On the Resolution of the Acceleration Vector According to Bishop Frame

Yıl 2021, , 26 - 32, 22.03.2021

Kahraman Esen Özen Murat Tosun

https://doi.org/10.32323/ujma.818071

Cited By: 1

Öz

In the second half of the 19th century, Siacci investigated the motion of a particle in space under the influence of any forces (Atti R Accad Sci. Torino \textbf{14}(1879)). In this study, Siacci obtained a resolution of the acceleration vector which is very useful when the angular momentum is conserved. On the other hand, Bishop introduced the Bishop frame which is well defined for every curves and so very convenient for mathematical researches in the third quarter of the 20th century (Am Math Monthly \textbf{82}(1975)). In this study, we discuss the Siacci's resolution of the acceleration vector according to Bishop frame of the trajectory of the moving particle. Also, we provide an illustrative example for the obtained results.

Anahtar Kelimeler

Kinematics of a particle, Plane and space curves, Bishop frame, Siacci

Teşekkür

This article is derived from the master's thesis titled "Siacci's theorem for the first and the second Bishop frame" which was carried out by K. E. \"Ozen under the supervision of M. Tosun at Sakarya University / Graduate School of Natural and Applied Sciences.

Kaynakça

• [1] F. Siacci, Moto per una linea piana, Atti R Accad Sci. Torino, 14 (1879), 750-760.
• [2] F. Siacci, Moto per una linea gobba, Atti R Accad Sci. Torino, 14 (1879), 946-951.
• [3] E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge University Press, Cambridge. Dover, New York, 1944.
• [4] N. Grossman, The Sheer Joy of Celestial Mechanics, Birkh¨auser, Basel, 1996.
• [5] J. Casey, Siacci’s resolution of the acceleration vector for a space curve, Meccanica, 46 (2011), 471-476.
• [6] Z. Küçüukarslan, M. Y. Yılmaz, M. Bektas¸, Siacci’s theorem for curves in Finsler manifold F3, Turkish J. Sci. Technol., 7 (2012), 181-185.
• [7] K. E. Özen, Siacci’s theorem for the first and the second Bishop frame, Master’s Thesis, Sakarya University, 2015.
• [8] K. E. Özen, M. Tosun, M. Akyig˜it, Siacci’s theorem according to Darboux frame, Analele Stiintifice ale Univ. Ovidius Constanta, Ser. Mat., 25 (2017), 155-165.
• [9] K. E. Özen, M. Gu¨ner, M. Tosun, A note on the acceleration and jerk in motion along a space curve, Analele Stiintifice ale Univ. Ovidius Constanta, Ser. Mat., 28 (2020), 151-164.
• [10] K. E. Özen, Siacci’s theorem for Frenet curves in Minkowski 3-space, Math. Sci. Appl. E-Notes, 8 (2020), 159-167.
• [11] R. L. Bishop, There is more than one way to frame a curve, Am. Math. Mon., 82 (1975), 246-251.
• [12] B. Bükcü, M. K. Karacan. The slant helices according to Bishop frame, Int. J. Comput. Math. Sci., 3 (2009), 67-70.
• [13] K. Eren, Geometry of coupled dispersionless equations by using Bishop frames. In: Hvedri, I. (ed.) TBILISI-MATHEMATICS, pp 38-47. Sciendo, Berlin (2020).
• [14] F. Doğan, Y. Yaylı, On the curvatures of tubular surface with Bishop frame, Commun. Fac. Sci. Univ. Ank. Series A1, 60 (2011), 59-69.
• [15] T. Körpınar, M. T. Sarıydın, E. Turhan, Associated curves according to Bishop frame in Euclidean 3 space, Adv. Model. Optim., 15 (2013), 713-717.
• [16] A. J. Hanson, H. Ma, Parallel transport approach to curve framing Indiana University, Techreports-TR425, 11 (1995), 3-7.
• [17] M. Masal, A. Z. Azak, Ruled surfaces according to Bishop frame in the Euclidean 3-space, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 89 (2019), 415-424.
• [18] A. Kelleci, M. Bektaş, M. Ergüt, The Hasimoto surface according to Bishop frame, Adıyaman U¨ niversitesi Fen Bilimleri Dergisi, 9 (2019), 13-22.
• [19] T. Shifrin, Differential Geometry: A First Course in Curves and Surfaces, University of Georgia, Preliminary Version, 2008.
• [20] T. Körpınar, S. Baş, On characterization of B-Focal curves in E3, Bol. Soc. Paran. Mat., 31 (2013), 175-178.