# NEW VERSION OF THE MAGNETIC CURVES ACCORDING TO THE BISHOP FRAME IN 𝔼3

• Sariaydin, Muhammed T. (Department of Mathematics, Selcuk University) ;
• Korpinar, Talat (Department of Mathematics, Mus Alparslan University)
• Accepted : 2018.12.18
• Published : 2019.03.25

#### Abstract

In this paper, it is investigated Lorentz force equations for $N_1$ and $N_2$-magnetic curves in 3-Dimensional Euclidean space. We give the Lorentz force in the Bishop frame in ${\mathbb{E}}^3$. Then, we obtain a new characterization for a magnetic field V. Also, we also give examples for each curve.

#### References

1. Backlund transformation and multi-soliton solutions, NPTEL Course, 112105165/lec38.,
2. Bishop, R. L. (1975). There is more than one way to frame a curve. The American Mathematical Monthly, 82(3), 246-251. https://doi.org/10.1080/00029890.1975.11993807
3. Bozkurt, Z., Gok, I., Yayli, Y., & Ekmekci, F. N. (2014). A new approach for magnetic curves in 3D Riemannian manifolds. Journal of Mathematical Physics, 55(5), 053501. https://doi.org/10.1063/1.4870583
4. Chern, S. S., & Tenenblat, K. (1986). Pseudospherical surfaces and evolution equations. Studies in Applied Mathematics, 74(1), 55-83. https://doi.org/10.1002/sapm198674155
5. Clelland, J. N., & Ivey, T.A. Backlund transformations and darboux integrability for nonlinear wave equations, arXiv:0707.4408v2.
6. Comtet, A. (1987). On the landau levels on the hyperbolic plane. Annals of physics, (Vol. 173, No. 1).
7. Druta-Romaniuc, S. L., & Munteanu, M. I. (2013). Killing magnetic curves in a Minkowski 3-space. Nonlinear Analysis: Real World Applications, 14(1), 383-396. https://doi.org/10.1016/j.nonrwa.2012.07.002
8. Druta-Romaniuc, S. L., & Munteanu, M. I. (2011). Magnetic curves corresponding to Killing magnetic fields in $E^3$. Journal of Mathematical Physics, (Vol. 52, No. 11).
9. Karacan, M. K., & Tuncer, Y. (2012). Backlund transformations according to bishop frame in Euclidean 3-space. In Siauliai Mathematical Seminar (Vol. 7, No. 15).
10. Korpinar, T., & Sariaydn, M.T. (in press). On the Magnetic curves corresponding to the Backlund transformation in the Euclidean 3-space.
11. Korpinar, T. New characterizations for minimizing energy of biharmonic particles in Heisenberg spacetime. International Journal of Theoretical Physics, 2014;53.9:3208-3218. https://doi.org/10.1007/s10773-014-2118-5
12. Korpinar, T. B-tubular surfaces in Lorentzian Heisenberg Group H3. Acta Scientiarum.Technology, 2015;37.1.
13. Korpinar, T. New characterization of b-m2 developable surfaces. Acta Scientiarum. Technology, 2015;37.2.
14. Korpinar, T, Turhan, E. A New Version of Inextensible Flows of Spacelike Curves with Timelike $B_2$ in Minkowski Space-Time $E^4_1$, Dynamical Systems, 2013;21.3:281-290.
15. Korpinar, T. A new version of energy for slant helix with bending energy in the Lie groups. Journal of Science and Arts, 2017;17.4:721-730.
16. Munteanu, M. I. (2013). Magnetic curves in a Euclidean space: one example, several approaches. Publications De I'Institut Mathematique, 94, 141-150. https://doi.org/10.2298/PIM1308141M
17. Munteanu, M. I., & Nistor, A. I. (2012). The classification of Killing magnetic curves in $S^2{\times}R$. Journal of Geometry and Physics, 62(2), 170-182. https://doi.org/10.1016/j.geomphys.2011.10.002
18. Munteanu, M. I. (2013). Magnetic curves in a Euclidean space: one example, several approaches. Publications De I'Institut Mathematique, 94, 141-150. https://doi.org/10.2298/PIM1308141M
19. Munteanu, M. I., & Nistor, A. I. (2017). On some closed magnetic curves on a 3-torus. Mathematical Physics, Analysis and Geometry, (Vol. 20, No. 2).
20. O'neill, B. (1983). Semi-Riemannian geometry with applications to relativity. (Vol. 103). Academic press.
21. Sunada, T. (1993). Magnetic flows on a Riemann surface. In Proc. KAIST Math. Workshop (Vol. 8, No. 93, p. 108).
22. Weiss, J. (1984). On classes of integrable systems and the Painleve property. Journal of Mathematical Physics, 25(1), 13-24. https://doi.org/10.1063/1.526009