  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.
