Geometry of Curves with Fractional Derivatives in Lorentz Plane

Year 2022, Issue: 38, 88 - 98, 31.03.2022

Meltem Öğrenmiş

https://doi.org/10.53570/jnt.1087800

Cited By: 3

Abstract

In this paper, the geometry of curves is discussed based on the Caputo fractional derivative in the Lorentz plane. Firstly, the tangent vector of a spacelike plane curve is defined in terms of the fractional derivative. Then, by considering a spacelike curve in the Lorentz plane, the arc length and fractional ordered frame of this curve are obtained. Later, the curvature and Frenet-Serret formulas are found for this fractional ordered frame. Finally, the relation between the fractional curvature and classical curvature of a spacelike plane curve is obtained. In the last part of the study, considering the timelike plane curve in the Lorentz plane, new results are obtained with the method in the previous section.

Keywords

Caputo fractional derivative, Tangent vectors, Frenet-Serret formulas, curvature

References

• D. Baleanu, K. Diethelm, E. Scalas, E., J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, New Jersey, 2012.
• A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
• K. B. Oldham, J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Dover Publications, New York, 2006.
• I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
• R. L. Bagley, R. A. Calico, Fractional Order State Equations for the Control of Viscoelastically Damped Structures, Journal of Guidance, Control, and Dynamics 14(2) (1991) 304–311.
• R. L. Bagley, P. J. Torvik, A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity, Journal of Rheology 27(3) (1983) 201–210.
• R. L. Bagley, P. J. Torvik, Fractional Calculus - A Different Approach to the Analysis of Viscoelastically Damped Structures, American Institute of Aeronautics and Astronautics 21(5) (1983) 741–748.
• R. L. Bagley, P. J. Torvik, On the Fractional Calculus Model of Viscoelastic Behaviour, Journal of Rheology 30(1) (1986) 133–155.
• M. Caputo, F. Mainardi, A New Dissipation Model Based on Memory Mechanism, Pure and Applied Geophysics 91(1) (1971) 134–147.
• R. C. Koeller, Applications of Fractional Calculus to the Theory of Viscoelasticity, Journal of Applied Mechanics 51(2) (1984) 299–307.
• T. Yajima, H. Nagahama, Differential Geometry of Viscoelastic Models with Fractional-Order Derivatives, Journal of Physics A: Mathematical and Theoretical 43(38) Article ID 385207 (2010) 9 pages.
• O. P. Agrawal, Formulation of Euler-Lagrange Equations for Fractional Variational Problems, Journal of Mathematical Analysis Applications 272(1) (2002) 368– 379.
• T. M. Atanackovic ́, S. Konjik, L. Oparnica, S. Pilipovic ́, Generalized Hamilton’s Principle with Fractional Derivatives, Journal of Physics A: Mathematical and Theoretical 43(25) Article ID 255203 (2010) 12 pages.
• D. Baleanu, T. Maaraba (Abdeljawad), F. Jarad, Fractional Variational Principles with Delay, Journal of Physics A: Mathematical and Theoretical 41(31) Article ID 315403 (2008) 8 pages.
• A. Gjurchinovski, T. Sandev, V. Urumov, Delayed Feedback Control of Fractional-Order Chaotic Systems, Journal of Physics A: Mathematical and Theoretical 43(44) Article ID 445102. (2010) 16 pages.
• I. Grigorenko, E. Grigorenko, Chaotic Dynamics of the Fractional Lorenz System, Physical Review Letter 91(3) Article ID 034101 (2003).
• V. E. Tarasov, Fractional Generalization of Gradient and Hamiltonian Systems, Journal of Physics A: Mathematical and General 38(6) (2005) 5929–5943.
• V. E. Tarasov, Fractional Generalization of Gradient Systems, Letters in Mathematical Physics 73(1) (2005) 49–58.
• T. Yajima, H. Nagahama, Geometric Structures of Fractional Dynamical Systems in Non-Riemannian Space: Applications to Mechanical and Electromechanical Systems, Annalen der Physik. (Berlin). 530(5) Article ID 01700391 (2018) 9 pages.
• K. A. Lazopoulos, A. K. Lazopoulos, Fractional Differential Geometry of Curves & Surfaces, Progress in Fractional Differentiation and Applications 2(3) (2016) 169–186.
• D. Baleanu, S. I. Vacaru, Constant Curvature Coefficients and Exact Solutions in Fractional Gravity and Geometric Mechanics, Central European Journal of Physics 9(5) (2011) 1267–1279.
• D. Baleanu, S. I. Vacaru, Fractional Almost Kahler-Lagrange Geometry. Nonlinear Dynamics 64(4) (2011) 365–373.
• D. Baleanu, S. I. Vacaru, Fractional Curve flows and Solitonic Hierarchies in Gravity and Geometric Mechanics, Journal of Mathematical Physics 52(5) Article ID 053514 (2011).
• D. Baleanu, S. I. Vacaru, Fedosov Quantization of Fractional Lagrange Spaces, International Journal of Theoretical Physics 50(1) (2011) 233–243.
• S. I. Vacaru, Fractional Nonholonomic Ricci Flows, Chaos, Solitons Fractals 45(9-10) (2012) 1266–1276.
• S. I. Vacaru, Fractional Dynamics from Einstein Gravity, General Solutions, and Black Holes, International Journal of Theoretical Physics 51(5) (2012) 1338–1359.
• M. E. Aydın, M. Bektaş, A. O. Öğrenmiş, A. Yokuş, Differential Geometry of Curves in Euclidean 3-Space with Fractional Order, International Electronic Journal of Geometry 14(1) (2021) 132-144.
• M. Öğrenmiş, Differential Geometry of Curves with Fractional Derivative, PhD Dissertation, Fırat University (2022) Elazığ, Turkey.
• D. Baleanu, J. J. Trujillo, A New Method of Finding the Fractional Euler-Lagrange and Hamilton Equations within Caputo Fractional Derivatives, Communications in Nonlinear Science and Numerical Simulation 15(5) (2010) 1111-1115.
• T. Yajima, S. Oiwa, K. Yamasaki, Geometry of Curves with Fractional-Order Tangent Vector and Frenet-Serret Formulas, Fractional Calculus and Applied Analysis 21(6) (2018) 1493–1505.
• B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York 1983.