Noether and Lie Symmetry for Singular Systems Involving Mixed Derivatives

Abstract

Singular systems play an important role in many fields, and some new fractional operators, which are general, have been proposed recently. Therefore, singular systems on the basis of the mixed derivatives including the integer order derivative and the generalized fractional operators are studied. Firstly, Lagrange equations within mixed derivatives are established, and the primary constraints are presented for the singular systems. Then the constrained Hamilton equations are constructed by introducing the Lagrange multipliers. Thirdly, Noether symmetry, Lie symmetry and the corresponding conserved quantities for the constrained Hamiltonian systems are investigated. And finally, an example is given to illustrate the methods and results.

1. Introduction

Fractional calculus is a hot topic recently. Many results have been obtained in fractional calculus and its applications [1,2,3,4,5,6,7]. Since fractional derivatives were used to deal with dissipative forces for nonconservative systems by Riewe [8,9] in 1996, fractional variational problems became popular. For example, Klimek [10] studied Lagrangian and Hamiltonian fractional sequential mechanics; Muslih and Baleanu [11] established the Hamiltonian formulation of the systems with linear velocities within the Riemann–Liouville fractional derivative; Agrawal [12], investigated the fractional variational calculus in terms of the Riesz fractional derivative; Luo [13] studied the fractional Birkhoffian mechanics in terms of the combined Riemann-Liouville fractional derivative and the combined Caputo fractional derivative; Song and Agrawal [14] presented the Euler-Lagrange equations involving the Caputo fractional derivative for singular systems, and so on [15,16]. Especially, in 2010, Agrawal [17] introduced a new kernel ${\kappa }_{\alpha }\left(t,\tau \right)$ (or ${\kappa }_{\alpha }\left(\tau ,t\right)$), on which the generalized fractional derivatives are defined. Only when the parameter set is specified, and the kernel ${\kappa }_{\alpha }\left(t,\tau \right)$ is equal to ${\left(t-\tau \right)}^{\alpha -1}/\Gamma \left(\alpha \right)$, can the Riemann-Liouville fractional derivative, the Caputo fractional derivative, the Riesz-Riemann-Liouville fractional derivative and the Riesz-Caputo fractional derivative be obtained. Besides, the kernel can also be replaced with other kernels, and the entire theories of classical and fractional variational calculus can be redeveloped. Therefore, the generalized fractional derivatives are more general.

Singular system is another keyword of this paper. Singular system plays an important role in field theory, because many important physical systems in field theory are singular ones, such as the Yang-Mills field, the gravitational field, the electromagnetic field, supersymmetry, superstring, supergravity, relativistic moving particles and so on [18]. Singular system has two forms, one is expressed by a Lagrangian, and the other is expressed by a Hamiltonian. When a singular system is expressed in the form of the Hamiltonian, there exist inherent constraints among the canonical variables, and the corresponding system is called a constrained Hamiltonian system [19,20]. The constrained Hamiltonian system also has many applications, such as in quantum field theories of anyons and theories of condensed matter [19,21,22].

In this paper, we intend to study the fractional calculus of variations for singular systems on the basis of generalized fractional derivatives. After the fractional differential equations of motion are established, the next step is to find solutions to them. The symmetry method in mechanics is one of the most effective methods. The symmetry method mainly contains three kinds of methods, namely, the Noether symmetry method, the Lie symmetry method and the Mei symmetry method [23]. This paper focuses on the Noether symmetry method and the Lie symmetry method. The Noether symmetry method was introduced by a German female mathematician Noether [24]. Noether symmetry is an invariance of the Hamilton action under the infinitesimal transformations of time and coordinates, and can lead to a conserved quantity. Lie symmetry is an invariance of the differential equations under the infinitesimal transformations of time and coordinates. Lie symmetry can also lead to a conserved quantity under certain conditions. Many results have been obtained with Noether symmetry and Lie symmetry, including both integer order calculus and fractional order calculus [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46], we only refer to them briefly here.

