1. Introduction

Fractional calculus is widely used in science and engineering [1–3] especially in physics and mechanics [4–12]. Researchers explore higher-order dynamical systems using methods like Dirac’s technique [13–15]. They use fractional Noether’s theorem and derivatives to study systems with higher-order derivatives [16–18]. Podolsky’s electrodynamics with second-order fractional derivatives is an example of fractional calculus [19]. In a recent study, a technique for systems with higher-order derivatives was introduced, deriving equations from the Degasperis-Procesi equation [20] as an example of second-order order. In another paper, researchers proposed a method for generating fields with fractional derivatives [21,22]. Other approaches calculate conserved variables using fractional Noether’s theorem [23], and Hamiltonian formalism with fractional derivatives is applied similarly to conventional mechanics [24]. Hamiltonian formalism has also been extended to classical fields with fractional derivatives, and its applicability has been demonstrated by solving two discrete problems and one continuous problem. The outcomes of these problems were consistent with Agrawal’s formalism. The work in [25] presented a simple numerical strategy to address the problems using the Riemann-Liouville fractional derivative. Two issues—time-invariant and time-varying—were solved using numerical solutions. The findings demonstrated that the solutions converged as the segment size increased in the time domain. The Riemann-Liouville fractional derivative is a comprehensive extension of ordinary calculus that encompasses most other fractional derivative definitions. It adheres to mathematical principles within fractional calculus. When used with Podolsky’s generalized electrodynamics, it provides the initial conditions with practical and mathematically sound derivatives. Caputo’s derivatives impose stricter regularity criteria and only apply to differentiable functions, while Riemann-Liouville fractional derivatives can extend to features without a first derivative. Researchers generally prefer the Riemann-Liouville definition over others, such as the Caputo or Atangana-Baleanu fractional derivatives. This method can be applied to Podolsky’s generalized electrodynamics model, which involves continuous systems with second-order fractional derivatives. The main contributions of this study are as follows: We noticed that there was perfect agreement between the Lagrangian formulation and the classical equations of motion that were obtained in our investigation. The advantage of using our formalism lies in the fact that the resulting Hamiltonian is linear in Pα, a characteristic that simplifies the analysis of the system and facilitates the application of fractional Hamiltonian mechanics. This linearity is particularly beneficial when dealing with systems governed by fractional derivatives, as it preserves the structure of the equations of motion while accommodating the non-local effects inherent in fractional calculus.

Fractional calculus has increasingly gained attention in various fields of science and engineering due to its ability to model non-locality and memory effects, which are not adequately captured by classical calculus methods. In electrodynamics, fractional derivatives, particularly the Riemann-Liouville formulation, offer a powerful tool for exploring complex interactions within systems characterized by non-local behavior and long-range interactions. For example, recent studies have successfully applied fractional calculus to address complex problems in fields such as fluid dynamics and electrodynamics [26,27].

In the context of fractional-order systems, there have been notable advances in both theoretical and practical aspects, such as the formulation of new models and the numerical methods used to solve them. For instance, a comparative study on fractional-order models for supercapacitors highlighted the effectiveness of fractional derivatives in capturing the behavior of time-varying systems [28]. Moreover, fractional electrodynamics has proven useful in extending classical theories to accommodate higher-order derivatives, which play a critical role in modern electrodynamics and quantum mechanical systems [29].

Podolsky’s electrodynamics, combined with fractional calculus, provides a promising framework for further exploring the dynamic behavior of electromagnetic fields, particularly under the influence of gravitational effects and other complex interactions [30,31]. This research seeks to extend the application of Riemann-Liouville fractional derivatives to second-order Lagrangian systems, further bridging the gap between classical and modern electrodynamics. The findings from recent studies have reinforced the importance of developing models that capture the intricacies of fractional systems, especially those that involve non-locality and memory [32,33]. In addition to these advancements, researchers have also explored the application of fractional calculus in nonlinear dynamical systems, such as the Hyperbolic Lane–Emden System [34], which extends the classical Lane–Emden equation to describe time-dependent phenomena in astrophysics and plasma physics. Similarly, the Coupled Lane–Emden–Klein–Gordon–Fock system with central symmetry [35] has been studied to understand the symmetries and conservation laws in systems combining Lane–Emden and Klein–Gordon–Fock equations, which are crucial in relativistic quantum mechanics. Furthermore, the Reductions and exact solutions of the (2+1)-dimensional breaking soliton equation [36] demonstrate how conservation laws can be used to derive exact solutions for nonlinear partial differential equations, providing insights into the behavior of solitons in higher-dimensional systems. These studies highlight the versatility of fractional calculus and its ability to address complex nonlinear phenomena across various scientific domains.

