Hildeberto Cabral

In this work we construct symmetric central configurations for the $n$-body problem, for $n = 4, 5, 6, 7, 8$. Geometric arguments are used in the proof of existence of these configurations. We observe that these examples of central configurations can be proved by other means, for instance, using arguments for the existence of symmetric solutions to variational problems as is done by J. Montaldi in [10]. However, we believe that our geometrical approach will be useful in tackling other questions of central configurations of the Newtonian $n$-body problem.

We study the mechanical system consisting of the following variant of the planar pendulum. The suspension point oscillates harmonically in the vertical direction, with small amplitude $\varepsilon$, about the center of a circumference which is located in the plane of oscillations of the pendulum. The circumference has a uniform distribution of electric charges with total charge $Q$ and the bob of the pendulum, with mass $m$, carries an electric charge $q$. We study the motion of the pendulum as a function of three parameters: $\varepsilon$, the ratio of charges $\mu = \frac qQ$ and a parameter $\alpha$ related to the frequency of oscillations of the suspension point and the length of the pendulum. As the speed of oscillations of the mass $m$ are small magnetic effects are disregarded and the motion is subjected only to the gravity force and the electrostatic force. The electrostatic potential is determined in terms of the Jacobi elliptic functions. We study the parametric resonance of the linearized equations about the stable equilibrium finding the boundary surfaces of stability domains using the Deprit – Hori method.