Curvature Dependent Energy of Surface Curves in Minkowski Space

Article Sidebar

PDF
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.

Main Article Content

Talat Korpınar, Rıdvan Cem Demirkol, Vedat Asil

Abstract

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.

Article Details

References

  1. C.M. Wood, On the Energy of a Unit Vector Field, Geom. Dedicata. 64 (1997), 319-330.
  2. O. Gil Medrano, Relationship between volume and energy of vector fields, Differ. Geom. Appl. 15 (2001) 137-152.
  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.
  4. P.M. Chacon and A.M. Naveira, Corrected Energy of Distribution on Riemannian Manifolds, Osaka J. Math. 41 (2004) 97-105.
  5. A. Altin, On the energy and Pseduoangle of Frenet Vector Fields in R v ?, Ukr. Math J. 63 (2011) 969-975.
  6. G. Kirchhoff, ber Das Gleichgewicht und die Bewegung einer elastichen Scheibe, Crelles J. 40 (1850) 51-88.
  7. E. Catmull and J. Clark, Recursively generated b-spline surfaces on arbitrary topological surfaces, Comput.-Aided Des. 10 (1978), 350-355.
  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.
  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.
  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.
  11. L. Euler, Additamentum ”˜de curvis elasticis', in Methodus Inveniendi Lineas Curvas Maximi Minimive Probprietate Gau- dentes, Lausanne, 1744.
  12. C.H. Sequin, CAD Tools for Aesthetic Engineering, Comput.-Aided Des. Appl. 1 (2004) 301-309.
  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.
  14. P. Joshi and C. Sequin, Energy Minimizer for Curvature-Based Surface Functional, CAD Conference, Waikiki, Hawaii. (2007) 607-617.
  15. A. Einstein, Zur Elektrodynamik bewegter K?rper, Annalen der Physik. 17 (1905), 891-921.
  16. A. Einstein, Relativity:The Special and General Theory, Henry Holt, New York, 1920.
  17. T. Roberts, S. Schleif and J.M. Dlugosz, What is the experimental basis of Special Relativity? Usenet Physics FAQ, 2007.
  18. A. Einstein, Does the inertia of a body depend on its energy content?, Annalen der Physik, 18 (1905) 639-641.
  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.
  20. R. Capovilla, C. Chryssomalakos and J. Guven, Hamiltonians for curves, J. Phys. A. 35 (2002) 6571-6587.
  21. M. Carmeli, Motion of a charge in a gravitational field, Phys. Rev. B. 138 (1965) 1003-1007.
  22. J. Weber, Relativity and Gravitation, Interscience, New York, 1961.
  23. G. Napoli, L. Vergori, Extrinsic Curvature Effects on Nematic Shells, Phys. Rev. Lett. 108 (2012), Article ID 207803.