Relaxed Elastic Line On An Oriented Surface In The Galilean Space

TL;DR

This paper develops Euler-Lagrange equations for elastic lines on surfaces in Galilean space, characterizing solutions and exploring their relation to geodesics, advancing the understanding of variational problems in non-Euclidean geometries.

Contribution

It introduces the formulation of elastic lines in Galilean space and characterizes their solutions using variational methods, a novel approach in this geometric context.

Findings

Derived Euler-Lagrange equations for elastic lines in Galilean space

Characterized solution curves of the energy functional

Investigated the relation between relaxed elastic curves and geodesics

Abstract

In this paper, we consider the classical variational problem in the Galilean space. we develop the Euler-Lagrange equations for a elastic line on an oriented surface in the Galilean 3-dimensional space $G_3$. Using the varia- tion method, we will try to give some characterization for the solution curve (the elastic line) of energy equation described by the total squared curvature function of a curve on an oriented surface in $G_3$. Finally, we will investigate whether or not the relaxed elastic curves are on a geodesic.