2. Preliminaries on the Generalized Operators

Generalized operators were introduced by Agrawal [17] in 2010. He defined the operator $K_M^{\alpha }$ as

$$K_M^{\alpha }f\left(t\right)=m{\displaystyle {\int }_{t_1}^t{\kappa }_{\alpha }\left(t,\tau \right)f\left(\tau \right)}\mathrm{d}\tau +\omega {\displaystyle {\int }_t^{t_2}{\kappa }_{\alpha }\left(\tau ,t\right)f\left(\tau \right)}\mathrm{d}\tau , \alpha >0,$$

(1)

where $t_1<t<t_2$, $M=<t_1,t,t_2,m,\omega >$ is a parameter set, $m$ and $\omega$ are two real numbers, ${\kappa }_{\alpha }\left(t,\tau \right)$ is a kernel which may depend on a parameter $\alpha$. It is easy to verify that the operator $K_M^{\alpha }$ is a linear operator and satisfies the following integration by parts formula,

$${\displaystyle {\int }_{t_1}^{t_2}g\left(t\right)K_M^{\alpha }f\left(t\right)}\mathrm{d}t={\displaystyle {\int }_{t_1}^{t_2}f\left(t\right)K_{M^{*}}^{\alpha }g\left(t\right)}\mathrm{d}t$$

(2)

where $M^{*}=<t_1,t,t_2,\omega ,m>$.

The operators $A_M^{\alpha }$ and $B_M^{\alpha }$ were defined by Agrawal as

$$A_M^{\alpha }f\left(t\right)=D^n{K}_M^{n-\alpha }f\left(t\right), n-1<\alpha <n,$$

(3)

$$B_M^{\alpha }f\left(t\right)=K_M^{n-\alpha }{D}^{n}f\left(t\right), n-1<\alpha <n,$$

(4)

where $D$ means the classical integer order derivative $\mathrm{d}/\mathrm{d}t$, $n$ is an integer. Both operators are also linear and they satisfy the following integration by parts formulae,

$${\displaystyle {\int }_{t_1}^{t_2}g\left(t\right)A_M^{\alpha }f\left(t\right)}\mathrm{d}t={\left(-1\right)}^n{\displaystyle {\int }_{t_1}^{t_2}f\left(t\right)B_{M^{*}}^{\alpha }g\left(t\right)}\mathrm{d}t+{\left.{\displaystyle \sum _{j=0}^{n-1}{\left(-D\right)}^{n-1-j}g\left(t\right)A_M^{\alpha +j-n}f\left(t\right)}\right|}_{t=t_1}^{t=t_2}$$

(5)

$${\displaystyle {\int }_{t_1}^{t_2}g\left(t\right)B_M^{\alpha }f\left(t\right)}\mathrm{d}t={\left(-1\right)}^n{\displaystyle {\int }_{t_1}^{t_2}f\left(t\right)A_{M^{*}}^{\alpha }g\left(t\right)}\mathrm{d}t+{\left.{\displaystyle \sum _{j=0}^{n-1}{\left(-1\right)}^j{A}_{M^{*}}^{\alpha +j-n}g\left(t\right)D^{n-1-j}f\left(t\right)}\right|}_{t=t_1}^{t=t_2}$$

(6)

where $M^{*}=<t_1,t,t_2,\omega ,m>$, $n$ is an integer, $n-1<\alpha <n$.

Specifically, if we let ${\kappa }_{\alpha }\left(t,\tau \right)={\left(t-\tau \right)}^{\alpha -1}/\Gamma \left(\alpha \right)$ and let $M=M_1=<t_1,t,t_2,1,0>$, $M=M_2=<t_1,t,t_2,0,1>$ and $M=M_3=<t_1,t,t_2,1/2,1/2>$, then the operator $A_M^{\alpha }$ reduces to the left Riemann-Liouville, the right Riemann-Liouville and the Riesz-Riemann-Liouville fractional derivative operators, respectively. Similarly, the operator $B_M^{\alpha }$ reduces to the left Caputo, the right Caputo and the Riesz-Caputo fractional derivative operators, respectively.