We investigated the second-order Lagrangian of systems with fractional derivatives to test our formalism and gain a better understanding. This conclusion was found to be in perfect agreement with the Hamiltonian model. These experiments were intended to pave the way for the eventual development of fractional relativistic special relativity. The use of the Riemann-Liouville function in the generic fractional Lagrangian technique provides a more adaptable model than classical modeling. This property can be used to explain the relationship between Dirac theory in ordinary Podolsky theory and higher-order field theories, typical fractional formulations, and restricted systems. Consequently, we believe that the Podolsky theory, in general, and the solution technique outlined in this study are innovative and encompass information that is markedly different from that contained in comparable standard classical equations. The RL fractional derivative, also known as the Riemann-Liouville fractional derivative, is popular in fractional calculus due to its simplicity and connection to the theoretical foundations. It offers a more thorough depiction of system behavior and depicts non-locality in fractional systems, such as fractional electrodynamics. RL fractional derivatives take into account both the past and present, in contrast to Caputo fractional derivatives, which permit non-locality modeling and long-range interactions. Taking into account the benefits and drawbacks of each operator, the fractional derivative operator selected will rely on the particular situation at hand as well as the level of modeling accuracy that is required. The goal of this work is to extend Dirac’s theory of restricted systems to higher-order field theories and provide a thorough examination of fractional Podolsky theory. The goal is to clarify the interactions between these two frameworks and investigate their ramifications in a more in-depth and thorough way.

This study presents a novel approach to addressing challenges posed by fractional derivatives in second-order Lagrangian systems, particularly in electrodynamics, by extending Riemann-Liouville fractional derivatives. It offers new insights into non-local behavior and memory effects, which have been overlooked in previous research, bridging fractional calculus with classical field theory. The integration of second-order fractional derivatives with classical electrodynamics overcomes singular Lagrangian limitations and extends Podolsky’s electrodynamics, broadening its applicability and laying the foundation for future research in fractional special relativity. This contribution demonstrates the advantages of fractional calculus in capturing complex real-world phenomena.

The remainder of this manuscript is organized as follows: In Section 2, we provide a brief overview of the definitions of fractional derivatives. Section 3 focuses on Podolsky’s fractional special relativity equation. In Section 4, we discuss second-order singular Lagrangian fractional constraints. Section 5 explores various applications of fractional calculus and suggests potential directions for future research. Section 6 presents a comparison of our results with those from other studies, highlighting both similarities and differences. Finally, Section 7 concludes the paper with some final remarks.

2. Overview of properties and definitions for Riemann-Liouville fractional derivative

In this section of our study, we will provide a brief overview of the fundamental properties and definitions that are used throughout this work. Specifically, we will be discussing the Riemann-Liouville derivative, which is also referred to as the fractional derivative. Its definition, as given in [37], is presented below:

(1)

The right Riemann- Liouville fractional derivative

(2)

where Γ denotes the Gamma function, and is the order of the derivative such that . If is an integer, these derivatives are defined in the usual sense, i.e.,

The Riemann-Liouville fractional derivative is a generalization of the ordinary derivative to non-integer orders. Here are some properties of the Riemann-Liouville fractional derivative:

1. Fractional Linearity: The fractional derivative, denoted by , follows the property of fractional linearity. For any real numbers and and functions and , we have:

1. Commutativity: The fractional derivative commutes with respect to the order of differentiation. In other words, for any two real numbers and , we have:

1. Fractional Integration: The Riemann-Liouville fractional derivative is related to fractional integration. The fractional integral operator of a function is defined as the inverse of the fractional derivative . That is, for any real number , we have:

1. Homogeneity Property: The fractional derivative has a homogeneity property, which means that for any real number and a constant , we have:

These are some of the key properties associated with the Riemann-Liouville fractional derivative.

