Evolution of curves with modified orthogonal frame in G3

This research investigates the dynamics of inextensible curve flows using a modified orthogonal frame within the context of three-dimensional Galilean space [Formula: see text]. The study begins with the formulation of a modified orthogonal frame, derived from torsion considerations in [Formula: see text]. Subsequently, we establish the mathematical relationship between this newly developed frame and the classical Frenet frame within the Galilean space framework. Using this relationship, we analyze the evolution equations governing space curves according to the modified orthogonal frame. This analysis encompasses both curvature and torsion aspects, providing a comprehensive understanding of curve behavior in [Formula: see text]. Through this innovative approach, we derive significant characterizations and results that contribute to the broader understanding of curve dynamics in Galilean space. These findings offer new perspectives on geometric flow problems and may have implications for various fields, including differential geometry and mathematical physics.