In this text we set $0<\alpha <1$. We begin with variational problems and the primary constraints.

3. Variational Problems and the Primary Constraints

3.1. The Variational Problem and the Primary Constraint with the Operator $A_M^{\alpha }$

Hamilton action with the operator $A_M^{\alpha }$ is defined as

$$I_A\left[q_A\left(\cdot \right)\right]={\displaystyle {\int }_{t_1}^{t_2}{L}_A\left(t,q_A,{\dot{q}}_A,A_M^{\alpha }{q}_A\right)}\mathrm{d}t$$

(7)

where $q_A=\left(q_{A1},q_{A2},\cdots ,q_{An}\right)$, $q_{Ai}\in C^2\left(\left[t_1,t_2\right];ℝ\right)$, $i=1,2,\cdots ,n$, ${\dot{q}}_A=\left({\dot{q}}_{A1},{\dot{q}}_{A2},\cdots ,{\dot{q}}_{An}\right)$, $A_M^{\alpha }{q}_A=\left(A_M^{\alpha }{q}_{A1},A_M^{\alpha }{q}_{A2},\cdots ,A_M^{\alpha }{q}_{An}\right)$, $0<\alpha <1$ and $L_A\left(\cdot ,\cdot ,\cdot ,\cdot \right)\in C^2\left(\left[t_1,t_2\right]\times {ℝ}^n\times {ℝ}^n\times {ℝ}^n;ℝ\right)$.

Then

$$\delta I_A=0$$

(8)

with

$$q_A\left(t_1\right)=q_{A1}, q_A\left(t_2\right)=q_{A2}, \delta {\dot{q}}_{Ai}=\frac{\mathrm{d}}{\mathrm{d}t}\delta q_{Ai}, \delta A_M^{\alpha }{q}_{Ai}=A_M^{\alpha }\delta q_{Ai}, i=1,2,\cdots ,n$$

(9)

is called the Hamilton principle with the operator $A_M^{\alpha }$, where $q_{A1}=\left(q_{A11},q_{A12},\cdots ,q_{A1n}\right)$, $q_{A2}=\left(q_{A21},q_{A22},\cdots ,q_{A2n}\right)$.

From Equations (5), (8) and (9), we obtain

$$\frac{\partial L_A}{\partial q_{Ai}}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_A}{\partial {\dot{q}}_{Ai}}-B_{M^{*}}^{\alpha }\frac{\partial L_A}{\partial A_M^{\alpha }{q}_{Ai}}+m{\kappa }_{1-\alpha }\left(t_2,t\right)\frac{\partial L_A\left(t_2\right)}{\partial A_M^{\alpha }{q}_{Ai}}-\omega {\kappa }_{1-\alpha }\left(t,t_1\right)\frac{\partial L_A\left(t_1\right)}{\partial A_M^{\alpha }{q}_{Ai}}=0$$

(10)

where $L_A\left(t_1\right)=L_A\left(t_1,q_A\left(t_1\right),{\dot{q}}_A\left(t_1\right),A_M^{\alpha }{q}_A\left(t_1\right)\right)$, $L_A\left(t_2\right)=L_A\left(t_2,q_A\left(t_2\right),{\dot{q}}_A\left(t_2\right),A_M^{\alpha }{q}_A\left(t_2\right)\right)$, $i=1,2,\cdots ,n$. Equation (10) is called the Lagrange equation with the operator $A_M^{\alpha }$.

