We study the 1-D motions of a charge under Coulomb force, within the electrostatic approximation. When the electrostatic force is attractive, no oscillating motion takes place. When repulsive, nonlinear oscillations will arise. In both cases dry friction has been taken into account and time equations have been solved providing time as elliptic integrals of first and second kind. A short phase plane analysis has been included. The oscillation period has been exactly computed and found to increase versus the initial speed of the mobile.
Giovanni Scarpello. Daniele Ritelli. "Nonlinear 1-D Oscillations of a Charge Particle Under Coulomb Forces and Dry Friction." J. Geom. Symmetry Phys. 34 77 - 85, 2014. https://doi.org/10.7546/jgsp-34-2014-77-85