Abstract
We consider the principle of least action in the context of fractional calculus. Namely, we derive the fractional Euler–Lagrange equation and the general equation of motion with the composition of the left and right fractional derivatives defined on infinite intervals. In addition, we construct an explicit representation of solutions to a model fractional oscillator equation containing the left and right Gerasimov–Caputo fractional derivatives with origins at plus and minus infinity. We derive a representation for the composition of the left and right derivatives with origins at plus and minus infinity in terms of the Riesz potential, and introduce special functions with which we give solutions to the model fractional oscillator equation with a complex coefficient. This approach can be useful for describing dissipative dynamical systems with the property of heredity.
Highlights
The basic framework for Lagrangian and Hamiltonian mechanics is the principle of least action.In problems of mechanics, the principle of stationary action is most commonly used, and is the most important among all extreme principles
The following question arises: is it possible to use a fractional Euler–Lagrange equation if the time variable falls outside the interval within which the action functional is considered? Any value can be chosen as a time interval, including a small physical time. In this case, studying any physical process in this interval is believed to be incorrect. This ambiguity in a sense can be eliminated by using the Gerasimov–Caputo fractional derivatives in the Euler–Lagrange equation, and we offer this in the present article
The rest of the paper is structured as follows: Section 2 contains the definitions of fractional integrals and derivatives; in Section 3, we derive the Euler–Lagrange equation and the equation of motion with fractional derivatives defined on infinite intervals; and, in Section 4, we construct solutions of a model one-dimensional equation for the fractional oscillator; Section 5 presents the main findings of this work
Summary
The basic framework for Lagrangian and Hamiltonian mechanics is the principle of least action. Mathematics 2020, 8, 2122 in very exotic differential equations of fractional order, with composition of the Riemann–Liouville or Caputo operators with different origins These equations and their possible application to modeling dynamic processes can be found in [12,13,14,15,16,17]. In this case, studying any physical process in this interval is believed to be incorrect This ambiguity in a sense can be eliminated by using the Gerasimov–Caputo fractional derivatives (fractional operators defined on an infinite interval [18,19]) in the Euler–Lagrange equation, and we offer this in the present article. The rest of the paper is structured as follows: Section 2 contains the definitions of fractional integrals and derivatives; in Section 3, we derive the Euler–Lagrange equation and the equation of motion with fractional derivatives defined on infinite intervals; and, in Section 4, we construct solutions of a model one-dimensional equation for the fractional oscillator; Section 5 presents the main findings of this work
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