Research ArticleNo Access

Elastic magnetic curves of ferromagnetic and superparamagnetic models on the surface

Talat Korpinar

Mathematics Department, Mus Alparslan University, Guzeltepe Campus, Mus, 49250, Turkey

E-mail Address: talatkorpinar@gmail.com

Search for more papers by this author

Ridvan Cem Demirkol

Mathematics Department, Mus Alparslan University, Guzeltepe Campus, Mus, 49250, Turkey

E-mail Address: rcdemirkol@gmail.com

Corresponding author.

Search for more papers by this author

, and

Vedat Asil

Mathematics Department, Firat University, Elazig, 23119, Turkey

E-mail Address: vasil@firat.edu.tr

Search for more papers by this author

https://doi.org/10.1142/S0219887821500377Cited by:11 (Source: Crossref)

Abstract

We are interested in defining new energy functionals and solving them by using the variational approach method and Darboux equations. That is, we aim to define a new class of elastic curves on the regular surface $Λ$ . We further improve an alternative method to find critical points of the bending energy functionals acting on a class of magnetic curves on $Λ$ . As a result, we classify these critical curves as elastic magnetic curves of the Darboux vector family.

Keywords:

AMSC: 53A04, 35B38

References

1. L. Euler, Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, Sive Solutio Problematis Isoperimetrici Lattisimo Sensu Accepti, Vol. 1/24 (Lausannæ, Genevæ, apud Marcum-Michaelem Bousquet & socios, 1744). Google Scholar
2. J. Radon, Zum Problem von Lagrange, Vier Vorträge, Gehalten im Mathematischen Seminar der Hamburgischen Universität (7.–24. Juli 1928), Vol. 6 (Vieweg+Teubner Verlag, 1928), pp. 273–299 (in German). Google Scholar
3. J. Langer and D. A. Singer , Lagrangian aspect of the Kirchhoff elastic rod, SIAM Rev. 38 (1986) 605–618. Crossref, Web of Science, Google Scholar
4. J. Langer and D. A. Singer , The total squared curvature of closed curves, J. Differ. Geom. 20(1) (1984) 1–22. Crossref, Web of Science, Google Scholar
5. D. A. Singer , Lectures on elastic curves and rods, AIP Conf. Proc. Am. Inst. Phys. 1002(1) (2008) 3–32. Crossref, Google Scholar
6. O. Ameline, S. Haliyo, X. Huang and J. A. Cognet , Classifications of ideal 3D elastica shapes at equilibrium, J. Math. Phys. 58(6) (2017) 062902. Crossref, Web of Science, Google Scholar
7. R. Huang , A note on the $p$ -elastica in a constant sectional curvature manifold, J. Geom. Phys. 49 (2004) 343–349. Crossref, Web of Science, Google Scholar
8. O. J. Garay and S. Montaldo , Elasticae in killing submersions, Med. J. Math. 13(5) (2016) 3281–3302. Web of Science, Google Scholar
9. B. Palmer and Á. Pámpano , Anisotropic bending energies of curves, Ann. Global Anal. Geom. 57(2) (2020) 257–287. Crossref, Web of Science, Google Scholar
10. K. S. Thorne and R. D. Blandford , Modern Classical Physics: Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics (Princeton University Press, 2017). Google Scholar
11. Ó. J. Garay , Extremals of the generalized Euler-Bernoulli energy and applications, in Proc. Tenth Int. Conf. Geometry, Integrability and Quantization, Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences (Varna, Bulgaria, 2009), pp. 56–87. Google Scholar
12. M. Barros, J. L. Cabrerizo, M. Fernández and A. Romero , Magnetic vortex filament flows, J. Math. Phys. 48(8) (2007) 082904. Crossref, Web of Science, Google Scholar
13. M. Barros, A. Ferrández and Ó. J. Garay , Dynamics of charges and solitons, J. Geom. Phys. 125 (2018) 12–22. Crossref, Web of Science, Google Scholar
14. T. Körpınar, R. C. Demirkol and Z. Körpınar , Soliton propagation of electromagnetic field vectors of polarized light ray traveling in a coiled optical fiber in Minkowski space with Bishop equations, Eur. Phys. J. D 73(9) (2019) 203. Crossref, Web of Science, Google Scholar
15. T. Körpınar, R. C. Demirkol and Z. Körpınar , Soliton propagation of electromagnetic field vectors of polarized light ray traveling in a coiled optical fiber in the ordinary space, Int. J. Geom. Meth. Modern Phys. 16(08) (2019) 1950117. Link, Web of Science, Google Scholar
16. T. Körpınar, R. C. Demirkol and Z. Körpınar , Soliton propagation of electromagnetic field vectors of polarized light ray traveling along with coiled optical fiber on the unit 2-sphere $𝕊^{2}$ , Rev. Mex. Fís. 65(6) (2019) 626–633. Crossref, Web of Science, Google Scholar
17. T. Körpınar and R. C. Demirkol , On the new approach for the energy of elastica, Acta Scient. Technol. 40 (2018) e35493–e35493. Crossref, Web of Science, Google Scholar
18. T. Körpınar and R. C. Demirkol , A new approach on the energy of elastica and non-elastica in Minkowski space $𝔼_{2}^{4}$ , Bullet. Braz. Math. Soc. 49(1) (2018) 159–177. Crossref, Web of Science, Google Scholar
19. Z. Özdemir, İ. Gök and F. N. Ekmekci , A geometric view of magnetic surfaces and magnetic curves, Int. Electron. J. Geom. 12(1) (2019) 126–134. Crossref, Web of Science, Google Scholar
20. A. Cebers and K. Erglis , Flexible magnetic filaments and their applications, Adv. Function Mater. 26(22) (2016) 3783–3795. Crossref, Web of Science, Google Scholar