Relaxed Hyperelastic Curves on a Non-Degenerate Surface

We study the variational problem belonging to a relaxed hyperelastic curve for non-null curve on a non-degenerate surface in Minkowski three-space ({E_{1}^{3}}) . Firstly, we derive the intrinsic equations for a relaxed hyperelastic curve and we give the necessary condition for being relaxed hyperelastic curve of any non-null geodesic on the surface in ({E_{1}^{3}}) . Then, we examine this formulation on non-null geodesics of pseudo-plane, pseudo-sphere ({S_{1}^{2}(r) }) , hyperbolic space ({H_{0}^{2}(r)}) and pseudo-cylinder ({C_{1}^{2}(r)}) .