Open Access
 Issue Wuhan Univ. J. Nat. Sci. Volume 28, Number 3, June 2023 207 - 216 https://doi.org/10.1051/wujns/2023283207 13 July 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## 0 Introduction

Given a Lagrangian , , , we define

(1)

where the matrix is called a Hessian matrix. If , then the Lagrangian is called regular. If , then the Lagrangian is called singular. For example, the Parra's Lagrangian[1]

(2)

the Deriglazov's Lagrangian[2]

(3)

the Cawley's Lagrangian[3]

(4)

and the Mittelstaedt's Lagrangian[4]

(5)

where , and are constants and represents potential energy, they are all singular. In fact, singular systems have a close relationship with the condensed matter theories, the gauge field theories, the quantum field theories of anyons, the particle physics and so forth[5-7]. Regarding the singular systems, Dirac[8] was the first one to study their canonical equation. Then, singular systems were also investigated in several classical mechanics textbooks[5-7], including both the canonical equation and the Noether theorem.

The Noether theorem was introduced by the German female mathematician Noether [9]. The Noether symmetry method is one of the methods used to solve the differential equations of motion. There are many results on the Noether theorem [10-12]. Fractional calculus has been popular recently. The fractional derivatives that are used most often are the Riemann-Liouville, Caputo and Riesz fractional derivatives. In 2007, Frederico and Torres [13] initiated the study of the fractional Noether theorem. Based on a bilinear operator (), they defined the fractional conserved quantity. Using this definition, fractional Noether theorems and their applications were discussed for several mechanics systems, such as the Lagrangian system[14-17], the Birkhoffian system [18,19], the Hamiltonian system [20], and the multidimensional Lagrangian system [21]. Two years later, with the idea of the definition of the classical conserved quantity, Atanacković et al [22] introduced another definition of the fractional conserved quantity (). They held the point that this definition is more reasonable than the former definition. Then, fractional Noether theorems of the different mechanics systems, such as the Birkhoffian system [23-25], the Hamiltonian system [26,27] and the nonconservative system [28,29] were obtained.

At present, two fractional singular systems, one concerning the mixed integer and Caputo fractional derivatives and the other concerning the Caputo fractional derivatives, have been established, including the fractional primary constraints and the fractional constrained Hamilton equations [30]. The next task is to find the solutions to them. Therefore, in this paper, we intend to make use of the Noether symmetry method to complete this study.

## 1 Preliminaries

We give the definitions of the Riemann-Liouville and the Caputo fractional derivatives as follows. Given a function and two constants and that satisfy , where is an integer, the Riemann-Liouville fractional derivative and the Caputo fractional derivative have the forms [31]

(6)

(7)

(8)

(9)

here, and represent the orders of the fractional derivatives. When , the fractional derivative operators reduce to the classical integer derivative operators, namely,

(10)

Throughout this paper, we assume that .

For the Lagrangian , where , , , are the generalized coordinates, are the generalized velocities, are the Caputo derivatives of , , , , and , we define the corresponding generalized momenta and the Hamiltonian as

(11)

Specifically, we assume that can always be described by a function that depends on the elements of , , and , namely, , .

In this case, we define the elements of the Hessian matrix as

(12)

if , , then the fractional primary constraints with the mixed derivatives have the forms [30]

(13)

where , , . The fractional constrained Hamilton equations with the mixed derivatives have the forms [30]

(14)

where , , , , are the Lagrange multipliers, , , and .

However, when the Lagrange multipliers cannot be solved, Eq. (14) is invalid, and the fractional constrained Hamilton equations with the mixed derivatives have another forms, which have been investigated in Ref. [30]. In this paper, we discuss only the case in which the Lagrange multipliers can be solved.

For the Lagrangian , where , , are the generalized coordinates, are the Caputo derivatives of , , , , and , we define the corresponding generalized momenta and the Hamiltonian as

(15)

We assume that the Lagrangian is singular, i.e., only elements of , , can be solved, where .

In this case, the fractional primary constraints with the Caputo fractional derivatives have the forms [30]

(16)

The fractional constrained Hamilton equations with the Caputo fractional derivatives have the forms [30]

(17)

where , , , are the Lagrange multipliers, , , and .

Similarly, when the Lagrange multipliers cannot be solved, Eq. (17) is invalid, and the fractional constrained Hamilton equations with the Caputo fractional derivatives have another forms, which have been investigated in Ref. [30]. In this paper, we discuss only the case in which the Lagrange multipliers can be solved.

## 2 Fractional Noether Theorem with Mixed Derivatives

Noether symmetry with the mixed derivatives is determined by the Noether symmetric transformations, under which the fractional Hamilton action with the mixed derivatives

(18)

remains invariant. Therefore, if we want to study the Noether theorem, we first need to give the infinitesimal transformations with the mixed derivatives. Then, we discuss the change of the fractional Hamilton action (Eq. (18)) under the given infinitesimal transformations. Finally, the condition which is called the fractional Noether identity with the mixed derivatives is obtained.

Here, the infinitesimal transformations have the forms

(19)

whose expansions are

(20)

where , , and are called the infinitesimal generators with the mixed derivatives, is a small parameter, and .

We denote the change of the fractional Hamilton action as , namely, . If we consider only the linear part of , then we have

(21)

where

That fractional Hamilton action remains invariant implies , therefore, from Eq. (21), we have

(22)

Equation (22) is called the fractional Noether identity with the mixed derivatives for the fractional constrained Hamiltonian system (Eq. (14)). The infinitesimal transformations in this case are called the Noether symmetric transformations, which determine the Noether symmetry.

In this paper, we adopt Atanacković's definition of the fractional conserved quantity. We review it first.

