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 
Mathematics
CLC number: O 316
Conserved Quantity for Fractional Constrained Hamiltonian System
^{1}
School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou
215009, Jiangsu, China
^{2}
College of Civil Engineering, Suzhou University of Science and Technology, Suzhou
215009, Jiangsu, China
^{†} To whom correspondence should be addressed. Email: songchuanjingsun@126.com
Received:
8
February
2022
Singular system has great relationship with gauge field theory, condensed matter theory and some other research areas. Based on the mixed integer and RiemannLiouville fractional derivatives, the fractional singular system is studied. Firstly, the fractional constrained Hamilton equation and the fractional inherent constraint are presented. Secondly, Lie symmetry and conserved quantity are analyzed, including determined equation, limited equation, additional limited equation and structural equation. And finally, an example is given to illustrate the methods and results.
Key words: fractional constrained Hamilton equation / inherent constraint / Lie symmetry / conserved quantity
Biography: SONG Chuanjing, female, Ph. D., Associate professor, research direction: mathematical methods in classical mechanics. Email: songchuanjing sun@126.com
Foundation item: Supported by the National Natural Science Foundation of China (12172241, 11802193, 11972241), the Natural Science Foundation of Jiangsu Province (BK20191454) and the "Qinglan Project" of Jiangsu Province
© Wuhan University 2022
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
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 higherorder singular systems, etc. Readers who are interested, please see Refs. [13] for details.
Recently, fractional calculus has gained much attention and has been applied to many fields ^{[410]}. 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^{ [1921]}. Especially, Frederico and Torres ^{[22]} studied the fractional variational problems within mixed integer and RiemannLiouville fractional derivatives and established the following fractional EulerLagrange equation
where , , is the generalized coordinate, is the generalized velocity, means the RiemannLiouville 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 ^{[2352]}. 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.[3339] 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 RiemannLiouville 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 RiemannLiouville fractional integrals are defined as
the left and the right RiemannLiouville fractional derivatives are defined as
the left and the right Caputo fractional derivatives are defined as
The left and the right RiemannLiouville fractional derivatives satisfy the following formulae for integration by parts
where .
Specially, when , we have
2 Fractional Constrained Hamiltonian System
2.1 Fractional EulerLagrange Equation
Hamilton action within mixed integer and RiemannLiouville fractional derivatives is
The isochronous variation
with , and is called Hamilton principle within mixed integer and RiemannLiouville fractional derivatives, where means the isochronous variation.
From Eq. (12), using the integration by parts formula (Eq. (8)), we have
where
Substituting Eqs. (14) and (15) into Eq. (13), considering the independence of , and the arbitrariness of the interval , we get
where . Eq. (16) is called the fractional EulerLagrange equation within mixed integer and RiemannLiouville 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
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
In this paper, we assume that , and , .
When , we have
where , , , , and . From Eq. (18), we get
If , then Eq. (21) is reasonable. When , will not depend on , i.e.,
where and . We write Eq. (22) as
where , . Renumber Eq. (23), we can obtain
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 EulerLagrange equation (Eq. (16)).
Similarly, when , we can get
Therefore, from Eqs. (24) and (25), we have
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
where
Because , we have
From Eq. (26), we have
Making use of the Lagrange multipliers , , , from Eqs. (27)  (30), we get
Equation (31) is called the fractional constrained Hamilton equation (also called fractional constrained canonical equation) within mixed integer and RiemannLiouville 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
Let , , , then we have
It follows from , , , that
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
whose expanding expressions are
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
For Eq. (37), we have
where , .
For Eq. (38), we have
where .
For Eq. (39), we have
where .
For the fractional primary constraint Eq. (26), we have
Lie symmetry requires that the coefficients of are zero in Eqs. (40)  (43), then we obtain
and
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 RiemannLiouville 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 RiemannLiouville 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
Equation (46) is called additional limited equation.
Definition 3 For the fractional constrained Hamiltonian system within mixed integer and RiemannLiouville 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
then the following Lie symmetry conserved quantity exists for the fractional constrained Hamiltonian system (31)
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
From Eq. (49) we have
Therefore, there exist two fractional primary constraints
From Eq. (34), we get
Then Eq. (31) gives the following fractional constrained Hamilton equation
The determined equation (44) gives
The limited equation (45) gives
The additional limited equation (46) gives
Taking calculation, we find that
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
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
5 Conclusion
Lie symmetry and conserved quantity are studied for the fractional constrained Hamiltonian system within mixed integer and RiemannLiouville 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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