Research Article
Fractional Killing-Yano Tensors and Killing
Vectors Using the Caputo Derivative in Some One- and
Two-Dimensional Curved Space
Ehab Malkawi
1
and D. Baleanu
2,3,4
1
Department of Physics, United Arab Emirates University, 15551 Al Ain, UAE
2
Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204,
Jeddah 21589, Saudi Arabia
3
Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, 06530, Balgat, Ankara, Turkey
4
Institute of Space Sciences, Magurele 76900, Bucharest, Romania
Correspondence should be addressed to Ehab Malkawi; emalkawi@uaeu.ac.ae
Received 30 January 2014; Accepted 25 February 2014; Published 24 March 2014
Academic Editor: Xiao-Jun Yang
Copyright © 2014 E. Malkawi and D. Baleanu. Tis is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Te classical free Lagrangian admitting a constant of motion, in one- and two-dimensional space, is generalized using the Caputo
derivative of fractional calculus. Te corresponding metric is obtained and the fractional Christofel symbols, Killing vectors, and
Killing-Yano tensors are derived. Some exact solutions of these quantities are reported.
1. Introduction
Te tool of the fractional calculus started to be successfully
applied in many felds of science and engineering (see, e.g., [1–
12] and the references therein). Fractals and its connection to
local fractional vector calculus represents another interesting
feld of application (see, e.g., [13, 14] and the references
therein). Several defnitions of the fractional diferentiation
and integration exist in the literature. Te most commonly
used are the Riemann-Liouville and the Caputo derivatives.
Te Riemann-Liouville derivative of a constant is not zero
while Caputo’s derivative of a constant is zero. Tis property
makes the Caputo defnition more suitable in all problems
involving the fractional diferential geometry [15, 16]. Te
Caputo diferential operator of fractional calculus is defned
as [1–8]
()≡
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
1
Γ(−)
×∫
(−)
−−1
()
, −1<<
() , =,
(1)
where Γ(⋅) is the Gamma function and >. In this work, we
consider the case =0, −1<≤. For the power function
, ∈, the Caputo fractional derivative satisfes
=
{
{
{
{
{
Γ(+1)
Γ(−+1)
−
0, =0,1,2,...,−1.
(2)
Te role played by Killing and Killing-Yano tensors for the
geodesic motion of the particle and the superparticle in a
curved background was a topic subjected to an intense debate
during the last decades [17–26]. In [27] a generalization
of exterior calculus was presented. Besides, the quadratic
Lagrangians are introduced by adding surface terms to a free-
particle Lagrangian in [28].
Motivated by the above mentioned results in diferential
geometry, we discuss in this paper the hidden symmetries
corresponding to the fractional Killing vectors and Killing-
Yano tensors on curved spaces deeply related to physical
systems.
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 290694, 4 pages
http://dx.doi.org/10.1155/2014/290694