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Fractional Killing-Yano Tensors and Killing Vectors Using the Caputo Derivative in Some One- and Two-Dimensional Curved Space
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.
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  a generalization of exterior calculus was presented. Besides, the quadratic Lagrangians are introduced by adding surface terms to a free-particle Lagrangian in .
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  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  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.
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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