Formulation of Euler-Lagrange and Hamilton equations involving fractional operators with regular kernel

Antonio Coronel-Escamilla,José Francisco Gómez-Aguilar,Dumitru Baleanu,Victor Hugo Olivares-Peregrino,Ricardo Fabricio Escobar-Jiménez,Arturo Abundez-Pliego

doi:10.1186/s13662-016-1001-5

Abstract

This paper presents alternative representations to traditional calculus of the Euler-Lagrangian equations, in the alternative representations these equations contain fractional operators. In this work, we consider two problems, the Lagrangian of a Pais-Uhlenbeck oscillator and the Hamiltonian of a two-electric pendulum model where the fractional operators have a regular kernel. The Euler-Lagrange formalism was used to obtain the dynamic model based on the Caputo-Fabrizio operator and the new fractional operator based on the Mittag-Leffler function. The simulations showed the effectiveness of these two representations for different values of γ.

Highlights

Fractional calculus (FC) has become an alternative mathematical method to describe models with non-local behavior
Atangana and Baleanu proposed two fractional operators with non-singular and non-local kernel, these novel operators preserve the benefits of the Caputo-Fabrizio operator [ – ]. This manuscript is focused on the fractional Euler-Lagrange equation of the Pais-Uhlenbeck oscillator (P-U) oscillator and the Hamiltonian of a two-electric pendulum model via the Caputo-Fabrizio operator and the new fractional operator based on the Mittag-Leffler function
Based on concepts in the Caputo-Fabrizio and Atangana-Baleanu-Caputo sense, a derivation of the special solution was achieved via an iterative approach and using an iterative methodology via the CrankNicholson scheme

Summary

Introduction

Fractional calculus (FC) has become an alternative mathematical method to describe models with non-local behavior. In [ ], using Liouville-Caputo derivatives the Euler-Lagrange equations corresponding an oscillator were stated as a series formulation; in [ ] the fractional simple pendulum was studied using a fractional space representation. This manuscript is focused on the fractional Euler-Lagrange equation of the P-U oscillator and the Hamiltonian of a two-electric pendulum model via the Caputo-Fabrizio operator and the new fractional operator based on the Mittag-Leffler function. The Atangana-Baleanu operator with fractional order in the Liouville-Caputo sense is given as

Results

Conclusion