Spinor Representations of Positional Adapted Frame in the Euclidean 3-Space

Yıl 2023, Cilt: 16 Sayı: 1, 62 - 72, 30.04.2023

Zehra İşbilir , Kahraman Esen Özen , Mehmet Güner

https://doi.org/10.36890/iejg.1179503

Cited By: 1

Öz

The main goal of this study is to bring together the spinors, which have a major place in several disciplines from mathematics to physics, and Positional Adapted Frame (PAF) which is a new type frame that attracts the attention of many researchers. In accordance with this purpose, we introduce the spinor representations for the trajectories endowed with PAF in the Euclidean 3-space $\mathbb{E}^3$, and construct the spinor equations of PAF vectors. Then, we find the relations between spinor representations of PAF and Serret-Frenet frame. Also we give some results and present some geometric interpretations with respect to this relationship. Moreover, we present an illustrative numerical example in order to support the given theorems and results.

Anahtar Kelimeler

Kinematics of a particle, positional adapted frame, spinors

Kaynakça

• [1] Balcı, Y., Erişir, T., Güngör, M. A.: Hyperbolic spinor Darboux equations of spacelike curves in Minkowski 3-space. J. Chungcheong Math. Soc. 28 (4), 525-535 (2015).
• [2] Bishop, R. L.: There is more than one way to frame a curve. Amer. Math. Monthly. 82 (3), 246-251 (1975).
• [3] Cartan, E.: The theory of spinors. Herman. Paris (1966), reprinted by Dover. New York (1981).
• [4] Chen, B. Y.: When does the position vector of a space curve always lie in its rectifying plane? Amer. Math. Monthly. 110 (2), 147-152 (2003).
• [5] Clifford, W. K.: Applications of Grassmann’s extensive algebra. Amer. J. Math. 1 (4), 350-358 (1878).
• [6] Darboux, G.: Leçons sur la thèorie gènèrale des surfaces I-II-III-IV. Gauthier-Villars. Paris (1896).
• [7] Dede, M.: A new representation of tubular surfaces. Houston J. Math. 45 (3), 707-720 (2019).
• [8] Dede, M., Ekici, C., Tozak, H.: Directional tubular surfaces. Int. J. Algebra. 9 (12), 527-535 (2015).
• [9] del Castillo, G. F. T.: 3-D Spinors, Spin-weighted functions and their applications. Birkhäuser. Boston (2003).
• [10] del Castillo, G. F. T, Sánchez Barrales, G.: Spinor formulation of the differential geometry of curves. Rev. Colomb. Mat. 38 (1), 27-34 (2004). [11] Erişir, T.: On spinor construction of Bertrand curves. AIMS Mathematics. 6 (4), 3583-3591 (2021).
• [12] Erişir, T., Güngör, M. A., Tosun, M.: Geometry of the hyperbolic spinors corresponding to alternative frame. Adv. Appl. Clifford Algebras. 25 (4), 799-810 (2015).
• [13] Erişir, T., Güngör, M. A.: On Fibonacci spinors. Int. J. Geom. Methods Mod. Phys. 17, 2050065 (2020).
• [14] Erişir, T., Karda˘ g, N. C.: Spinor representations of involute evolute curves in E3. Fundam. J. Math. Appl. 2 (2), 148-155 (2019).
• [15] Erişir, T., Öztaş, H. K.: Spinor equations of successor curves. Univers. J. Math. Appl. 5 (1), 32-41 (2022).
• [16] Gürbüz, N. E.: The evolution of an electric field with respect to the type-1 PAF and the PAFORS frames in R31 . Optik. 250 (1), 168285 (2022).
• [17] Hladik, J.: Spinors in physics. Springer Science & Business Media. New York (1999).
• [18] Iyer, B. R., Vishveshwara, C. V.: Frenet-Serret description of gyroscopic precession. Physical Review D. 48 (12), 5706 (1993).
• [19] İlarslan, K., Nesoviç, E.: Some characterizations of osculating curves in the Euclidean spaces. Demonstr. Math. 41 (4), 931-939 (2008).
• [20] İşbilir, Z., Özen, K. E., Tosun, M.: Mannheim partner P-trajectories in the Euclidean 3-Space E3. Honam Math. J. 44 (3), 419-431 (2022).
