Open Access
Issue
Wuhan Univ. J. Nat. Sci.
Volume 27, Number 3, June 2022
Page(s) 201 - 210
DOI https://doi.org/10.1051/wujns/2022273201
Published online 24 August 2022

© Wuhan University 2022

Licence Creative CommonsThis 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

As we all know, there are two ways to describe a mechanical system, the Lagrangian Method and the Hamiltonian Method. The two methods are transformed through Legendre transformation. However, if the system studied is a singular one, there will exist inherent constraints when it is transformed from the Lagrangian to the Hamiltonian, and the system is called a constrained Hamiltonian system (also called constrained canonical system). The constrained Hamiltonian system has great relationship with gauge field theory, which dominates particle physics. Besides, the constrained Hamiltonian systems can also be applied to the theory of condensed matter and some other research areas. Therefore, the study of the constrained Hamiltonian system is a hot topic, and much progress has been made, especially on the aspect of symmetry. For example, the standard constrained canonical systems and their quantization, standard canonical symmetries, symmetries of the path integral, canonical symmetries for higher-order singular systems, etc. Readers who are interested, please see Refs. [1-3] for details.

Recently, fractional calculus has gained much attention and has been applied to many fields [4-10]. Because fractional derivatives are elegant tools to deal with the dissipative forces for the nonconservative systems [11,12], fractional calculus of variations was studied by many scholars, such as Klimek [13], Agrawal [14,15], Muslih et al [16], Rabei et al [17], Malinowska and Torres [18], and so on [19-21]. Especially, Frederico and Torres [22] studied the fractional variational problems within mixed integer and Riemann-Liouville fractional derivatives and established the following fractional Euler-Lagrange equation

(1)

where , , is the generalized coordinate, is the generalized velocity, means the Riemann-Liouville fractional derivative of , , , .

After the differential equations of motion are established, one needs to find the solutions to them. Symmetry methods play the key role in solving differential equations [23-52]. Particularly, Lie symmetry, which means the invariance of the differential equations of motion, is one of the most important methods to help reduce the freedoms of differential equations (see Refs.[33-39] for a review). Therefore, in this paper, we intend to establish the Lagrange and Hamilton equations for the fractional singular Lagrangian , and then study Lie symmetry and conserved quantity for the corresponding dynamic system.

It is noted that Eq. (1) in Ref. [22] is not correct for the Lagrangian . Therefore, we begin with the fractional calculus of variations within mixed integer and Riemann-Liouville fractional derivatives. In this paper, we set .

1 Preliminaries

Several definitions and properties of fractional calculus are listed [10, 14, 21].

Let be a function, , then the left and the right Riemann-Liouville fractional integrals are defined as

(2)

(3)

the left and the right Riemann-Liouville fractional derivatives are defined as

(4)

(5)

the left and the right Caputo fractional derivatives are defined as

(6)

(7)

The left and the right Riemann-Liouville fractional derivatives satisfy the following formulae for integration by parts

(8)

(9)

where .

Specially, when , we have

(10)

2 Fractional Constrained Hamiltonian System

2.1 Fractional Euler-Lagrange Equation

Hamilton action within mixed integer and Riemann-Liouville fractional derivatives is

(11)

The isochronous variation

(12)

with , and is called Hamilton principle within mixed integer and Riemann-Liouville fractional derivatives, where means the isochronous variation.

From Eq. (12), using the integration by parts formula (Eq. (8)), we have

(13)

where

(14)

(15)

Substituting Eqs. (14) and (15) into Eq. (13), considering the independence of , and the arbitrariness of the interval , we get

(16)

where . Eq. (16) is called the fractional Euler-Lagrange equation within mixed integer and Riemann-Liouville fractional derivatives.

Remark 1   Equation (16) is different from Eq. (1), which discards the second terms of Eq. (15).

For the fractional Lagrangian system (Eq. (16)), define the element of the Hessian matrix as

(17)

If , the Hessian matrix is said to be degenerated and the corresponding Lagrangian is called singular. Furthermore, if the fractional Lagrangian system (Eq. (16)) is a singular one, then there exist inherent constraints when it is transformed into a fractional constrained Hamiltonian system.

2.2 Fractional Inherent Constraint

Define the generalized momenta and the Hamiltonian as

(18)

(19)

In this paper, we assume that , and , .

When , we have

(20)

where , , , , and . From Eq. (18), we get

(21)

If , then Eq. (21) is reasonable. When , will not depend on , i.e.,

(22)

where and . We write Eq. (22) as

(23)

where , . Renumber Eq. (23), we can obtain

(24)

Equation (24) is the fractional inherent constraint, and we call it the fractional primary constraint. It is noted that the fractional primary constraint comes from the definition of the generalized momenta (Eq. (18)) rather than the fractional Euler-Lagrange equation (Eq. (16)).

Similarly, when , we can get

(25)

Therefore, from Eqs. (24) and (25), we have

(26)

Equation (26) is the inherent constraint and is called fractional primary constraint.

