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

Authors: Ehab Malkawi; D. Baleanu
Journal: Abstract and Applied Analysis
 (2014)

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## 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.


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## Body

## 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]
(1)aDxαf(x)≡{1Γ(n-α)×∫ax‍(x-u)n-α-1dnf(u)dundu,n-1<α<ndndxnf(x),α=n,
where Γ(·) is the Gamma function and x>a. In this work, we consider the case a=0, n-1<α≤n. For the power function xp, p∈R, the Caputo fractional derivative satisfies
(2)Dxαxp={Γ(p+1)Γ(p-α+1)xp-α0,p=0,1,2,…,n-1.
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 orderα is defined as
(3)a∂xαf(x,y)≡{1Γ(n-α)×∫ax‍(x-u)n-α-1∂nf(u,y)∂undu,n-1<α<n∂n∂xnf(x,y).α=n
Again in this work we consider the case a=0, n-1<α≤n, and we drop the term a 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]
(4)L=12x˙2+λ˙2x˙.
The Lagrangian can be rewritten as
(5)L=12giju˙iu˙j,
where gij=[1110]. The fractional Lagrangian of order q is given by
(6)L=12gijDtquiDtquj,
where we consider the Caputo fractional derivative.We generalize the Christoffel symbols in the fractional case, of ordern-1<q<n, as
(7)qΓβμγ=12gαγ(∂μqgαβ+∂βqgαμ-∂γqgβμ),
where the partial derivatives of order q are defined in the fractional case.We notice that because the metric is constant, all the Christoffel symbols vanish,(8)qΓμνγ=0.


#### 2.1.1. Fractional Killing Vectors and Killing-Yano Tensors

The Killing vectors can be calculated from the generalized equations, namely,(9)Vα;βq+Vβ;αq=0,
where Vα;βq is the fractional covariant derivative defined as
(10)Vα;βq=∂βqVα+gαμqΓδβμgδλVλ.
Because all the Christoffel symbols vanish, it is easy to show that
(11)V1;1q=∂1qV1=0,V2;2q=∂2qV2=0,V1;2q+V2;1q=V1,2q+V2,1q=∂2qV1+∂1qV2=0,
For 0<q≤1, a solution of the above equations is V1=-cyq, V2=cxq, where c is a constant. While for q>1, we have the general solution
(12)V1=-cyq+∑k=0n-1‍(akxk+bkyk),V2=cxq+∑k=0n-1‍(ak′xk+bk′yk),
where c, ak, bk, ak′, bk′ are constants.The fractional Killing-Yano antisymmetric tensorqfμν can be calculated using the condition
(13)qfμν;λ+qfλν;μ=0,
where qfμν;λ is the fractional covariant derivative of the Killing-Yano tensor qfμν defined as
(14)qfμν;λ=∂λqfμν-fανqΓλμα-fμαqΓλνα.
We find that
(15)∂λqfμν=0
for all values of λ, ν, μ. A solution is f11=f22=0 and f12=c=-f21, where c is a constant and for 0<q≤1. While for q>1, that is, n≥2, we have the general solution
(16)f12=-f21=∑k=0n-1‍(akxk+bkyk),
where ak, bk 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]
(17)L=12(x˙2+y˙2)+λ˙3(xy˙-yx˙).
The fractional Lagrangian is given by
(18)L=12gijDαqiDαqj,
where gij is given by
(19)gij=[10-y01x-yx0].The inverse matrix of the metric is(20)gij=1x2+y2[x2xy-yxyy2x-yx-1].We generalize the Christoffel symbols in the fractional case, of ordern-1<q<n, as
(21)qΓβμγ=12gαγ(∂μqgαβ+∂βqgαμ-∂γqgβμ).One can show that(22)qΓμμγ=0
for γ,μ=1,2,3, while
(23)qΓ12γ=g3γ2(∂1qg32+∂2qg31),qΓ13γ=g2γ2(∂1qg32-∂2qg31),qΓ23γ=g1γ2(∂2qg13+∂1qg23).