• [21] İşbilir, Z., Yazıcı, B. D., Tosun, M.: The spinor representations of framed Bertrand curves. Filomat. 37 (9), 2831–2843 (2023).
• [22] Keskin, O., Yaylı, Y.: An application of N-Bishop frame to spherical images for direction curves. Int. J. Geom. Methods Mod. Phys. 14 (11), 1750162 (2017).
• [23] Ketenci, Z., Erişir, T., Güngör, M. A.: Spinor equations of curves in Minkowski space. V. Congress of the Turkic World Mathematicians, June 05-07 2014, Issyk, Kyrgyzstan.
• [24] Ketenci, Z., Erişir, T., Güngör, M. A.: A construction of hyperbolic spinors according to Frenet frame in Minkowski space. Journal of Dynamical Systems and Geometric Theories. 13 (2), 179-193 (2015).
• [25] Kişi, İ., Tosun, M.: Spinor Darboux equations of curves in Euclidean 3-space. Mathematica Moravica. 19 (1), 87-93 (2015).
• [26] Koenderink, J. J.: Solid shape. MIT Press. Cambridge. Massachusetts (1990).
• [27] Lounesto, P.: Clifford algebras and spinors. In: Clifford Algebras and their Applications in Mathematical Physics. Vol. 183. Springer, Dordrecht (1986).
• [28] Lounesto, P.: Clifford algebras and spinors. Cambridge University Press. Cambridge (2001).
• [29] Özen, K. E.,İşbilir, Z., Tosun, M.: Characterization of Tzitzéica curves using positional adapted frame. Konuralp Journal of Mathematics. 10 (2), 260-268 (2022).
• [30] Özen, K. E., Tosun, M.: A new moving frame for trajectories with non-vanishing angular momentum. J. Math. Sci. Model. 4 (1), 7-18 (2021).
• [31] Özen, K. E., Tosun, M.: Trajectories generated by special Smarandache curves according to positional adapted frame. Karamano˘glu Mehmetbey University Journal of Engineering and Natural Sciences. 3 (1), 15-23 (2021).
• [32] Pauli, W.: Zur quantenmechanik des magnetischen elektrons. Zeitschrift für Physik. 43, 601-623 (1927).
• [33] Shifrin, T.: Differential geometry: A first course in curves and surfaces. University of Georgia. Preliminary Version (2008).
• [34] Soliman, M. A., Abdel-All, N. H., Hussien, R. A., Youssef, T.: Evolution of space curves using type-3 Bishop frame. Casp. J. Math. Sci. 8 (1), 58-73 (2019).
• [35] Solouma, E. M.: Characterization of Smarandache trajectory curves of constant mass point particles as they move along the trajectory curve via PAF. Bull. Math. Anal. Appl. 13 (4), 14-30 (2021).
• [36] Şenyurt, S., Çalışkan, A.: Spinor formulation of Sabban frame of curve on S2. Pure Mathematical Sciences. 4 (1), 37-42 (2015).
• [37] Tomonaga, S. I.: The story of spin. University of Chicago Press. Chicago (1997).
• [38] Ünal, D., Kişi, İ., Tosun, M.: Spinor Bishop equations of curves in Euclidean 3-space. Adv. Appl. Clifford Algebras. 23 (3), 757-765 (2013).
• [39] Vaz, J., da Rocha, R.: An introduction to Clifford algebras and spinors. Oxford University Press. London (2016).
• [40] Vivarelli, M. D.: Development of spinor descriptions of rotational mechanics from Euler’s rigid body displacement theorem. Celestial Mechanics. 32 (3), 193-207 (1984).
• [41] Wang, X. F., Zou, Z. J., Li, T. S., Luo, W. L.: Adaptive path following controller of underactuated ships using Serret-Frenet frame. Journal of Shanghai Jiaotong University (Science). 15 (3), 334-339 (2010).
• [42] Yazıcı, B. D., İşbilir, Z., Tosun, M.: Spinor representation of framed Mannheim curves. Turk. J. Math. 46 (7), 2690-2700 (2022).
• [43] Yılmaz, S., Turgut, M.: A new version of Bishop frame and an application to spherical images. J. Math. Anal. Appl. 371 (2), 764-776 (2010).