Define the generalized momenta and the Hamiltonian as

$$p_{Ai}=\frac{\partial L_A\left(t,q_A,{\dot{q}}_A,A_M^{\alpha }{q}_A\right)}{\partial {\dot{q}}_{Ai}}, p_{Ai}^{\alpha }=\frac{\partial L_A\left(t,q_A,{\dot{q}}_A,A_M^{\alpha }{q}_A\right)}{\partial A_M^{\alpha }{q}_{Ai}},$$

(11)

$$H_A=p_{Ai}{\dot{q}}_{Ai}+p_{Ai}^{\alpha }\cdot A_M^{\alpha }{q}_{Ai}-L_A\left(t,q_A,{\dot{q}}_A,A_M^{\alpha }{q}_A\right), i=1,2,\cdots ,n.$$

(12)

In this paper, we assume that $A_M^{\alpha }{q}_{Ai}=u_{Ai}\left(t,q_A,{\dot{q}}_A,p_A^{\alpha }\right)$ (or $A_M^{\alpha }{q}_A=u_A\left(t,q_A,{\dot{q}}_A,p_A^{\alpha }\right)$), where $p_A^{\alpha }=\left(p_{A1}^{\alpha },p_{A2}^{\alpha },\cdots ,p_{An}^{\alpha }\right)$, $u_A=\left(u_{A1},u_{A2},\cdots ,u_{An}\right)$.

Define the elements $H_{Aij}$, $i,j=1,2,\cdots ,n$, of the Hessian matrix $\left[H_{Aij}\right]$ as

$$H_{Aij}=\frac{{\partial }^2{L}_A}{\partial {\dot{q}}_{Ai}\partial {\dot{q}}_{Aj}}, i,j=1,2,\cdots ,n,$$

(13)

then the Lagrangian is called regular if $\det \left[H_{Aij}\right]\ne 0$, and if $\det \left[H_{Aij}\right]=0$, then the Lagrangian $L_A$ is called singular. In this text, we assume that $\det \left[H_{Aij}\right]=0$ and $\mathrm{rank}\left[H_{Aij}\right]=R$, $0\le R<n$. In the sequel, we will discuss two cases, i.e., $1\le R<n$ and $R=0$.

Firstly, when $1\le R<n$, which means that only ${\dot{q}}_{A\sigma }$, $\sigma =1,2,\cdots ,R$, can be determined from Equation (11) while ${\dot{q}}_{A\rho }$, $\rho =R+1,R+2,\cdots ,n$, are random. From Equation (11), we express ${\dot{q}}_{A\sigma }$, $\sigma =1,2,\cdots ,R$, as

$${\dot{q}}_{A\sigma }=f_A^{\sigma }\left(t,q_A,p_A^{\alpha },p_{AE},{\dot{q}}_{AF}\right), (\mathrm{or} {\dot{q}}_{AE}=f_A^E\left(t,q_A,p_A^{\alpha },p_{AE},{\dot{q}}_{AF}\right)),$$

(14)

where $p_{A{\rm E}}=\left(p_{A1},p_{A2},\cdots ,p_{AR}\right)$, ${\dot{q}}_{AE}=\left({\dot{q}}_{A1},{\dot{q}}_{A2},\cdots ,{\dot{q}}_{AR}\right)$, ${\dot{q}}_{AF}=\left({\dot{q}}_{AR+1},{\dot{q}}_{AR+2},\cdots ,{\dot{q}}_{An}\right)$, $f_A^E=\left(f_A^1,f_A^2,\cdots ,f_A^R\right)$, $1\le R<n$.

From Equations (11) and (14), we have

$$p_{Ai}=g_{Ai}\left(t,q_A,f_A^E\left(t,q_A,p_A^{\alpha },p_{AE},{\dot{q}}_{AF}\right),{\dot{q}}_{AF},p_A^{\alpha }\right)=g_{Ai}\left(t,q_A,p_A^{\alpha },p_{AE},{\dot{q}}_{AF}\right), i=1,2,\cdots ,n.$$

(15)

For Equation (15), if $i=1,2,\cdots ,R$, $1\le R<n$, then Equation (15) always holds. If $i=R+1,R+2,\cdots ,n$, $1\le R<n$, then from the assumption $\mathrm{rank}\left[H_{Aij}\right]=R$, $1\le R<n$, we have

$$p_{A\rho }=g_{A\rho }\left(t,q_A,p_{AE},p_A^{\alpha }\right), (\mathrm{or} p_{AF}=g_{AF}\left(t,q_A,p_{AE},p_A^{\alpha }\right)),$$

