Gerson Araujo

In this study, we analyze a planar mathematical pendulum with a suspension point that oscillates harmonically in the vertical direction. The bob of the pendulum is electrically charged and is located between two wires with a uniform distribution of electric charges, both equidistant from the suspension point. The dynamics of this phenomenon is investigated. The system has three parameters, and we analyze the parametric stability of the equilibrium points, determining surfaces that separate the regions of stability and instability in the parameter space. In the case where the parameter associated with the charges is equal to zero, we obtain boundary curves that separate the regions of stability and instability for the Mathieu equation.

We consider a planar pendulum with an oscillating suspension point and with the bob carrying an electric charge $q$. The pendulum oscillates above a fixed point with a charge $Q.$ The dynamics is studied as a system in the small parameter $\epsilon$ given by the amplitude of the suspension point. The system depends on two other parameters, $\alpha$ and $\beta,$ the first related to the frequency of the oscillation of the suspension point and the second being the ratio of charges. We study the parametric stability of the linearly stable equilibria and use the Deprit-Hori method to construct the boundary surfaces of the stability/instability regions.

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.