Relativistic pendulum equation

Prof. David Arcoya
Universidad de Granada

Abstract: We survey the results of the joint paper with Antonio Ambrosetti (Differential and
Integral equations, 33 (2020), 92-112). We consider the equation

(u'/√(1−u'²))'+F'(u)=0

modeling, if F'(u) = sin u, the motion of the free relativistic planar pendulum. Using the
critical point theory for non-smooth functionals due to Szulkin, we prove the existence of
non-trivial T periodic solutions provided T is sufficiently large.
We also show the existence of periodic solutions to the free and forced relativistic spherical
pendulum, where F' is substituted by F'(u)−h²G'(u) ∼ sin u−h²cos u/sin³ u, h ∈ R.