José Laudelino de Menezes Neto

We study the dynamics of a simple pendulum attached to the center of mass of a satellite in an elliptic orbit. We consider the case where the pendulum lies in the orbital plane of the satellite. We find two linearly stable equilibrium positions for the Hamiltonian system describing the problem and study their parametric stability by constructing the boundary curves 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.