Definition 1 [22] A quantity is called a fractional conserved quantity if and only if holds.

Theorem 1   If the infinitesimal generators , , and satisfy the fractional Noether identity (Eq. (22)), then a fractional conserved quantity with the mixed derivatives exists for the fractional constrained Hamiltonian system (Eq. (14)), as follows:

(23)

Proof   It is obtained from Eqs. (14), (23) that

where

The proof is completed.

Furthermore, if the fractional Hamilton action (Eq. (18)) does not remain invariant under the infinitesimal transformations with the mixed derivatives, for instance, if , where and is called a gauge function with the mixed derivatives, then from Eq. (21), we obtain

(24)

Equation (24) is called the fractional Noether-quasi identity with the mixed derivatives for the fractional constrained Hamiltonian system (Eq. (14)). The infinitesimal transformations in this case are called the Noether-quasi symmetric transformations with the mixed derivatives, which determine the Noether-quasi symmetry with the mixed derivatives. Then, a fractional conserved quantity can also be obtained from the Noether-quasi symmetry.

Theorem 2   If the infinitesimal generators , , , and a gauge function satisfy the fractional Noether-quasi identity (Eq. (24)), then a fractional conserved quantity with the mixed derivatives exists for the fractional constrained Hamiltonian system (Eq. (14))

(25)

Proof   The intended result can be obtained from Eqs. (14), (24) and (25).

Remark 1   The Noether-quasi symmetry with the mixed derivatives is more general than the Noether symmetry with the mixed derivatives. In fact, by setting , Theorem 2 reduces to Theorem 1.

An example is presented to illustrate the results and methods above.

Example 1 For the Lagrangian

(26)

find its conserved quantity.

For this Lagrangian , there exist two fractional primary constraints [30]

(27)

In addition, all the Lagrange multipliers can be obtained [30]:

(28)

The fractional constrained Hamilton equations can also be established [30]:

(29)

The fractional Noether-quasi identity (Eq. (24)) gives

(30)

Through computation, we can verify that

(31)

is a solution to Eq. (30). Finally, Theorem 2 gives the fractional conserved quantity

(32)

## 3 Fractional Noether Theorem with only Caputo Fractional Derivatives

Noether symmetry with the Caputo fractional derivative is determined by the Noether symmetric transformations under which the fractional Hamilton action with the Caputo fractional derivatives

(33)

remains invariant. Similarly, if we want to study the Noether theorem, we first need to give the infinitesimal transformations with the Caputo fractional derivative; then, we discuss the change of the fractional Hamilton action (Eq. (33)) under the given infinitesimal transformations. Finally, the condition called the fractional Noether identity with the Caputo fractional derivatives is obtained.

Here, the infinitesimal transformations have the forms

(34)

whose expansions are

(35)

where , and are called the infinitesimal generators with the Caputo fractional derivatives, is a small parameter, and .

We denote the change of the fractional Hamilton action (Eq. (33)) as ; namely, . If we consider only the linear part of , then we have

(36)

where

That fractional Hamilton action (Eq. (33)) remains invariant implies ; therefore, from Eq. (36), we have

(37)

Equation (37) is called the fractional Noether identity with the Caputo fractional derivatives for the fractional constrained Hamiltonian system (Eq. (17)). The infinitesimal transformations in this case are called the Noether symmetric transformations with the Caputo fractional derivatives, which determine the Noether symmetry.

Theorem 3   If the infinitesimal generators , , and satisfy the fractional Noether identity (Eq. (37)), then a fractional conserved quantity with the Caputo fractional derivatives exists for the fractional constrained Hamiltonian system (Eq. (17)) as follows:

(38)

Proof   It is obtained from Eqs. (17), (37) and (38) that

where

The proof is completed.

Let , where and is a gauge function with the Caputo fractional derivatives; then, from Eq. (37), we obtain

(39)

Equation (39) is called the fractional Noether-quasi identity with the Caputo fractional derivatives for the fractional constrained Hamiltonian system (Eq. (17)). The infinitesimal transformations in this case are called Noether-quasi symmetric transformations, which determine the Noether-quasi symmetry. Then, a fractional conserved quantity with the Caputo fractional derivatives can be obtained.

Theorem 4   If the infinitesimal generators , , , and a gauge function satisfy the fractional Noether-quasi identity (Eq. (39)), then a fractional conserved quantity with the Caputo fractional derivatives exists for the fractional constrained Hamiltonian system (Eq. (17))

(40)

Proof   The intended result can be obtained from Eqs. (17), (39) and (40).

Remark 2   Based on the Caputo fractional derivatives, the Noether-quasi symmetry is more general than the Noether symmetry. In fact, by setting , Theorem 4 reduces to Theorem 3.

Remark 3   Based on the Caputo fractional derivatives, if let , then the fractional primary constraint (Eq. (16)), the fractional constrained Hamilton equation (Eq. (17)) and the Noether theorem (Theorem 3) reduce to the corresponding classical integer-order cases, which are consistent with the results in Ref. [6].

An example is presented to illustrate the results and methods above.

Example 2 For the Lagrangian

(41)

find its conserved quantity.

For this Lagrangian , there exist two fractional primary constraints [30]:

(42)

The fractional constrained Hamilton equations can also be established [30]:

(43)

The fractional Noether-quasi identity (Eq. (39)) gives

(44)

Through computation, we can verify that

(45)

is a solution to Eq. (44). Finally, Theorem 4 gives the fractional conserved quantity

(46)

## 4 Conclusion

Noether theorems for the singular systems with the mixed derivatives and with only Caputo fractional derivatives are studied for the first time. Theorems 1-4 are all new work. Besides, the constrained Hamiltonian system on time scales is another topic deserved to be done.

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