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

Closed $1/2$-Elasticae in the Hyperbolic Plane