Siacci's Theorem for Frenet Curves in Minkowski 3-Space

Year 2020, Volume: 8 Issue: 1, 159 - 167, 20.03.2020

Kahraman Esen Özen

https://doi.org/10.36753/mathenot.693053

Cited By: 5

Abstract

For motion of a material point along a space curve, due to Siacci [1], a resolution of the acceleration vector is well known. In this resolution, the acceleration vector is stated as the sum of two special oblique components in the osculating plane to the curve. In this paper, we have studied the Siacci’s theorem for non-relativistic particles moving along the Frenet curves at very low speeds relative to the speed of light in Minkowski 3-space. Moreover, an illustrative example is given to show how the aforesaid theorem works. This theorem is a new contribution to the field and it may be useful for some specific applications in mathematical and computational physics.

Keywords

Kinematics of a particle, Minkowski 3-space, Plane and space curves, Siacci

References

• Siacci, F.: Moto per una linea gobba, Atti R Accad Sci. Torino, 14, 946-951 (1879).
• Babaarslan, M., Yaylı, Y.: On spacelike constant slope surfaces and Bertrand curves in Minkowski 3-space, Annals of the Alexandru Ioan Cuza University-Mathematics, (2015). DOI:10.1515/aicu-2015-0009.
• Ekici, C., Öztürk, H.: On time-like ruled surfaces in Minkowski 3-space, Universal Journal of Applied Science, 1(2), 56-63 (2013).
• Lopez, R.: Differential geometry of curves and surfaces in Lorentz-Minkowski space, International Electronic Journal of Geometry, 7(1), 44-107 (2014).
• Choi, J. H., Kimb, Y. H., Ali, A. T.: Some associated curves of Frenet non-lightlike curves in E31 , J. Math. Anal. Appl., 394(2), 712-723 (2012).
• Altunkaya, B., Kula, L.: Characterizations of slant and spherical helices due to pseudo-Sabban frame, Fundamental Journal of Mathematics and Applications, 1(1), 49-56 (2018).
• Samancı, H. K.: Introduction to timelike uniform B-spline curves in Minkowski-3 space, Journal of Mathematical Sciences and Modelling, 1(3), 206-210 (2018).
• Ersoy, S., Eren, K.: Timelike tangent developable surfaces and Bonnet surfaces, Abstract and Applied Analysis, 2016, 1-7 (2016).
• Yıldız, Ö. G., Hızal, S., Akyi˜ git, M.: Type I+ helicoidal surfaces with prescribed weighted mean or Gaussian curvature in Minkowski space with density, An. St. Univ. Ovidius Constanta, 26(3), 99-108 (2018).
• Siacci, F.: Moto per una linea piana, Atti R Accad Sci. Torino, 14, 750-760 (1879).
• Casey, J.: Siacci’s resolution of the acceleration vector for a space curve, Meccanica, 46(2), 471-476 (2011).
• Whittaker, E. T.: A Treatise on the analytical dynamics of particles and rigid bodies. 4th edn. Cambridge University Press. Cambridge. Dover, New York (1944).
• Grossman, N.: The sheer joy of celestial mechanics. Birkhäuser. Basel (1996).
• Küçükarslan, Z., Yılmaz, M. Y., Bekta¸s, M.: Siacci’s theorem for curves in Finsler manifold F3, Turkish Journial of Science and Technology, 7(2), 181-185 (2012).
• Özen, K. E., Tosun, M., Akyi˜ git, M.: Siacci’s theorem according to Darboux frame, An. St. Univ. Ovidius Constanta, 25(3), 155-165 (2017).