Relaxed Hyperelastic Curves on a Non-Degenerate Surface

Abstract

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 $${C_{1}^{2}(r)}$$ .