2.3 Fractional Constrained Hamilton Equation

From Eqs. (18) and (19), we obtain

(27)

where

(28)

Because , we have

(29)

From Eq. (26), we have

(30)

Making use of the Lagrange multipliers , , , from Eqs. (27) - (30), we get

(31)

Equation (31) is called the fractional constrained Hamilton equation (also called fractional constrained canonical equation) within mixed integer and Riemann-Liouville fractional derivatives.

Remark 2   When the element in is omitted, the fractional constrained Hamilton equation (Eq. (31)) will reduce to the classical constrained Hamilton equation in Ref. [2].

2.4 Consistency Condition

In this section, the main purpose is to solve the Lagrange multipliers , , .

Let , , then Poisson bracket is defined as

(32)

Let , , , then we have

(33)

It follows from , , , that

(34)

where , , . Eq. (34) is called the consistency condition of the fractional primary constraint.

3 Lie Symmetry and Conserved Quantity

3.1 Lie Symmetry

Lie symmetry means the invariance of the differential equations of motion (Eq. (31)) under the infinitesimal transformations

(35)

whose expanding expressions are

(36)

where is a small parameter, , , and are called infinitesimal generators of the infinitesimal transformations. In this paper, we only consider the linear part of and neglect its higher order.

Expanding Eq. (31) and denoting

(37)

(38)

(39)

For Eq. (37), we have

(40)

where , .

For Eq. (38), we have

(41)

where .

For Eq. (39), we have

(42)

where .

For the fractional primary constraint Eq. (26), we have

(43)

Lie symmetry requires that the coefficients of are zero in Eqs. (40) - (43), then we obtain

(44)

and

(45)

Equation (44) is called determined equation and Eq. (45) is called limited equation.

Definition 1   For the fractional constrained Hamiltonian system within mixed integer and Riemann-Liouville fractional derivatives (31), if the infinitesimal generators , , and satisfy the determined equation (44), then the corresponding symmetry is called Lie symmetry.

Definition 2   For the fractional constrained Hamiltonian system within mixed integer and Riemann-Liouville fractional derivatives (31), if the infinitesimal generators , , and satisfy both the determined equation (44) and the limited equation (45), then the corresponding symmetry is called weak Lie symmetry.

However, if we consider the deduction process of the fractional constrained Hamilton equation (31), then another condition will be exposed on the infinitesimal generators , , and , and a new Lie symmetry will be defined.

Substituting Eq. (36) into Eq. (30), we get

(46)

Equation (46) is called additional limited equation.

Definition 3   For the fractional constrained Hamiltonian system within mixed integer and Riemann-Liouville fractional derivatives (31), if the infinitesimal generators , , and satisfy the determined equation (44), the limited equation (45) and the additional limited equation (46), then the corresponding symmetry is called strong Lie symmetry.

3.2 Conserved Quantity

Lie symmetry may not lead to a conserved quantity. However, under some conditions, a conserved quantity can be deduced from Lie symmetry. Of course, different conditions lead to different results.

Definition 4   A quantity is called a conserved quantity if and only if .

Theorem 1   For the infinitesimal generators , , and , which meet the determined equation (44), if there exists a gauge function satisfying the following structural equation

(47)

then the following Lie symmetry conserved quantity exists for the fractional constrained Hamiltonian system (31)

(48)

Proof   Using Eqs. (26), (31) and (47), we have

This proof is completed.

Theorem 2   For the infinitesimal generators , , and , which meet the determined equation (44) and the limited equation (45), if there exists a gauge function satisfying the structural equation (47), then a weak Lie symmetry conserved quantity (Eq. (48)) exists for the fractional constrained Hamiltonian system (31).

Theorem 3   For the infinitesimal generators , , and , which meet the determined equation (44), the limited equation (45) and the additional limited equation (46), if there exists a gauge function satisfying the structural equation (47), then a strong Lie symmetry conserved quantity (Eq. (48)) exists for the fractional constrained Hamiltonian system (31).

Remark 3   When the element in is omitted, Theorem 1 - Theorem 3 will reduce to the integer order cases, which are consistent with the results in Ref. [39].

4 An Example

We study Lie symmetry for the following fractional singular Lagrangian system

(49)

From Eq. (49) we have

(50)

(51)

Therefore, there exist two fractional primary constraints

(52)

From Eq. (34), we get

(53)

Then Eq. (31) gives the following fractional constrained Hamilton equation

(54)

The determined equation (44) gives

(55)

The limited equation (45) gives

(56)

The additional limited equation (46) gives

(57)

Taking calculation, we find that

(58)

satisfy the structural equation (47) and they also meet the determined equation (55) under the condition . Therefore, Lie symmetry conserved quantity can be obtained from Theorem 1 as

(59)

Because the solution (58) also meets the limited equation (56) and the additional limited equation (57), Eq. (59) is also a strong Lie symmetry conserved quantity.

Specially, if the element in is omitted, then this example will reduce to the integer order case, in which the strong Lie symmetry conserved quantity has the form

(60)

5 Conclusion

Lie symmetry and conserved quantity are studied for the fractional constrained Hamiltonian system within mixed integer and Riemann-Liouville fractional derivatives here. The study of the fractional singular system just begins, and a lot of work is deserving to be done, such as, other symmetry methods, singular systems with only fractional derivatives, singular systems on time scales, and so on.

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