3. Podolsky’s fractional special relativity equation

When the electric charge is assumed to be a globally conserved scalar, charge density transforms as the zero component of a four-vector. This 4-vector represents the current density where . Global charge conservation implies local charge conservation. The continuity equation for electric charge is obtained.

According to the assertion, i can have values of 1, 2, and 3 when μ is equal to zero. That holds true for all Lorentz referential as well. To make the equation more widely applicable.

To its Lorentz-transformed shape-variant form, we need to account for the effects of relativity. To put it another way, the equation must be expressed in a fashion that complies with the rules of physics in every inertial frame of reference, independent of the frames relative motion. Here a mathematical technique known as the Lorentz transformation aids in the conversion between several reference frames that are moving in relation to one another at a constant velocity. It enables us to take into consideration the impacts of relativistic phenomena like length contraction and time dilation that may have an impact on how physical systems behave. The charge density can then be expressed in terms of J after the equation has been generalized to its Lorentz-transformed shape-variant form. This is thus because the charge density in the rest frame of the charge distribution is represented by J, a scalar function. We may then use J to compute the charge density in any frame of reference, independent of its relative speed, by formulating the equation in a way that is consistent with the laws of physics as they apply in all inertial frames of reference.

(3)

is the zero component of a four-vector if is a second-order tensor. It is not difficult to demonstrate that the electric field may be represented by the second-order tensor component shown below. if i=1,2,3. Because the right side of (3) is the zero component of a 4-vector, we obtain by the following second-order tensor components:

(4)

where, ,

Take as an example:

Similarly, if we repeat the processes for Equation (5), we get:

(5)

Using the method of mathematical induction with the aid of Equation (23), it can be concluded that Equation (22) implies the following statement.

(6)

The term “metric versus variant tensor” refers to the special relativity tensor, which in our usage contains the following elements:

The phrase “metric versus variant tensor” refers to a tensor in special relativity that is denoted as and has specific components. It is used to distinguish between two types of tensors in relativity: metric tensors and variant tensors. The components of this tensor can be described in a different manner to help understand its characteristics, such as symmetry, positive definiteness, and nondegeneracy.

Consequently, all sides of (6) change into the zero component of a 4-vector. As a result of generalizing Lorentz’s laws to include all 4-vectors and making them form invariant Lorentz transformations (6), we get:

(7)

Taking the fractional derivative of Eq. (6) now yields:

(8)

Equation (8) is said to be antisymmetric. As a consequence of the previous statement, Equation (7) is zero. This is the most direct solution we will use. Moving on, we are asked to choose the last three components from the equation . To accomplish this, we can rewrite the equation as:

Here, we have used the notation to represent the partial derivative with respect to the th coordinate. Now, we can select the last three components by replacing and with the desired coordinates:

These equations represent the last three components of the electromagnetic field strength tensor, which are often referred to as the “magnetic field” components.

Take as an example ,

Similarly, if we use the same techniques to solve equation , we get:

The magnetic induction field is represented by . One of the equations in Podolsky’s pair is the left (A4) equation, which is the relativistic version of the left (A4) equation (see Appendix in S1 File). Use generalization to find the missing equation (A2). If i, j, and k are an even permutation (odd), the Levi-Civita symbol ijk = +1 (-1) and the 1, 2, and 3 cancel out if any two indices are equal.

(9)

where

The standard equations for integer order are generated if the above equation has an if statement of . These are listed in the following order:

The fourth-order totally antisymmetric tensor, represented by the symbol , is a mathematical tool that can transform equations. When its determinant value is evaluated, the result is +1. Using its components (which is equivalent to ), the tensor can transform Equation (8) and simplify mathematical problems.

(10)

After establishing Eq. (9) invariance under the Lorentz transformation, we may now explore Einstein’s general theory of relativity. As a result, it is evident that, the covariant metric tensor of special relativity; its covariant tensor is denoted by the formula

4. Second-order singular Lagrangian fractional restrictions

The essential features of the generalized electrodynamics in [38] suggested will be discussed here. On the Lagrangian, it is based.

(11)

where

So, we can write the Lagrangian (11) as:

To recreate the Podolsky Lagrangian density in Riemann-Liouville fractional form, we use the following relations:

The fractional electromagnetic lagrangian density formulation is as follows:

The expression you provided relates the field strength tensor to the potentials and through the covariant derivative operator . The indices μ and ν represent spacetime indices, while and represent internal indices associated with the gauge symmetry group. The constant and has dimensions of length raised to the power of .

