Periodic solutions of the forced conservative pendulum with relativistic acceleration and derivative dependent coefficient

We prove that the periodic problem for the forced pendulum equation with relativistic acceleration and derivative dependent coefficient ( u ′ 1 − u ′ 2 ) ′ + a 1 − u ′ 2 sin ⁡ u = f ( t ) , u ( 0 ) = u ( T ) , u ′ ( 0 ) = u ′ ( T ) , has at least two geometrically distinct solutions when f has mean value zero. After showing that the problem is the Euler-Lagrange equation of some functional I defined when | u ′ | ∞ < 1 , we extend I into a functional I R defined on the Sobolev space H T 1 whose critical points are critical points of I . We then apply a multiplicity theorem based on Lusternik-Schnirelman category to obtain the existence of two geometrically distinct critical points of I R .