position of the center of mass of the three charges with respect to the circumcenter of the triangle configuration, which does not depend directly of the charges. If the center of mass is outside or on the circumference of a well defined radius ρ, then spectral stability occurs. In addition, we analyze the existence of resonances within the spectral region of stability under this geometric interpretation, determining resonance curves of order 2, 3, 4, . . ., some of them with multiple resonances.

Dynamics in the Charged Restricted Circular Three-Body Problem

“…In order to apply Han-Li-Yi's Theorem [14], we incorporate to (18) the terms associated to the action L dropped in the process of normalization and undo the time scalings, so we get Hamiltonian (3) in the local coordinates that have been introduced through the process. Specifically we arrive at…”

“…The charged three-body problem was introduced in [13] and has been studied by several authors [1,2]. Some interesting results regarding this dynamical system from different points of view can be found, for example, in [4,6,25,18,7,8,16,17].…”