Parametric Stability of a Charged Pendulum with an Oscillating Suspension Point Located above Two Electric Charges of the Same Intensity and Sign, Equally Distant from the Suspension Point

In this study, we analyze a planar mathematical pendulum whose suspension point oscillates vertically according to a harmonic law. The pendulum bob is electrically charged and positioned slightly above two electric charges of equal sign and intensity, which are equidistant from the suspension point and separated by a distance of $2d$. Here, $d$ denotes the distance from each charge to the orthogonal projection of the suspension point onto the horizontal line where the charges lie. We formulate the Hamiltonian structure of this mechanical system, identify two equilibrium points, and examine the system’s linear stability. The dynamics are governed by three dimensionless parameters: $\mu$ which relates to the electric charges; $\varepsilon$, associated with the amplitude of oscillation of the suspension point; and $\alpha$, determined by the frequency of the system. We then investigate the parametric stability of the equilibrium points. Finally, we present the boundary surfaces that separate regions of stability and instability in the parameter space. For specific values of $\mu$, we derive cross-sectional curves that delineate these regions, using results from the Krein – Gelfand – Lidskii theorem and the Deprit – Hori method.