#### Abstract

This paper focuses on the exploration of fractional Birkhoffian mechanics and its fractional Noether theorems under quasi-fractional dynamics models. The quasi-fractional dynamics models under study are nonconservative dynamics models proposed by El-Nabulsi, including three cases: extended by Riemann–Liouville fractional integral (abbreviated as ERLFI), extended by exponential fractional integral (abbreviated as EEFI), and extended by periodic fractional integral (abbreviated as EPFI). First, the fractional Pfaff–Birkhoff principles based on quasi-fractional dynamics models are proposed, in which the Pfaff action contains the fractional-order derivative terms, and the corresponding fractional Birkhoff’s equations are obtained. Second, the Noether symmetries and conservation laws of the systems are studied. Finally, three concrete examples are given to demonstrate the validity of the results.

#### 1. Introduction

Symmetry theory plays an important role in mathematics, physics, and mechanics, and the study of symmetry properties of dynamic systems has become a very effective method to solve some practical problems. The most important and common symmetries are mainly of two kinds, namely, Noether symmetry and Lie symmetry. Noether’s symmetry theory originated in 1918 and was first put forward by the famous mathematician Emmy Noether [1]. In this method, the relationship between symmetry and conserved quantity was established by using the invariance of Hamilton action under the infinitesimal group transformation of time and generalized coordinates. Candotti [2] and Desloge [3] applied Noether’s theorem to classical mechanics. Djukić [4] established Noether’s theorem for nonconservative systems. Liu [5] generalized Noether’s theorem to nonholonomic mechanical systems. In 1979, Lutzky [6] applied the Lie method [7] of invariance of differential equations under infinitesimal group transformations to differential equations of motion for dynamical systems and started the study of Lie symmetry and conserved quantity of mechanical systems. Ibragimov [8] and Bluman [9] elaborated the role of Lie algebra and Lie group in studying the invariance of differential equations. Zhao [10] extended Lie symmetry theory to nonconservative mechanical systems. Mei [11, 12] systematically studied Noether symmetry, Lie symmetry of constrained mechanical systems, and corresponding conserved quantities. Recently, some new progress has been made in the study of these two symmetries (cf. [13–24] and references therein).

Fractional calculus is an important mathematical tool in science and engineering [25–28]. In recent decades, the research of fractional calculus has developed greatly, and its application fields have expanded to automatic control, quantum mechanics, and mechanical systems [29–35]. Riewe [36, 37] introduced the fractional variational problem for the first time in the study of nonconservative mechanics. In 2005, El-Nabulsi established a dynamical model of nonconservative systems under the framework of fractional calculus [38] based on the definition of Riemann–Liouville fractional integral (ERLFI). El-Nabulsi expanded the idea of dynamics modeling and successively put forward the dynamical models of nonconservative systems, which are extended by exponentially fractional integral (EEFI) and extended by periodic laws fractional integral (EPFI) [39, 40], respectively. The equations obtained from quasi-fractional dynamics models are similar to dynamical equations of classical conservative systems, which contain the generalized fractional external forces corresponding to dissipative forces, but the term with the fractional derivative does not show up. Different from other models, the fractional time integration of quasi-fractional dynamics models only needs one parameter. In this way, it simplifies the calculation of complex fractional calculus and provides a modeling method for nonconservative systems. Therefore, the quasi-fractional dynamics models can be used to study complex dynamical systems more conveniently. Frederico and Torres [41] first presented fractional Noether’s theorems. Since then, studies on fractional Noether symmetry and conservation laws have been extensively developed [42–49]. In addition, Torres and Frederico studied Noether’s theorems of fractional action-like variation problems [50, 51]. In recent years, nonconservative dynamical systems based on quasi-fractional dynamical models have been studied deeply, and the corresponding dynamical equations and Noether conservation laws have been obtained [52–55]. However, most of the previous studies on the variational problems of quasi-fractional dynamics models are confined to Lagrangian framework and Hamiltonian framework.

