Noether Theorem for Fractional Singular Systems

: Noether theorems for two fractional singular systems are discussed. One system involves mixed integer and Caputo fractional derivatives, and the other involves only Caputo fractional derivatives. Firstly, the fractional primary constraints and the fractional con‐ strained Hamilton equations are given. Then, the fractional Noether theorems of the two fractional singular systems are established, includ‐ ing the fractional Noether identities, the fractional Noether quasi-identities and the fractional conserved quantities. Finally, the results ob‐ tained are illustrated by two examples.


Introduction
Given a Lagrangian L = L ( tqq̇), q = ( q 1 q 2 q n ) , q̇= ( q̇1 q̇2 q̇n ) , we define where the matrix [ H ij ] is called a Hessian matrix. If det [ H ij ] ¹ 0, then the Lagrangian is called regular. If det [ H ij ] = 0, then the Lagrangian is called singular. For example, the Parra s Lagrangian [1] L = 1 2 m ( q̇2 1 + q̇2 2 + l 2 q̇2 3 + 2lq̇1q̇3 cos q 3 + 2lq̇2q̇3 sin q 3 ) + V ( q 1 q 2 q 3 ) the Deriglazov s Lagrangian [2] L = q 2 2 q̇2 1 + q 2 the Cawley s Lagrangian [3] where l, m and μ are constants and V 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][6][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][6][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][11][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 D (D (I ) = 0), 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][15][16][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 (dI/dt = 0). 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][24][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.

Preliminaries
We give the definitions of the Riemann-Liouville and the Caputo fractional derivatives as follows. Given a function f (t) and two constants α and β that satisfy n -1 ≤ αβ < n, where n is an integer, the Riemann-Liouville fractional derivative and the Caputo fractional derivative have the forms [31] RL here, α and β represent the orders of the fractional derivatives. When αβ ® 1, the fractional derivative operators reduce to the classical integer derivative operators, namely, Throughout this paper, we assume that 0 < αβ < 1.
For the Lagrangian L M = L M ( tq M q̇M C and 0 < α < 1, we define the corresponding generalized momenta and the Hamiltonian as In this case, we define the elements of the Hessian matrix as if rank [ H Mij ] = R, 0 ≤ R < n, then the fractional primary constraints with the mixed derivatives have the forms [30] ϕ Ma ( tq Mj p Mj p α where a = 12n -R, 0 ≤ R < n, j = 12n. The fractional constrained Hamilton equations with the mixed derivatives have the forms [30] where 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.
, and 0 < α < 1, we define the corresponding generalized momenta and the Hamiltonian as We assume that the Lagrangian In this case, the fractional primary constraints with the Caputo fractional derivatives have the forms [30] ϕ The fractional constrained Hamilton equations with the Caputo fractional derivatives have the forms [30] C where 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.

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 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 whose expansions are where ξ M0 , ξ Mi , η Mi and η α Mi are called the infinitesimal generators with the mixed derivatives, θ M is a small parameter, and i = 12n.
We denote the change of the fractional Hamilton action as DI M , namely, DI M = IˉM -I M . If we consider only the linear part of θ M , then we have That fractional Hamilton action remains invariant implies DI M = 0, therefore, from Eq. (21), we have 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 C is called a fractional conserved quantity if and only if dC dt = 0 holds. then a fractional conserved quantity with the mixed derivatives exists for the fractional constrained Hamiltonian system (Eq. (14)), as follows: Proof It is obtained from Eqs. (14), (23) that d dt The proof is completed.
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 ξ M0 , ξ Mi , η Mi , η α Mi and a gauge function G M 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)) 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 G M = 0, Theorem 2 reduces to Theorem 1.
An example is presented to illustrate the results and methods above.

Example 1 For the Lagrangian
find its conserved quantity. For this Lagrangian L M , there exist two fractional primary constraints [30] In addition, all the Lagrange multipliers can be obtained [30] : The fractional constrained Hamilton equations can also be established [30] : The fractional Noether-quasi identity (Eq. (24)) gives Through computation, we can verify that is a solution to Eq. (30). Finally, Theorem 2 gives the fractional conserved quantity ù û ú ú ú ú = const (32)

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 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 where ξ C0 , ξ Ci and η Ci are called the infinitesimal generators with the Caputo fractional derivatives, θ C is a small parameter, and i = 12n. We denote the change of the fractional Hamilton action (Eq. (33)) as DI C ; namely, DI C = IˉC -I C . If we consider only the linear part of θ C , then we have That fractional Hamilton action (Eq. (33)) remains invariant implies DI C = 0; therefore, from Eq. (36), we have 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 ξ C0 , ξ Ci , η Ci and η α Ci 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: Proof It is obtained from Eqs. (17), (37) and (38) that d dt The proof is completed.
Let DI C = IˉC -I C = -∫ t 1 t 2 d dt ( DG C ) dt, where DG C = θ C G C and G C = G C ( tq Cj p Cj ) is a gauge function with the Caputo fractional derivatives; then, from Eq. (37), we obtain 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 Noetherquasi 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 ξ C0 , ξ Ci , η Ci , and a gauge function G C satisfy the fractional Noetherquasi identity (Eq. (39)), then a fractional conserved quantity with the Caputo fractional derivatives exists for the fractional constrained Hamiltonian system (Eq. (17))