When , the Lagrangian generates a gauge theory, which is a linear field theory with gauge symmetry. This theory reduces to the familiar Maxwell theory, which describes the electromagnetic field, in the limit where the fields are weak and the electric charge is small. In this limit, the field strength tensor reduces to the familiar expression , which relates the field strength to the vector potential .

In summary, the expression you provided relates the field strength tensor to the potentials and through the covariant derivative operator . When , the Lagrangian generates a gauge theory, which reduces to Maxwell theory in the weak-field limit.

These are the motion-related Euler-Lagrange equations.

(12)

If it goes to 1, then Eq. (12) becomes:

In which According to the potentials, the aforementioned equations read

The Lagrangian has the following expression:

where represents the magnetic field and is the electric field, given by .

In summary, the above equation defines the Lagrangian in terms of the magnetic and electric fields, as well as the vector potential and . An alternative expression for the Lagrangian is

The motion equations are formulated as follows:

The energy-momentum tensor contains the following formula according to Poisson bracket formalism:

(13)

The energy-momentum tensor can be constructed simple to demonstrate that it satisfies the conservation equation, which is given by or ( in the presence of external sources represented by the current . However, it should be noted that the energy-momentum tensor does have a trace component, which means that the sum of its diagonal elements is not necessarily zero.

The absence of conformal invariance in the theory is intimately linked to the behavior of the potentials (x). These potentials do not exhibit behavior that is entirely consistent with that of massless fields, as indicated by Equation (13) and the related comments. Equation (12) confirms the existence of this deviation.

(14)

In general, the tensor , defined as the energy-momentum tensor in the theory of relativity, may not be positive definite. However, in the specific case of electrostatics, with the fields and , it can be shown that is positive. Assuming that approaches zero faster than 1/r at infinity, we can obtain an expression for the total energy , which is given by integrating overall space. This expression can be rewritten as:

(15)

where is a constant. For a point charge the electrostatic potential is given by

(16)

It is important to note that this result makes it straightforward to show that the total energy, as stated in Equation (15), has a finite value and equals The calculation for (r) presented above should also take into account a Yukawa-type potential with a mass parameter of in addition to the standard Coulomb potential.

We note that the Lagrangian is singular with respect to the Hamiltonian formalism.

The Lagrangian density of a continuous system is based on the second-order derivatives of the variables in the dynamical field, represented by . The indices used are Greek (0–3) and Latin (1–3), with the measure for both. To accommodate the Euler-Lagrange equations of motion generated by this Lagrangian, the Podolsky Lagrangian formalism must be extended.

And that the momenta, conjugated, respectively, to and , are

(17)

If we write the aforementioned expression using the definitions of and , we obtain the following:

The primary constraints are and

Applying the definition of π, we can express the accelerations of point with respect to the frame of fractional form as .

(18)

The canonical Hamiltonian is defined as follows:

Using equation above as a basis, we can arrive at the following:

To obtain the canonical Hamiltonian by Dirac’s method, we add an arbitrary linear combination of the principal constraints to the existing Hamiltonian. This result in the equation

Since c = 1, 2 and

We have the consistency conditions

where is an arbitrary constant. This leads to a second constraint.

(19)

It is important to note that these constraints are of the highest caliber and are the only ones that can be verified easily. Finally, the extended Hamiltonian is expressed as:

The Hamiltonian is responsible for the system’s temporal evolution and complete gauge freedom.

The dynamical variables’ equations of motion, given by , are as follows:

(20)

Simply put, equals plus additive arbitrary functions. In addition, gives

(21)

As a result, both and are arbitrary, while we have again obtained Equation (18).

The subject at hand is the study of the Hamiltonian equations of motion. First, we see that the Hamiltonian in question exhibits the property of being linear in , which is a property of the Hamiltonian approach for second-order systems. As a result, the equations of motion for essentially tell us that, from a Hamiltonian perspective, is defined in terms of up to an additive arbitrary function. However, the equations of motion for A(x) reproduce equations , whereas we obtain = C for ). This suggests that both and are arbitrary functions.

