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In this paper, we consider the classical variational problem in the Galileanspace. we develop the Euler-Lagrange equations for a elastic line on anoriented 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 curvaturefunction of a curve on an oriented surface in $G_3$. Finally, we willinvestigate whether or not the relaxed elastic curves are on a geodesic.

Relaxed Elastic Line On An Oriented Surface In The Galilean Space