  We firstly discuss classical stability for a dynamical system of two ions
levitated in a 3D Radio-Frequency (RF) trap, assimilated with two coupled
oscillators. We obtain the solutions of the coupled system of equations that
characterizes the associated dynamics. In addition, we supply the modes of
oscillation and demonstrate the weak coupling condition is inappropriate in
practice, while for collective modes of motion (and strong coupling) only a
peak of the mass can be detected. Phase portraits and power spectra are
employed to illustrate how the trajectory executes quasiperiodic motion on the
surface of torus, namely a Kolmogorov-Arnold-Moser (KAM) torus. In an attempt
to better describe dynamical stability of the system, we introduce a model that
characterizes dynamical stability and the critical points based on the Hessian
matrix approach. The model is then applied to investigate quantum dynamics for
many-body systems consisting of identical ions, levitated in 2D and 3D ion
traps. Finally, the same model is applied to the case of a combined 3D
Quadrupole Ion Trap (QIT) with axial symmetry, for which we obtain the
associated Hamilton function. The ion distribution can be described by means of
numerical modeling, based on the Hamilton function we assign to the system. The
approach we introduce is effective to infer the parameters of distinct types of
traps by applying a unitary and coherent method, and especially for identifying
equilibrium configurations, of large interest for ion crystals or quantum
logic.
