  We study critical trajectories in the hyperbolic plane for the
$1/2$-Bernoulli's bending energy with length constraint. Critical trajectories
with periodic curvature are classified into three different types according to
the causal character of their momentum. We prove that closed trajectories arise
only when the momentum is a time-like vector. Indeed, for suitable values of
the Lagrange multiplier encoding the conservation of the length during the
variation, we show the existence of countably many closed trajectories with
time-like momentum, which depend on a pair of relatively prime natural numbers.