In the case of the momenta variables and , the following results are obtained:

(22)

(23)

(24)

Equation (22) is the concept of given by Equation (17), which when combined with (19) yields constraint . The demonstration that the gauge transformations in the theory are generated by the constraint’s equations can be performed using the approach detailed in [39]. The point of origin for the generator is as follows:

The first-class (main) limitations are and , respectively. Two coefficients, and , are presented in a differential equation.

Where

In our case the resulting generator is

, it follows that

Consequently, the theory’s gauge transformations produced by are correct.

Using Hamilton-Jacobi formalism, we can derive the following:

In the case of the total differential equation is:

Since is arbitrary, it is completely equivalent to Equation (21). With respect to , the total differential equation is:

The fractional form of the above equation is the same as the fractional form of the equation obtained from Dirac’s method. and can be determined as follows:

(25)

and

(26)

Finally, for we have:

(27)

Equations (23), (24), and (25) are completely equivalent to equations (22), (23), and (26); as a result, equations (24) and (25) have the secondary limit that is absent from the total differential equations.

5. Fractional calculus applications and research suggestions

Fractional derivatives present two key advantages over classical derivatives: enhanced flexibility and non-local behavior. Their fractional order allows for a more adaptable fit to real-world data compared to classical derivatives. This study is aimed at scientists who are investigating fractional calculus and possible applications in the real world. It also provides recommendations for additional research into the generalized electrodynamic fields of Podolsky.

6. Comparison of results with other studies: Similarities and differences

This study compares our findings with the work of other researchers in the field of fractional electrodynamics. Our results regarding the Riemann-Liouville constraints in second-order Lagrangian models align significantly with research by [40], which highlights the effectiveness of fractional derivatives in addressing challenges related to non-local models. His recent work further explores the impact of these derivatives, demonstrating their value in capturing memory effects and long-range interactions, findings that are consistent with the focus of our study.

The work in [41] provided comparative insights into fractional-order models for capacitors, particularly emphasizing numerical strategies. Their analysis of the convergence of solutions in fractional-order systems parallels our numerical approach in electrodynamics. The robustness and accuracy of their numerical methods, particularly in time-varying problems, reflect similar strengths in our methodology, highlighting the broad applicability of fractional calculus across different domains. Furthermore, our extension of the Hamilton-Jacobi formalism aligns with the findings of [42]. In their work on time-fractional electrodynamics. Their research expands the Hamilton-Jacobi framework by incorporating fractional terms, which resonates with our approach to enhancing the understanding of the interplay between classical and fractional systems. Both studies contribute to a deeper comprehension of how fractional models can bridge gaps in traditional electrodynamics theories.

7. Conclusions

The canonical structure of generalized electrodynamics is established by two mathematical techniques: Dirac’s theory of confined systems and the Second-Order Lagrangian Fractional Model. These methods allow for the investigation of fractional dynamical variables, the construction of Hamiltonian constraints, and the precise deduction of gauge transformations. A thorough analysis of Podolsky’s electrodynamics is also provided, with an emphasis on the extended superposition principle and special relativity. In this study, we used Dirac’s theory to investigate Popolsky’s generalized electrodynamics. We found the Hamiltonian constraints and created a generator for second-order Lagrangian fractions. We focused on a specific system with second-order fractional differential equations to better understand the topic. The novelty of this work lies in the integration of fractional dynamics with established electrodynamic principles, offering fresh insights into the behavior of systems characterized by memory effects and non-local interactions. Our findings reveal that classical results are obtained when the fractional formulation is limited to derivatives of integer orders. This underscores the relevance of our approach, as it not only confirms previous theories but also extends them by providing a framework that accommodates a broader range of dynamical behaviors, thereby advancing the understanding of fractional electrodynamics beyond existing literature. While these mathematical techniques provide valuable insights, they also introduce significant challenges that must be addressed. One of the numerous challenges in the study was the complexity of second-order fractional derivatives, particularly the Riemann-Liouville derivative, which complicates modeling and solution derivation. The non-local nature of fractional calculus makes practical applications even more challenging. Despite improving Podolsky’s electrodynamics, the model’s complexity makes it unsuitable for standard engineering applications, requiring advanced numerical methods and posing the risk of non-physical solutions such as unbounded energy states.

Supporting information