#### 2.2.1. Fractional Killing Vectors

The Killing vectors can be calculated from the generalized equations(24)Vα;βq+Vβ;αq=0,
where Vα;βq is the fractional covariant derivative defined as
(25)Vα;βq=∂βqVα+gαμqΓδβμgδλVλ.
It is easy to show that
(26)V1;1q=∂1qV1=0,V2;2q=∂2qV2=0,V3;3q=∂3qV3=0,V1;2q+V2;1q=V1,2q+V2,1q=∂2qV1+∂1qV2=0,V1;3q+V3;1q=∂3qV1+∂1qV3+g2λVλ∂2qg13=0,V2;3q+V3;2q=∂3qV2+∂2qV3+g1λVλ∂1qg23=0.A solution forV1 and V2 can be easily found for any fractional order q, that is, n-1<q<n, namely,
(27)V1=cyq+∑k=0n-1‍(akxk+bkyk),V2=-cxq+∑k=0n-1‍(ckxk+dkyk),
where c, ak, bk, ck, dk are constants. The solution to V3 is not easy to find for 0<q≤1. However, for n≥2, that is, 1<q, the equations simplify because
(28)∂2qg13=∂1qg23=0.
In this case a general solution is obtained as
(29)V3=∑k=0n-1‍(ak′xk+bk′yk),
where ak′, bk′ are constants.


#### 2.2.2. Fractional Killing-Yano Tensors

The fractional antisymmetric Killing-Yano tensors can be derived using the condition that(30)qfμν;λ+qfλν;μ=0,
where qfμν;λ is the fractional covariant derivative of the Killing-Yano tensor qfμν defined as
(31)qfμν;λ=∂λqfμν-fανqΓλμα-fμαqΓλνα.
For the fractional order 0<q<1, it is difficult to find an analytic solution. However, for the order q>1, the Christoffel symbols vanish; we find that
(32)∂λqfμν=0
for all values of λ, ν, μ. A solution is that f11=f22=f33=0 and f12, f13, f23 are a linear combination of xk, yk where k=0,1,2,…,n-1, namely,
(33)f12=-f21=∑k=0n-1‍(akxk+bkyk),f13=-f31=∑k=0n-1‍(ak′xk+bk′yk),f23=-f32=∑k=0n-1‍(ckxk+dkyk),
where ak, bk, ak′, bk′, ck, dk are constants.


## 2.1. One-Dimensional Case

Consider the one-dimensional free Lagrangian, admitting a constant of motion; that is, momentum [28]
(4)L=12x˙2+λ˙2x˙.
The Lagrangian can be rewritten as
(5)L=12giju˙iu˙j,
where gij=[1110]. The fractional Lagrangian of order q is given by
(6)L=12gijDtquiDtquj,
where we consider the Caputo fractional derivative.We generalize the Christoffel symbols in the fractional case, of ordern-1<q<n, as
(7)qΓβμγ=12gαγ(∂μqgαβ+∂βqgαμ-∂γqgβμ),
where the partial derivatives of order q are defined in the fractional case.We notice that because the metric is constant, all the Christoffel symbols vanish,(8)qΓμνγ=0.


