# Normal Fermi-Walker Derivative in $E_{1}^{3}$

Document Type: Research articles

Authors

Ankara University

10.22080/cjms.2020.10693.1295

Abstract

In this paper, firstly, in $E_1^3$, we defined normal Fermi-Walker derivative and applied for the adapted frame. Normal Fermi-Walker parallelism, normal non-rotating frame, and Darboux vector expressions of normal Fermi-Walker derivative by normal Fermi-Walker derivative are given for adapted frame. Being conditions of normal Fermi-Walker derivative and normal non-rotating frame are examined for frames throughout spacelike, timelike, lightlike curves. It is shown that the vector field which takes part in [17] is normal Fermi-Walker parallel by the normal Fermi-Walker derivative throughout the spacelike, timelike, and lightlike general helix. Also, we show that the Frenet frame is a normal non-rotating frame using the normal Fermi-Walker derivative. Afterward, we testified that the adapted frame is a normal non-rotating frame throughout the spacelike, timelike, and lightlike general helix.

Keywords

Article Title [Persian]

### References

[1] R. Balakrishnan, Space curves, anholonomy and nonlinearity, Prama J. Phys. 64(4)(2005), 607-615.
[2] I. M. Benn, and R. W. Tucker, Wave mechanics and inertial guidance, Phys. Rev. D. 39(6)(1989), 1594-1601.
[3] M.V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. Roy. Soc. London A. 1960, 392.
[4] C. Calin, and M. Crasmareanu, Slant Curves and Particles in threedimensional Warped Products and their Lancret invariants, Bull. Australian Math. Soc. 88(1)(2013), 128-142.
[5] M. Crasmareanu, and C. Frigioiu, Unitary vector fields are Fermi-Walker transported along Rytov-Legendre curves, Int. J. Geom. Methods Mod. Phy. 12(2015), 1550111.
[6] R. Dandolof, Berry’s phase and Fermi-Walker parallel transport, Phys. Lett. A. 139 (1,2)(1989), 19-20.
[7] E. Fermi, Sopra i fenomeni che avvengono in vicinanza di una linea oraria, Atti Accad. Naz. Lincei Cl. Sci. Fiz. Mat. Nat. 31(1922), 184–306.
[8] S. W. Hawking, and G. F. R. Ellis, The Large Scale Structure of Spacetime, Cambridge University Press, 1973.
[9] F. Karaku¸s, and Y. Yaylı, On the Fermi-Walker derivative and non-rotating frame, Int. Journal of Geometric Methods in Modern Physics. (9,8)(2012), 1250066.
[10] F. Karaku¸s, and Y. Yaylı, The Fermi- Walker derivative in Lie groups, Int. J. Geom. Methods Mod. Phy. 10(7)(2013), Article ID 1320011:10p.
[11] F. Karaku¸s, and Y. Yaylı, The Fermi-Walker derivative in Minkowski 3-Space $E^3$
1 , 2nd International Eurasian Conference On Mathematics Sciences And Applications, Proceedings, 2013.
[12] F. Karaku¸s, and Y. Yaylı, The Fermi derivative in the hypersurfaces, Int. J. Geom. Methods in Mod. Phys. 12(1)(2015), Article ID 1550002:12p.
[13] F. Karaku¸s, and Y. Yaylı, On the Surface the Fermi- Walker derivative in Minkowski 3-Space $E^3$ 1 , Adv. Appl. Clifford Algebras Springer. International Publishing. (2015), 1-12.
[14] F. Karaku¸s, and Y. Yaylı, The Fermi-Walker derivative on the Spherical Indicatrix of Spacelike curve in Minkowski 3-Space $E^3$, 1 , Adv. Appl. Clifford Algebras. Springer International Publishing. (2016), Article DOI 10.1007/s00006-015-0635-9.
[15] R. L´opez, Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space, Int. Electron. J. Geom. 7(1)(2014), 44-107.
[16] P. D. Scofield, Curves of Constant Precession, The American Math. Monthly. (102)6(1995), 531-537.
[17] B. Uzunoglu, I. Gok, and Y. Yayli, A new approach on curves of constant precession, Applied Mathematics and Computation. 275(2016), 317–323.