Relativistic brachistochrones under electric or gravitational uniform fields

This paper analyzes the planar frictionless brachistochronic motion of a punctual charge moving in vacuo under the sole effect of a uniform electrostatic field accelerating it to relativistic speeds. Following the double assumption of minimizing the absolute (newtonian) and the local (relativistic) time, different fastest trajectory families are calculated in closed form, sometimes through the I and II kind elliptic integrals. The problem is furthermore treated and solved replacing the electrostatic field with the gravitational one.