Equation of lightpath in fractal spaces

Document Type : Research Paper

Authors

1 Department of Physics, Urmia Branch, Islamic Azad University, Urmia, Iran

2 Physics and Accelerators Research School, Nuclear Science and Technology Research Institute, Tehran, Iran

Abstract

The fractal dimension of Koch and Cesàro curves has different values. These two curves are not differentiable and integrable in the sense of ordinary calculus. This paper shows how fractal calculus can be applied to these two curves. To do this, we need to obtain the corresponding staircase function. Calculating the path integral on these two fractal curves is the starting point for formulating and extracting Lagrange equations. Fractal integration and fractal derivation on these two fractal paths are performed analytically, and the results are plotted in a MATLAB environment. In addition, we present an application by finding light path equations in this fractal space using the generalized Fermat's principle in fractal calculus.

Keywords

References

[1] L. F. Abbott and M. B. Wise, Dimension of a quantum-mechanical path, Amer. J. Phys., 49 (1981) 37–39.
[2] M. L. Boas, Mathematical methods in the physical sciences, John Wiley & Sons, Inc., (2006).
[3] N. Delfan, A. Pishkoo, M. Azhini and M. Darus, Using fractal calculus to express electric potential and electric field in terms of staircase and characteristic functions, Eur. J. Pure Appl., 13 (2020) 19–32.
[4] R. A. El-Nabulsi and A. K. Golmankhaneh, Generalized heat diffusion equations with variable coefficients and its fractalization from black-scholes equation, Commun. Theor. Phys, 73 (2021).
[5] R. P. Feynman and A. R. Hibbs, Quantum mechanics and path integrals, McGraw-Hill, New York, (1965).
[6] A. K. Golmankhaneh, On the calculus of parameterized fractal curves, Turk. J. Phys., 41 (2017) 418–425.
[7] A. K. Golmankhaneh, A. K. Golmankhaneh and D. Baleanu, Lagrangian and Hamiltonian mechanics on fractals subset of real-line, Int. J. Theor. Phys., 52 (2013) 4210–4217.
[8] A. K. Golmankhaneh and K. Welch, Equilibrium and non-equilibrium statistical mechanics with generalized fractal derivatives: A review, Mod. Phys. Lett. A, 36 (2021) 31 pp.
[9] A. K. Golmankhaneh and R. T. Sibatov, Fractal Stochastic Processes on Thin Cantor-Like Sets, mathematics, 9 (2021).
[10] V. Lakshminarayanan, A. Ghatak and K. Thyagarajan, Lagrangian optics. Springer Science & Business Media, (2002).
[11] A. Parvate and A. D. Gangal, Calculus on fractal subsets of real-line I: Formulation, Fractals, 17 (2009) 53–81.
[12] A. Parvate and A. D. Gangal, Calculus on fractal subsets of real line II: Conjugacy with ordinary calculus, Fractals, 19 (2011) 271–290.
[13] A. Parvate, S. Satin and A. D. Gangal, Calculus on fractal curves in Rn , Fractals, 19 (2011) 15–27.
[14] A .Pishkoo and M. Darus, Using fractal calculus to solve fractal Navier-Stokes equations and simulation of laminar static mixing in COMSOL Multiphysics, Fractal and Fractional, 5 (2021) 16 pp.
[15] H. Goldstein, C. Poole and J. Safko, Classical Mechanics, Pearson; 3rd edition, London, (2001).
 
[16] ا. پیشکو و ع. خلیلی گلمانخانه, حساب دیفرانسیل و انتگرال فرکتالی؛ از نظریه تا کاربرد، ریاضی و جامعه، 5 (1399) 55--68.
[17] م. ح. اکرمی، حسابان کسری از نظریه تا کاربرد، ریاضی و جامعه، 4 (1396) 56--69.

Files

History

  • Receive Date: 25 June 2021
  • Revise Date: 05 December 2021
  • Accept Date: 12 December 2021
  • Publish Date: 23 August 2021

Share

How to cite

Statistics