Title: Curvature Dependent Energy of Surface Curves in Minkowski Space
Author(s): Talat Korpınar, Rıdvan Cem Demirkol, Vedat Asil
Pages: 254-263
Cite as:
Talat Korpınar, Rıdvan Cem Demirkol, Vedat Asil, Curvature Dependent Energy of Surface Curves in Minkowski Space, Int. J. Anal. Appl., 16 (2) (2018), 254-263.


In this paper, we firstly introduce kinematics properties of the moving particle lying on a surface S. We assume that the particle corresponds to a different type of surface curves such that they are characterized by using the Darboux vector field W in Minkowski spacetime. Based on this result, we present geometrical understanding of the energy of the particle in each Darboux vector fields whether they lie on a spacelike surface or a timelike surface. Then, we also determine the bending elastic energy functional for the same particle on a surface S by assuming the particle has a bending feature of elastica. Finally, we prove that bending energy formula can be represented by the energy of the particle in each Darboux vector field W.

Full Text: PDF



  1. C.M. Wood, On the Energy of a Unit Vector Field, Geom. Dedicata. 64 (1997), 319-330. Google Scholar

  2. O. Gil Medrano, Relationship between volume and energy of vector fields, Differ. Geom. Appl. 15 (2001) 137-152. Google Scholar

  3. P.M. Chacon, A.M. Naveira and J.M. Weston, On the Energy of Distributions, with Application to the Quaternionic Hopf Fibrations, Monatsh. Math. 133 (2001) 281-294. Google Scholar

  4. P.M. Chacon and A.M. Naveira, Corrected Energy of Distribution on Riemannian Manifolds, Osaka J. Math. 41 (2004) 97-105. Google Scholar

  5. A. Altin, On the energy and Pseduoangle of Frenet Vector Fields in R v ?, Ukr. Math J. 63 (2011) 969-975. Google Scholar

  6. G. Kirchhoff, ber Das Gleichgewicht und die Bewegung einer elastichen Scheibe, Crelles J. 40 (1850) 51-88. Google Scholar

  7. E. Catmull and J. Clark, Recursively generated b-spline surfaces on arbitrary topological surfaces, Comput.-Aided Des. 10 (1978), 350-355. Google Scholar

  8. T. Lopez-Leon, V. Koning, K.B.S. Devaiah, V. Vitelli and A.A. Fernandez-Nieves, Frustrated nematic order in spherical geometries, Nature Phys. 7 (2011) 391-394. Google Scholar

  9. T. Lopez-Leon, A.A. Fernandez-Nieves, M. Nobili and C. Blanc, Nematic-Smectic Transition in Spherical Shells, Phys. Rev. Lett. 106 (2011) 247802. Google Scholar

  10. J. Guven J, D.M. Valencia and J. Vazquez-Montejo, Environmental bias and elastic curves on surfaces, Phys. A: Math. Theory. 47 (2014) Article ID 355201. Google Scholar

  11. L. Euler, Additamentum ‘de curvis elasticis’, in Methodus Inveniendi Lineas Curvas Maximi Minimive Probprietate Gau- dentes, Lausanne, 1744. Google Scholar

  12. C.H. Sequin, CAD Tools for Aesthetic Engineering, Comput.-Aided Des. Appl. 1 (2004) 301-309. Google Scholar

  13. D. Zorin, Curvature-based energy for simulation and variational modelling, Proceedings of the International Conference on Shape Modelling and Applications. SMI’05 (2005) 196-204. Google Scholar

  14. P. Joshi and C. Sequin, Energy Minimizer for Curvature-Based Surface Functional, CAD Conference, Waikiki, Hawaii. (2007) 607-617. Google Scholar

  15. A. Einstein, Zur Elektrodynamik bewegter K?rper, Annalen der Physik. 17 (1905), 891-921. Google Scholar

  16. A. Einstein, Relativity:The Special and General Theory, Henry Holt, New York, 1920. Google Scholar

  17. T. Roberts, S. Schleif and J.M. Dlugosz, What is the experimental basis of Special Relativity? Usenet Physics FAQ, 2007. Google Scholar

  18. A. Einstein, Does the inertia of a body depend on its energy content?, Annalen der Physik, 18 (1905) 639-641. Google Scholar

  19. M.K. Saad, H.S. Abdel-Aziz, G. Weiss and M.A. Soliman, Relation among Darboux frames of null Bertrand curves in Pseudo-Euclidean space, 1st Int. WLGK11, 2011. Google Scholar

  20. R. Capovilla, C. Chryssomalakos and J. Guven, Hamiltonians for curves, J. Phys. A. 35 (2002) 6571-6587. Google Scholar

  21. M. Carmeli, Motion of a charge in a gravitational field, Phys. Rev. B. 138 (1965) 1003-1007. Google Scholar

  22. J. Weber, Relativity and Gravitation, Interscience, New York, 1961. Google Scholar

  23. G. Napoli, L. Vergori, Extrinsic Curvature Effects on Nematic Shells, Phys. Rev. Lett. 108 (2012), Article ID 207803. Google Scholar


Copyright © 2021 IJAA, unless otherwise stated.