Curvature and torsion dependent energy of elastica and nonelastica for a lightlike curve in the Minkowski space

  • T. Körpinar Muş Alparslan Univ., Turkey
  • R. C. Demirkol Muş Alparslan Univ., Turkey


UDC 515.1

We firstly describe conditions for being elastica or nonelastica for a lightlike elastic Cartan curve in the Minkowski space $\mathbb{E}_{1}^{4}$ by using the Bishop orthonormal vector frame and associated Bishop components.  Then we compute the energy of the lightlike elastic and nonelastic Cartan curve in the Minkowski space $\mathbb{E}_{1}^{4}$ and investigate its relationship with the energy of the same curve in Bishop vector fields in $\mathbb{E}_{1}^{4}$.  Here, energy functionals are computed in terms of Bishop curvatures of the lightlike Cartan curve lying in the Minkowski space $\mathbb{E}_{1}^{4}$.


A. Einstein, Zur Elektrodynamik bewegter Korper, Ann. Phys., 17, 891 – 921 (1905).

A. Einstein, Relativity. The special and general theory, Henry Holt, New York(1920).

A. Altin, On the energy and pseduoangle of frenet vector fields in $R^n_v$ , Ukr. Mat. J., 63, № 6, 969 – 975 (2011),

T. Korpınar, ¨ New Characterization for minimizing energy of biharmonic particles in Heisenberg spacetime<.em>, Int. J. Phys., 53, 3208 – 3218 (2014),

T. Korpınar, R. C. Demirkol, V. Asil, ¨ A new version on the energy of lightlike curve in Minkowski space $E^4_1$(submitted).

L. G. Hughston, W. T. Shaw, Classical strings in ten dimensions, Proc. Roy. Soc. London Ser. A, 414, 423 – 431 (1987).

L. G. Hughston, W. T. Shaw, Constraint-free analysis of relativistic strings, Classical Quatum Gravity, 5, No 3, 69 – 72 (1988),

L. G. Hughston, W. T. Shaw, Spinor parametrizations of minimal surfaces, The Mathematics of Surfaces, III, Oxford Univ. Press, New York, p. 359 – 372(1989)

W. T. Shaw, Twistors and strings, Mathematics and general relativity (Santa Cruz, CA, 1986), 337--363, Contemp. Math., 71, Amer. Math. Soc., Providence, RI, (1988),

H. Urbantke, On Pinl’s representation of lightlike curves in $n$ dimensions, Relativity today (Budapest, 1987), 34--36, World Sci. Publ., Teaneck, NJ, (1988)

???????????????A. E. H. Love, A treatise on the mathematical theory of elasticity, (2013).

˙Ilarslan K., A. U¸cum, E. Nesovic, On geneeralized spacelike Manheim curves in Minkowski space-time, Proc., Mat. Acad. Sci., Sect. A, Phys. Sci., 86, № 2, 249 – 258 (2016),

P. M. Chacon, A. M. Naveira, Corrected energy of distribution on riemannian manifolds, J. Osaka Math. 41, No. 1, 97 – 105 (2004),

E. Bretin, J.-O. Lachaud, E. Oudet, Regularization of discrete contour by Willmore energy, J. Math. Imaging and Vision, 40, № 2, 214 – 229 (2011),

T. Schoenemann, F. Kahl, S. Masnou, D. Cremers, A linear framework for region-based image segmentation and inpainting involving curvature penalization, Int. J. Comput. Vision, 99, № 1, 53 – 68 (2012),

D. Mumford, Elastica and computer vision, Algebraic geometry and its applications (West Lafayette, IN, 1990), 491--506, Springer, New York, (1994)

G. Citti, A. Sarti, Cortical based model of perceptual completion in the roto-translation space, J. Math. Imaging and Vision, 24, № 3, 307 – 326 (2006),

J. Guven, D. M. Valencia, J. Vazquez-Montejo, Environmental bias and elastic curves on surfaces, J. Phys. A, Math. Theory, 47. no. 35, 355201, 29 pp. (2014),

L. Euler, Additamentum ‘de curvis elasticis’, Methodus Inveniendi Lineas Curvas Maximi Minimive Probprietate Gaudentes, Lausanne (1744.)

D. A. Singer, Lectures on Elastic Curves and Rods, Curvature and variational modeling in physics and biophysics, 3--32, AIP Conf. Proc., 1002, Amer. Inst. Phys., Melville, NY, (2008),

How to Cite
KörpinarT., and DemirkolR. C. “Curvature and Torsion Dependent Energy of Elastica and Nonelastica for a Lightlike Curve in the Minkowski Space”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 8, Aug. 2020, pp. 1095-0, doi:10.37863/umzh.v72i8.847.
Research articles