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New geometric magnetic energy according to geometric Frenet formulas

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Abstract

Geometric calculus is a special case of the Non-Newtonian Calculus introduced by Grossman and Katz (Non-Newtonian calculus, Lee Press, Pigeon Cove, 1972). Also, it is a more convenient calculation method for situations where the geometric increment is more meaningful than the arithmetic increment. In this study, geometric curves are defined and geometric Frenet–Serret formulas for these curves are presented. Furthermore, we give applications of these concepts to geometric magnetic curves.

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References

  • Altin, A.: On the energy and pseduoangle of Frenet vector fields in \(R_{v}^{n}\). Ukr. Math. J. 63, 969–976 (2011)

    Article  Google Scholar 

  • Barros, M., Romero, A., Cabrerizo, J.L., Fernández, M.: The Gauss–Landau–Hall problem on Riemannian surfaces. J. Math. Phys. 46, 1 (2005)

    Article  MathSciNet  Google Scholar 

  • Barros, M., Cabrerizo, J.L., Fernandez, M., Romero, A.: Magnetic vortex filament flows. J. Math. Phys. 48, 082904 (2007)

    Article  MathSciNet  Google Scholar 

  • Baş, S.: A new version of spherical magnetic curves in the De-Sitter space \(S_{1}^{2}\). Symmetry 10, 606 (2018)

    Article  ADS  Google Scholar 

  • Boeckx, E., Vanhecke, L.: Harmonic and minimal vector fields on tangent and unit tangent bundles. Differ. Geom. Appl. 13, 77–93 (2000)

    Article  MathSciNet  Google Scholar 

  • Bozkurt, Z., Gök, I., Yaylı, Y., Ekmekci, F.N.: A new approach for magnetic curves in 3D Riemannian manifolds. J. Math. Phys. 55, 053501 (2014)

    Article  MathSciNet  ADS  Google Scholar 

  • Cabrerizo, J.L.: Magnetic fields in 2D and 3D sphere. J. Nonlinear Math. Phys. 20, 440 (2013)

    Article  MathSciNet  Google Scholar 

  • Cabrerizo, J.L., Fernandez, M., Gomez, J.S.: On the existence of almost contact structure and the contact magnetic field. Acta Math. Hung. 125, 191–199 (2009)

    Article  MathSciNet  Google Scholar 

  • Chacon, P.M., Naveira, A.M., Osaka, J.: Corrected energy of distributions on Riemannian Manifolds. Osaka J. Math. 41, 97–105 (2004)

    MathSciNet  Google Scholar 

  • Druta-Romaniuc, S.L., Munteanu, M.I.: Magnetic curves corresponding to killing magnetic fields in \(\cal{E} ^{3}\). J. Math. Phys. 52, 113506:1 (2011)

    Article  MathSciNet  Google Scholar 

  • Druţă-Romaniuc, S.L., Munteanu, M.I.: Killing magnetic curves in a Minkowski 3-space. Nonlinear Anal. Real World Appl. 14, 383–396 (2013)

    Article  MathSciNet  Google Scholar 

  • Ekinci, A., Baş, S., Körpınar, T., Körpınar, Z.: Optical new geometric magnetic curves according to geometric Frenet formulas (in review) (2023)

  • Gil-Medrano, O.: Relationship between volume and energy of vector field. Differ. Geom. Appl. 15, 137–152 (2001)

    Article  MathSciNet  Google Scholar 

  • Gluck, H., Ziller, W.: On the volume of the unit vector fields on the three sphere. Comment Math. Helv. 6, 177 (1986)

    Article  MathSciNet  Google Scholar 

  • Gonzalez-Davila, J.C., Vanhecke, L.: Examples of minimal unit vector fields. Ann. Glob. Anal. Geom. 18, 385 (2000)

    Article  MathSciNet  Google Scholar 

  • Grossman, M., Katz, R.: Non-Newtonian Calculus. Lee Press, Pigeon Cove (1972)

    Google Scholar 

  • Johnson, D.L.: Volume of flows. Proc. Am. Math. Soc. 104, 923–931 (1988)

    Article  MathSciNet  Google Scholar 

  • Körpınar, T.: New characterization for minimizing energy of biharmonic particles in Heisenberg spacetime. Int. J. Phys. 53, 3208–3218 (2014)

    Article  MathSciNet  Google Scholar 

  • Körpınar, T.: On \(\textbf{T}\)-magnetic biharmonic particles with energy and angle in the three dimensional Heisenberg group \(\mathbb{H}\). Adv. Appl. Clifford Algebras 28, 9 (2018a)

    Article  MathSciNet  Google Scholar 

  • Körpınar, T.: A new version of normal magnetic force particles in 3D Heisenberg space. Adv. Appl. Clifford Algebras 28, 83 (2018b)

    Article  MathSciNet  Google Scholar 

  • Körpınar, T., Demirkol, R.C.: Frictional magnetic curves in 3D Riemannian manifolds. Int. J. Geom. Methods Mod. Phys. 15, 1850020–1 (2018)

    Article  MathSciNet  Google Scholar 

  • Körpınar, T., Demirkol, R.C.: Electromagnetic curves of the linearly polarized light wave along anoptical fiber in a 3D semi-Riemannian manifold. J. Mod. Opt. 66, 857–867 (2019)

    Article  ADS  Google Scholar 

  • Körpınar, T., Demirkol, R.C.: Electromagnetic curves of the linearly polarized light wave along anoptical fiber in a 3D Riemannian manifold with Bishop equations. Optik 200, 163334 (2020a)

    Article  ADS  Google Scholar 

  • Körpınar, T., Demirkol, R.C.: Electromagnetic curves of the polarized light wave along the optical fiber in De-Sitter 2-space \(S_{1}^{2}\). Indian J. Phys. 95, 147–156 (2020b)

    Article  ADS  Google Scholar 

  • Körpinar, T., Demirkol, R.C., Körpınar, Z.: 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, 1–11 (2019)

    Article  Google Scholar 

  • Körpinar, T., Demirkol, R.C., Khalil, E.M., Korpinar, Z., Baleanu, D.: Quasi binormal Schrodinger evolution of wave polarizatıon field of light wıth repulsive type. Phys. Scr. 96(4), 045104 (2021)

    Article  ADS  Google Scholar 

  • Özdemir, Z.: A new calculus for the treatment of Rytov’s law in the optical fiber. Optik 216, 164892 (2020)

    Article  ADS  Google Scholar 

  • Sunada, T.: Magnetic flows on a Riemann surface. In: Proceedings of KAIST Mathematics workshop, p. 93 (1993)

  • Wiegmink, G.: Total bending of vector fields on Riemannian manifolds. Math. Ann. 303, 325–344 (1995)

    Article  MathSciNet  Google Scholar 

  • Wood, C.M.: On the energy of a unit vector field. Geom. Dedicata 64, 319–330 (1997)

    Article  MathSciNet  Google Scholar 

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Correspondence to Zeliha Körpinar.

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Ekinci, A., Bas, S., Körpinar, T. et al. New geometric magnetic energy according to geometric Frenet formulas. Opt Quant Electron 56, 81 (2024). https://doi.org/10.1007/s11082-023-05569-z

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  • DOI: https://doi.org/10.1007/s11082-023-05569-z

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