### 2.1.1. Fractional Killing Vectors and Killing-Yano Tensors

The Killing vectors can be calculated from the generalized equations, namely,(9)Vα;βq+Vβ;αq=0,
where Vα;βq is the fractional covariant derivative defined as
(10)Vα;βq=∂βqVα+gαμqΓδβμgδλVλ.
Because all the Christoffel symbols vanish, it is easy to show that
(11)V1;1q=∂1qV1=0,V2;2q=∂2qV2=0,V1;2q+V2;1q=V1,2q+V2,1q=∂2qV1+∂1qV2=0,
For 0<q≤1, a solution of the above equations is V1=-cyq, V2=cxq, where c is a constant. While for q>1, we have the general solution
(12)V1=-cyq+∑k=0n-1‍(akxk+bkyk),V2=cxq+∑k=0n-1‍(ak′xk+bk′yk),
where c, ak, bk, ak′, bk′ are constants.The fractional Killing-Yano antisymmetric tensorqfμν can be calculated using the condition
(13)qfμν;λ+qfλν;μ=0,
where qfμν;λ is the fractional covariant derivative of the Killing-Yano tensor qfμν defined as
(14)qfμν;λ=∂λqfμν-fανqΓλμα-fμαqΓλνα.
We find that
(15)∂λqfμν=0
for all values of λ, ν, μ. A solution is f11=f22=0 and f12=c=-f21, where c is a constant and for 0<q≤1. While for q>1, that is, n≥2, we have the general solution
(16)f12=-f21=∑k=0n-1‍(akxk+bkyk),
where ak, bk are constants.


## 2.1.1. Fractional Killing Vectors and Killing-Yano Tensors

The Killing vectors can be calculated from the generalized equations, namely,(9)Vα;βq+Vβ;αq=0,
where Vα;βq is the fractional covariant derivative defined as
(10)Vα;βq=∂βqVα+gαμqΓδβμgδλVλ.
Because all the Christoffel symbols vanish, it is easy to show that
(11)V1;1q=∂1qV1=0,V2;2q=∂2qV2=0,V1;2q+V2;1q=V1,2q+V2,1q=∂2qV1+∂1qV2=0,
For 0<q≤1, a solution of the above equations is V1=-cyq, V2=cxq, where c is a constant. While for q>1, we have the general solution
(12)V1=-cyq+∑k=0n-1‍(akxk+bkyk),V2=cxq+∑k=0n-1‍(ak′xk+bk′yk),
where c, ak, bk, ak′, bk′ are constants.The fractional Killing-Yano antisymmetric tensorqfμν can be calculated using the condition
(13)qfμν;λ+qfλν;μ=0,
where qfμν;λ is the fractional covariant derivative of the Killing-Yano tensor qfμν defined as
(14)qfμν;λ=∂λqfμν-fανqΓλμα-fμαqΓλνα.
We find that
(15)∂λqfμν=0
for all values of λ, ν, μ. A solution is f11=f22=0 and f12=c=-f21, where c is a constant and for 0<q≤1. While for q>1, that is, n≥2, we have the general solution
(16)f12=-f21=∑k=0n-1‍(akxk+bkyk),
where ak, bk 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]
(17)L=12(x˙2+y˙2)+λ˙3(xy˙-yx˙).
The fractional Lagrangian is given by
(18)L=12gijDαqiDαqj,
where gij is given by
(19)gij=[10-y01x-yx0].The inverse matrix of the metric is(20)gij=1x2+y2[x2xy-yxyy2x-yx-1].We generalize the Christoffel symbols in the fractional case, of ordern-1<q<n, as
(21)qΓβμγ=12gαγ(∂μqgαβ+∂βqgαμ-∂γqgβμ).One can show that(22)qΓμμγ=0
for γ,μ=1,2,3, while
(23)qΓ12γ=g3γ2(∂1qg32+∂2qg31),qΓ13γ=g2γ2(∂1qg32-∂2qg31),qΓ23γ=g1γ2(∂2qg13+∂1qg23).


### 2.2.1. Fractional Killing Vectors

The Killing vectors can be calculated from the generalized equations(24)Vα;βq+Vβ;αq=0,
where Vα;βq is the fractional covariant derivative defined as
(25)Vα;βq=∂βqVα+gαμqΓδβμgδλVλ.
It is easy to show that
(26)V1;1q=∂1qV1=0,V2;2q=∂2qV2=0,V3;3q=∂3qV3=0,V1;2q+V2;1q=V1,2q+V2,1q=∂2qV1+∂1qV2=0,V1;3q+V3;1q=∂3qV1+∂1qV3+g2λVλ∂2qg13=0,V2;3q+V3;2q=∂3qV2+∂2qV3+g1λVλ∂1qg23=0.A solution forV1 and V2 can be easily found for any fractional order q, that is, n-1<q<n, namely,
(27)V1=cyq+∑k=0n-1‍(akxk+bkyk),V2=-cxq+∑k=0n-1‍(ckxk+dkyk),
where c, ak, bk, ck, dk are constants. The solution to V3 is not easy to find for 0<q≤1. However, for n≥2, that is, 1<q, the equations simplify because
(28)∂2qg13=∂1qg23=0.
In this case a general solution is obtained as
(29)V3=∑k=0n-1‍(ak′xk+bk′yk),
where ak′, bk′ are constants.


