Closed 1/2-Elasticae in the 2-Sphere

TL;DR

This paper investigates closed critical trajectories on the sphere for a specific bending functional, revealing infinitely many solutions characterized by pairs of coprime natural numbers and linking their geometry to these parameters.

Contribution

It establishes the existence of infinitely many closed solutions for the 1/2-Bernoulli's bending functional on the sphere, with a detailed geometric characterization.

Findings

Existence of infinitely many closed trajectories depending on coprime pairs

A geometric description relating the pairs to trajectory shapes

Connection between the Lagrange multipliers and trajectory properties

Abstract

We study critical trajectories in the sphere for the $1/2$-Bernoulli's bending functional with length constraint. For every Lagrange multiplier encoding the conservation of the length during the variation, we show the existence of infinitely many closed trajectories which depend on a pair of relatively prime natural numbers. A geometric description of these numbers and the relation with the shape of the corresponding critical trajectories is also given.