Curvature and Torsion Dependent Energy of Elastica and Nonelastica for a Lightlike Curve in the Minkowski Space

We first describe the conditions for being elastica or nonelastica for a lightlike elastic Cartan curve in the Minkowski space E14\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbbm{E}}_1^4 $$\end{document} by using the Bishop orthonormal vector frame and associated Bishop components. Then we compute the energy of the lightlike elastic and nonelastic Cartan curves in the Minkowski space E14\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbbm{E}}_1^4 $$\end{document} and investigate its relationship with the energy of the same curves in Bishop vector fields in E14\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbbm{E}}_1^4 $$\end{document}. In this case, the energy functionals are computed in terms of the Bishop curvatures of the lightlike Cartan curve lying in the Minkowski space E14\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbbm{E}}_1^4 $$\end{document}.