It is well known that Birkhoffian mechanics is a new stage in the development of Hamiltonian mechanics [56–58]. Under canonical transformation, Hamilton canonical equation remains unchanged, but under general noncanonical transformation, it becomes Birkhoff’s equation. Santilli [57] and Mei [58] both pointed out that Birkhoffian mechanics is the most general possible mechanics, which can be applied to hadron physics, space mechanics, statistical mechanics, biophysics, engineering, and other fields. Zhang and Zhai [59] has proposed the fractional Pfaff–Birkhoff principle and fractional Birkhoff’s equations and proved that the fractional Hamilton principle is the special case of the fractional Pfaff–Birkhoff principle, and the fractional Hamilton equations and the fractional Lagrange equations are the special cases of the fractional Birkhoff’s equations. Zhang and Zhou [52] proposed the quasi-fractional Pfaff-Birkhoff principle and derived corresponding quasi-fractional Birkhoff’s equations which is based on the quasi-fractional model given by [38]. Up to now, some results have been obtained on Noether symmetry of fractional or quasi-fractional Birkhoffian systems, such as [52, 59–66]. However, the results of these quasi-fractional Birkhoffian systems are limited to the Pfaff action containing only integral-order derivative terms. Here, we will further extend fractional Birkhoffian mechanics on the basis of three quasi-fractional dynamical models given in [38–40], where for the Pfaff actions, we consider contain fractional-order derivative terms. The quasi-fractional Lagrangian system and quasi-fractional Hamiltonian system are special cases of the results presented in this paper.

The text is organized as follows. In Section 2, the fractional Pfaff–Birkhoff principles under quasi-fractional dynamics models are presented and Birkhoff’s equations are given, and nonisochronous variational formulae of the Pfaff action are driven. In Section 3, fractional Noether symmetries are well defined and their criteria are established. In Section 4, fractional Noether theorems are proved. For illustrating the application of the methods and results in this text, three examples are given in Section 5. In Section 6, we come to the conclusions.

#### 2. Fractional Birkhoff’s Equations and Variation of Fractional Pfaff Action under Quasi-Fractional Dynamics Models

For an introduction to fractional calculus and its basic theory, please refer to the monographs [27, 28].

##### 2.1. Fractional Birkhoffian System Based on ERLFI

We consider a fractional Birkhoffian system determined by Birkhoff’s variables , whose Birkhoff’s functions are , the Birkhoffian is , is the order of fractional derivative, and .

Under the model of ERLFI, we define the Pfaff action aswhere is the fractional derivative term.

The variational principle,with commutative relation,and boundary conditions,is called the fractional Pfaff–Birkhoff principle based on ERLFI.

According to principle (2), we drive

Equation (5) is the fractional Birkhoff’s equations based on ERLFI.

If , equation (5) becomes Birkhoff’s equations based on ERLFI. If and , equation (5) becomes classical Birkhoff’s equations [58].

Take the infinitesimal transformations:and their first-order extensionswhere is the infinitesimal parameter and and are the generating functions.

Under transformation (6), the Pfaff action (1) is transformed into

And, we have

Let be nonisochronous variation of , which is the main line part of relative to , and we obtain

Sincethen we obtain

By using formula (7), we obtain

Equations (10) and (13) are two mutually equivalent formulas derived from Pfaff action (1).

##### 2.2. Fractional Birkhoffian System Based on EEFI

Under the model of EEFI, we define the Pfaff action as

The fractional Pfaff–Birkhoff principle isunder commutative relation,and boundary conditions,

The fractional Birkhoff’s equations are

If , equation (18) becomes Birkhoff’s equations based on EEFI. If and , equation (18) becomes classical Birkhoff’s equations [58].

According to formula (6), action (14) is transformed intoand we have

So, the nonisochronous variation of action is

Equation (21) can also be written as

By using formula (7), we have

Equations (21) and (23) are two mutually equivalent formulas derived from Pfaff action (14).

##### 2.3. Fractional Birkhoffian System Based on EPFI

Under the model of EPFI, we define the Pfaff action as

The fractional Pfaff–Birkhoff principle isunder commutative relation,and boundary conditions,

The fractional Birkhoff’s equations are

If , equation (28) becomes Birkhoff’s equations based on EPFI. If and , equation (28) becomes the classical Birkhoff’s equations [58].

According to formula (6), action (24) is transformed intoand we have

So, we have

Equation (31) can also be written as

By using formula (7), we have

Equations (31) and (33) are two mutually equivalent formulas derived from Pfaff action (24).

#### 3. Fractional Noether Symmetries under Quasi-Fractional Dynamics Models

Next, we will define the Noether symmetries of the system under three quasi-fractional dynamics models and establish their criteria.

##### 3.1. Fractional Noether Symmetries Based on ERLFI

*Definition 1. *If the Pfaff action (1) satisfies the equalitythen transformation (6) is said to be Noether symmetric for system (5).

According to Definition 1, using formulas (10) and (13), we have the following.

*Criterion 1. *If transformation (6) is Noether symmetric, then the equation,needs to be satisfied.

Equation (35) can be written as *r* equations:

If , equation (36) gives the fractional Noether identity based on ERLFI.

*Criterion 2. *If transformation (7) is Noether symmetric, then the following equations,need to be satisfied.

*Definition 2. *If the Pfaff action (1) satisfies the equalitywhere is the gauge function, then transformation (6) is said to be Noether quasi-symmetric for system (5).

