Abstract and Applied Analysis

Volume 2014 (2014), Article ID 290694, 4 pages

http://dx.doi.org/10.1155/2014/290694

Research Article

Fractional Killing-Yano Tensors and Killing Vectors Using the Caputo Derivative in Some One- and Two-Dimensional Curved Space

Ehab Malkawi¹ and D. Baleanu2,3,4

¹Department of Physics, United Arab Emirates University, 15551 Al Ain, UAE

²Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia

³Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, 06530, Balgat, Ankara, Turkey

⁴Institute of Space Sciences, Magurele 76900, Bucharest, Romania

Received 30 January 2014; Accepted 25 February 2014; Published 24 March 2014

Academic Editor: Xiao-Jun Yang

Copyright © 2014 Ehab Malkawi and D. Baleanu. This 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.

Abstract.

The classical free Lagrangian admitting a constant of motion, in one- and two-dimensional space, is generalized using the Caputo derivative of fractional calculus. The corresponding metric is obtained and the fractional Christoffel symbols, Killing vectors, and Killing-Yano tensors are derived. Some exact solutions of these quantities are reported.

1. Introduction

The tool of the fractional calculus started to be successfully applied in many fields 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 field of application (see, e.g., [13, 14] and the references therein). Several definitions of the fractional differentiation and integration exist in the literature. The most commonly used are the Riemann-Liouville and the Caputo derivatives. The Riemann-Liouville derivative of a constant is not zero while Caputo's derivative of a constant is zero. This property makes the Caputo definition more suitable in all problems involving the fractional differential geometry [15, 16]. The Caputo differential operator of fractional calculus is defined as [1–8]

where is the Gamma function and . In this work, we consider the case , . For the power function , , the Caputo fractional derivative satisfies

The 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 differential 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.

The Caputo partial differential operator of fractional order is defined as

Again in this work we consider the case , , and we drop the term in the notation.

2. The Main Results

In the following, we present the Killing vectors and Killing-Yano tensors corresponding to some curved spaces with some physical significance.

2.1. One-Dimensional Case

Consider the one-dimensional free Lagrangian, admitting a constant of motion; that is, momentum [28]

The Lagrangian can be rewritten as

where . The fractional Lagrangian of order is given by

where we consider the Caputo fractional derivative.

We generalize the Christoffel symbols in the fractional case, of order , as

where the partial derivatives of order are defined in the fractional case.

We notice that because the metric is constant, all the Christoffel symbols vanish,

2.1.1. Fractional Killing Vectors and Killing-Yano Tensors

The Killing vectors can be calculated from the generalized equations, namely,

where is the fractional covariant derivative defined as

Because all the Christoffel symbols vanish, it is easy to show that

For , a solution of the above equations is , , where is a constant. While for , we have the general solution

where , , , , are constants.

The fractional Killing-Yano antisymmetric tensor can be calculated using the condition

where is the fractional covariant derivative of the Killing-Yano tensor defined as

We find that

for all values of , , . A solution is and , where is a constant and for . While for , that is, , we have the general solution

where , are constants.

2.2. Two-Dimensional Case

Below we consider the classical free Lagrangian, in two dimensions, admitting a constant of motion; that is, angular momentum [28]

The fractional Lagrangian is given by

where is given by

The inverse matrix of the metric is

We generalize the Christoffel symbols in the fractional case, of order , as

One can show that

for , while

2.2.1. Fractional Killing Vectors

The Killing vectors can be calculated from the generalized equations

where is the fractional covariant derivative defined as

It is easy to show that

A solution for and can be easily found for any fractional order , that is, , namely,

where , , , , are constants. The solution to is not easy to find for . However, for , that is, , the equations simplify because

In this case a general solution is obtained as

where , are constants.

2.2.2. Fractional Killing-Yano Tensors

The fractional antisymmetric Killing-Yano tensors can be derived using the condition that

where is the fractional covariant derivative of the Killing-Yano tensor defined as

For the fractional order , it is difficult to find an analytic solution. However, for the order , the Christoffel symbols vanish; we find that

for all values of , , . A solution is that and , , are a linear combination of , where , namely,

where , , , , , are constants.

3. Conclusion

In this work, we investigate the existence of fractional Killing vectors and Killing-Yano tensors for the geometry induced by fractionalizing the classical free Lagrangian admitting a constant of motion. We discuss the cases of one-dimensional and two-dimensional curved space. We use the Caputo definition of the fractional derivative to calculate the fractional Christoffel symbols and consequently we provide explicit solution to the fractional Killing vectors and Killing-Yano tensors.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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