A New Approach on the Energy of Elastica and Non-Elastica in Minkowski Space E\(_{2}^{4}\)

  • Talat Körpinar
  • Rıdvan Cem DemirkolEmail author


In this work, we firstly describe conditions for being elastica for a moving particle corresponding to different type of space curves in Minkowski space \(\mathsf{E}_2^4\). Then, we investigate the energy on the elastic curves corresponding to a particular particle in the space and we also exploit its relationship with energy on the same particle in the Frenet vector fields. Finally, we characterize non-elastic curves in \(\mathsf{E}_2^4\) and we compute their energy to see the distinction between energies for the curves of elastic and non-elastic case in Minkowski space \(\mathsf{E}_2^4\).


Energy Minkowski space Elastic curves Frenet vectors 

Mathematics Subject Classification

53C80 53A10 74B05 53A35 


  1. Altin, A.: On the energy and pseduoangle of Frenet vector fields in \(R_v^n \). Ukr. Math. J. 63(6), 969–975 (2011)CrossRefzbMATHGoogle Scholar
  2. Bretin, E., Lachaud, J.-O., Oudet, E.: Regularization of discrete contour by Willmore energy. J. Math. Imaging Vis. 40(2), 214–229 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Chacon, P.M., Naveira, A.M.: Corrected energy of distrubution on Riemannian manifolds. Osaka J. Math. 41, 97–105 (2004)MathSciNetzbMATHGoogle Scholar
  4. Citti, G., Sarti, A.: Cortical based model of perceptual completion in the Roto-translation space. J. Math. Imaging Vis. 24(3), 307–326 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Duggal, K.L., Jin, D.H.: Null curves and hypersurfaces of semi Riemannian manifolds. World Scientific Publisher, London (2007)Google Scholar
  6. Einstein, A.: Zur Elektrodynamik bewegter Körper. Annalen der Physik 17, 891–921 (1905)CrossRefzbMATHGoogle Scholar
  7. Einstein, A.: Relativity. The Special and General Theory. New York, Henry Holt (1920)zbMATHGoogle Scholar
  8. Euler, L.: Additamentum ‘de curvis elasticis’, in Methodus Inveniendi Lineas Curvas Maximi Minimive Probprietate Gaudentes, Lausanne (1744)Google Scholar
  9. Guven, J., Valencia, D.M., Vazquez-Montejo, J.: Environmental bias and elastic curves on surfaces. Phys. A. Math Theor. 47, 355201–355231 (2014)Google Scholar
  10. Ilarslan, K., Nesovic, E.: Spacelike and timelike normal curves in Minkowski space-time. Pub. de L’Institut Math. 85(99), 111–118 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Körpinar, T.: New characterization for minimizing energy of biharmonic particles in Heisenberg spacetime. Int. J. Phys. 53, 3208–3218 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Körpinar T., Demirkol, R.C., Asil, V.: New Characterizations on the energy of parallel vector fields in Minkowski Space (2017)Google Scholar
  13. Love, A.E.H.: A treatise on the mathematical theory of elasticity. Cambridge University Press, Cambridge (2013)Google Scholar
  14. Mumford, D.: Elastica and Computer Vision, Algebraic Geometry and its Applications. Springer, New-York (1994)zbMATHGoogle Scholar
  15. Petrovic-Torgasev, M., İlarslan, K., Nesovic, E.: On partially null and pseudo null curves in the semi-Euclidean space \({\sf R}_2^4\). J. Geom. 84, 106–116 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Sasaki, M.: Null Cartan curves in \({\sf R}_2^4 .\) Toyoma Math. J. 32, 31–39 (2009)Google Scholar
  17. Schoenemann, T., Kahl, F., Masnou, S., Cremers, D.: A linear framework for region-based image segmentation and inpainting involving curvature penalization. Int. J. Comput. Vis. 99(1), 53–68 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Singer, D.A.: Lectures on elastic curves and rods. Dept. of Mathematics Case Western Reserve University, Cleveland (2007)Google Scholar
  19. Terzopoulost, D., Platt, J., Barr, A., Fleischert, K.: Elastically Deformable Models. Comput. Graph. 21(4), 205–214 (1987)Google Scholar

Copyright information

© Sociedade Brasileira de Matemática 2017

Authors and Affiliations

  1. 1.Department of MathematicsMus Alparslan UniversityMusTurkey

Personalised recommendations