According to Definition 2, using formulas (10) and (13), we have the following.

*Criterion 3. *. If transformation (6) is Noether quasi-symmetric, then the equation,needs to be satisfied.

Equation (39) can be written as *r* equations:

If , equation (40) also gives the fractional Noether identity based on ERLFI.

*Criterion 4. *If transformation (7) is Noether quasi-symmetric, then the following equations,need to be satisfied.

##### 3.2. Fractional Noether Symmetries Based on EEFI

*Definition 3. *If the Pfaff action (14) satisfies the equalitythen transformation (6) is said to be Noether symmetric for system (18).

According to Definition 3, using formulas (21) and (23), we have

*Criterion 5. *If transformation (6) is Noether symmetric, then the equation,needs to be satisfied.

Equation (43) can be written as *r* equations:

If , equation (44) gives the fractional Noether identity based on EEFI.

*Criterion 6. *If transformation (7) is Noether symmetric, then the following equations,need to be satisfied.

*Definition 4. *If the Pfaff action (14) satisfies the equalitywhere is the gauge function, then transformation (6) is said to be Noether quasi-symmetric for system (18).

According to Definition 4, using formulas (21) and (23), we have the following.

*Criterion 7. *If transformation (6) is Noether quasi-symmetric, then the equation,needs to be satisfied.

Equation (47) can be written as *r* equations:

If , equation (48) also gives the fractional Noether identity based on EEFI.

*Criterion 8. *If transformation (7) is Noether quasi-symmetric, then the following equations,need to be satisfied.

##### 3.3. Fractional Noether Symmetries Based on EPFI

*Definition 5. *If the Pfaff action 24 satisfies the equalitythen transformation (6) is said to be Noether symmetric for system (28).

According to Definition 5, using formulas (31) and (33), we have the following.

*Criterion 9. *If transformation (6) is Noether symmetric, then the equation,needs to be satisfied.

Equation (51) can be written as *r* equations:

If , equation (52) gives the fractional Noether identity based on EPFI.

*Criterion 10. *If transformation (7) is Noether symmetric, then the following equations,need to be satisfied.

*Definition 6. *If the Pfaff action (24) satisfies the equalitywhere is the gauge function, then transformation (6) is said to be Noether quasi-symmetric for system (28).

According to Definition 6, using formulas (21) and (23), we have the following.

*Criterion 11. *If transformation (6) is Noether quasi-symmetric, then the equation,needs to be satisfied.

Equation (55) can be written as *r* equations:

If , equation (56) gives the fractional Noether identity based on EPFI.

*Criterion 12. *If transformation (7) is Noether quasi-symmetric, then the following equations,need to be satisfied.

#### 4. Fractional Noether’s Theorems under Quasi-Fractional Dynamics Models

Now, we prove Noether’s theorems for fractional Birkhoffian systems under three quasi-fractional dynamics models.

##### 4.1. Fractional Noether’s Theorems Based on ERLFI

Theorem 1. *If transformation (7) is Noether symmetric of system (5) based on ERLFI, thenare linearly independent conserved quantities.*

*Proof. *From Definition 1, we get , namely,By substituting (5) into the above formula and considering the arbitrariness of the integral interval and the independence of , we obtainSo, Theorem 1 is proved.

Theorem 2. *If transformation (7) is Noether quasi-symmetric of system (5) based on ERLFI, thenare linearly independent conserved quantities.*

*Proof. *Combining Definition 2 and formula (13), using equation (5), and considering the arbitrariness of the integral interval and the independence of , the conclusion is obtained.

##### 4.2. Fractional Noether’s Theorems Based on EEFI

Theorem 3. *If transformation (7) is Noether symmetric of system (18) based on EEFI, thenare linearly independent conserved quantities.*

Theorem 4. *If transformation (7) is Noether quasi-symmetric of system (18) based on EEFI, thenare linearly independent conserved quantities.*

##### 4.3. Fractional Noether’s Theorems Based on EPFI

Theorem 5. *If transformation (7) is Noether symmetric of system (28) based on EPFI, thenare linearly independent conserved quantities.*

Theorem 6. *If transformation (7) is Noether quasi-symmetric of system (28) based on EPFI, thenare linearly independent conserved quantities.*

Obviously, if , then Theorems 1–6 give Noether’s theorems for quasi-fractional Birkhoffian systems. If and , Theorems 1–6 give Noether’s theorems for classical Birkhoffian systems [58].

#### 5. Examples

##### 5.1. Example 1

Consider a fractional Birkhoffian system based on ERLFI. The Pfaff action iswhere the Birkhoffian is , and Birkhoff’s functions are .

From equation (5), Birkhoff’s equations are

According to (40), the Noether identity gives

Let