On the relativistic pendulum-type equation

Abstract

In the first part of this paper, we consider the equation $$ \Big ( \frac{u'}{\sqrt{1-u'^2}} \Big )'+F'(u)=0 $$ modeling, if $F'(u)=\sin u$, the motion of the free relativistic planar pendulum. Using critical point theory for non-smooth functionals, we prove the existence of non-trivial $T$ periodic solutions provided $T$ is sufficiently large.

In the second part, we show the existence of periodic solutions to the free and forced relativistic spherical pendulum, where $F'$ is substituted by $$ F'(u)-h^2\, G'(u)\sim \sin u -h^2 \frac {\cos u}{\sin^3u} , \ \ \ h\in \mathbb R . $$