The Motion of a Bead Sliding on a Wire in Fractional Sense

D. Baleanua,b, A. Jajarmi c, J.H. Asadd and T. Blaszczyke aDepartment of Mathematics and Computer Science, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey bInstitute of Space Sciences, P.O. Box MG-23, 76900, Magurele, Bucharest, Romania cDepartment of Electrical Engineering, University of Bojnord, P.O. Box 94531-1339, Bojnord, Iran dDepartment of Physics, College of Arts and Sciences, Palestine Technical University, P.O. Box 7, Tulkarm, Palestine eInstitute of Mathematics, Czestochowa University of Technology, al. Armii Krajowej 21, 42-201 Częstochowa, Poland

Received: February 26, 2017

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In this study, we consider the motion of a bead sliding on a wire which is bent into a parabola form. We first introduce the classical Lagrangian from the system model under consideration and obtain the classical Euler-Lagrange equation of motion. As the second step, we generalize the classical Lagrangian to the fractional form and derive the fractional Euler-Lagrange equation in terms of the Caputo fractional derivatives. Finally, we provide numerical solution of the latter equation for some fractional orders and initial conditions. The method we used is based on a discretization scheme using a Grünwald-Letnikov approximation for the fractional derivatives. Numerical simulations verify that the proposed approach is efficient and easy to implement.

DOI: 10.12693/APhysPolA.131.1561 PACS/topics: motion of a bead on a wire, Euler-Lagrange equation, fractional derivative, Grünwald-Letnikov approximation