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

TL;DR

This paper classifies and proves the existence of closed critical trajectories with periodic curvature in the hyperbolic plane for a specific bending energy, revealing they occur only with time-like momentum and depend on coprime integers.

Contribution

It provides a complete classification of critical trajectories and establishes the existence of infinitely many closed solutions with time-like momentum in the hyperbolic plane.

Findings

Closed trajectories only occur with time-like momentum.

Existence of countably many closed trajectories depending on coprime integers.

Trajectories are classified into three types based on momentum's causal character.

Abstract

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.