### 2.2.2. Fractional Killing-Yano Tensors

The fractional antisymmetric Killing-Yano tensors can be derived using the condition that(30)qfμν;λ+qfλν;μ=0,
where qfμν;λ is the fractional covariant derivative of the Killing-Yano tensor qfμν defined as
(31)qfμν;λ=∂λqfμν-fανqΓλμα-fμαqΓλνα.
For the fractional order 0<q<1, it is difficult to find an analytic solution. However, for the order q>1, the Christoffel symbols vanish; we find that
(32)∂λqfμν=0
for all values of λ, ν, μ. A solution is that f11=f22=f33=0 and f12, f13, f23 are a linear combination of xk, yk where k=0,1,2,…,n-1, namely,
(33)f12=-f21=∑k=0n-1‍(akxk+bkyk),f13=-f31=∑k=0n-1‍(ak′xk+bk′yk),f23=-f32=∑k=0n-1‍(ckxk+dkyk),
where ak, bk, ak′, bk′, ck, dk are constants.


## 2.2.1. Fractional Killing Vectors

The Killing vectors can be calculated from the generalized equations(24)Vα;βq+Vβ;αq=0,
where Vα;βq is the fractional covariant derivative defined as
(25)Vα;βq=∂βqVα+gαμqΓδβμgδλVλ.
It is easy to show that
(26)V1;1q=∂1qV1=0,V2;2q=∂2qV2=0,V3;3q=∂3qV3=0,V1;2q+V2;1q=V1,2q+V2,1q=∂2qV1+∂1qV2=0,V1;3q+V3;1q=∂3qV1+∂1qV3+g2λVλ∂2qg13=0,V2;3q+V3;2q=∂3qV2+∂2qV3+g1λVλ∂1qg23=0.A solution forV1 and V2 can be easily found for any fractional order q, that is, n-1<q<n, namely,
(27)V1=cyq+∑k=0n-1‍(akxk+bkyk),V2=-cxq+∑k=0n-1‍(ckxk+dkyk),
where c, ak, bk, ck, dk are constants. The solution to V3 is not easy to find for 0<q≤1. However, for n≥2, that is, 1<q, the equations simplify because
(28)∂2qg13=∂1qg23=0.
In this case a general solution is obtained as
(29)V3=∑k=0n-1‍(ak′xk+bk′yk),
where ak′, bk′ are constants.


## 2.2.2. Fractional Killing-Yano Tensors

The fractional antisymmetric Killing-Yano tensors can be derived using the condition that(30)qfμν;λ+qfλν;μ=0,
where qfμν;λ is the fractional covariant derivative of the Killing-Yano tensor qfμν defined as
(31)qfμν;λ=∂λqfμν-fανqΓλμα-fμαqΓλνα.
For the fractional order 0<q<1, it is difficult to find an analytic solution. However, for the order q>1, the Christoffel symbols vanish; we find that
(32)∂λqfμν=0
for all values of λ, ν, μ. A solution is that f11=f22=f33=0 and f12, f13, f23 are a linear combination of xk, yk where k=0,1,2,…,n-1, namely,
(33)f12=-f21=∑k=0n-1‍(akxk+bkyk),f13=-f31=∑k=0n-1‍(ak′xk+bk′yk),f23=-f32=∑k=0n-1‍(ckxk+dkyk),
where ak, bk, ak′, bk′, ck, dk 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.



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*Source: 290694-2014-03-24.xml*