Abstract
In this new work, the free motion of a coupled oscillator is investigated. First, a fully description of the system under study is formulated by considering its classical Lagrangian, and as a result, the classical Euler-Lagrange equations of motion are constructed. After this point, we extend the classical Lagrangian in fractional sense, and thus, the fractional Euler-Lagrange equations of motion are derived. In this new formulation, we consider a recently introduced fractional operator with Mittag-Leffler nonsingular kernel. We also present an efficient numerical method for solving the latter equations in a proper manner. Due to this new powerful technique, we are able to obtain remarkable physical thinks; indeed, we indicate that the complex behavior of many physical systems is realistically demonstrated via the fractional calculus modelling. Finally, we report our numerical findings to verify the theoretical analysis.
Highlights
There are two main approaches in the classical mechanics to get the equations of motion for a dynamical system: Newtonian and Lagrangian
We investigate the dynamical behavior of the fractional Euler-Lagrange equations (FELEs) of motion for the coupled oscillator expressed by Equations (17)–(18) considering different values of the fractional order q
This paper studied the concept of the fractional calculus (FC) to evaluate the equations of motion for a coupled oscillator
Summary
There are two main approaches in the classical mechanics to get the equations of motion for a dynamical system: Newtonian and Lagrangian. The second approach was invented by Joseph Louis Lagrange, a French Mathematician. This approach is considered as a useful technique to find the equations of motion for many kinds of physical processes [1]. The fractional calculus (FC) is a branch of mathematical analysis, which deals with the non-integer integral and derivative operators. Over the past few decades, the classical mechanics has been extended by using the new aspects of the FC. In Riewe [10], the non-conservative Lagrangian systems were studied by Riewe using the concept of the FC. In Laskin [12], he developed the fractional quantum mechanics
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