(16)

where $\rho =R+1,R+2,\cdots ,n$, $p_{AF}=\left(p_{AR+1},p_{AR+2},\cdots ,p_{An}\right)$, $g_{AF}=\left(g_{AR+1},g_{AR+2},\cdots ,g_{An}\right)$, $1\le R<n$. Equation (16) has another form

$${\varphi }_A\left(t,q_A,p_A,p_A^{\alpha }\right)=p_{AF}-g_{AF}\left(t,q_A,p_{AE},p_A^{\alpha }\right)=0,$$

(17)

where ${\varphi }_A=\left({\varphi }_{A1},{\varphi }_{A2},\cdots ,{\varphi }_{An-R}\right)$, $1\le R<n$.

Secondly, when $R=0$, which means that no ${\dot{q}}_{Ai}$, $i=1,2,\cdots ,n$, can be determined from Equation (11). Then from Equation (11) and the assumption $\mathrm{rank}\left[H_{Aij}\right]=R$, $R=0$, we have

$$p_{Ai}=g_{Ai}\left(t,q_A,p_A^{\alpha }\right), (\mathrm{or} p_A=g_A\left(t,q_A,p_A^{\alpha }\right)),$$

(18)

where $i=1,2,\cdots ,n$, $p_A=\left(p_{A1},p_{A2},\cdots ,p_{An}\right)$, $g_A=\left(g_{A1},g_{A2},\cdots ,g_{An}\right)$. Then Equation (18) gives

$${\varphi }_A\left(t,q_A,p_A,p_A^{\alpha }\right)=p_A-g_A\left(t,q_A,p_A^{\alpha }\right)=0,$$

(19)

where ${\varphi }_A=\left({\varphi }_{A1},{\varphi }_{A2},\cdots ,{\varphi }_{An}\right)$.

Incorporating Equations (17) and (19), we get

$${\varphi }_A\left(t,q_A,p_A,p_A^{\alpha }\right)=0,$$

(20)

where ${\varphi }_A=\left({\varphi }_{A1},{\varphi }_{A2},\cdots ,{\varphi }_{An-R}\right)$, $0\le R<n$. Equation (20) is called primary constraint with the operator $A_M^{\alpha }$.

Remark 1.

Let${\kappa }_{\alpha }\left(t,\tau \right)={\left(t-\tau \right)}^{\alpha -1}/\Gamma \left(\alpha \right)$, when$M=M_1$,$M=M_2$and$M=M_3$, Equations (10) and (20) give the Lagrange equations and the primary constraints in terms of the left Riemann-Liouville, the right Riemann-Liouville and the Riesz-Riemann-Liouville fractional derivatives, respectively.

3.2. The Variational Problem and the Primary Constraint with the Operator $B_M^{\alpha }$

Hamilton action with the operator $B_M^{\alpha }$ is

$$I_B\left[q_B\left(\cdot \right)\right]={\displaystyle {\int }_{t_1}^{t_2}{L}_B\left(t,q_B,{\dot{q}}_B,B_M^{\alpha }{q}_B\right)}\mathrm{d}t,$$

(21)

where $q_B=\left(q_{B1},q_{B2},\cdots ,q_{Bn}\right)$, $q_{Bi}\in C^2\left(\left[t_1,t_2\right];ℝ\right)$, $i=1,2,\cdots ,n$, ${\dot{q}}_B=\left({\dot{q}}_{B1},{\dot{q}}_{B2},\cdots ,{\dot{q}}_{Bn}\right)$, $B_M^{\alpha }{q}_B=\left(B_M^{\alpha }{q}_{B1},B_M^{\alpha }{q}_{B2},\cdots ,B_M^{\alpha }{q}_{Bn}\right)$, $L_B\left(\cdot ,\cdot ,\cdot ,\cdot \right)\in C^2\left(\left[t_1,t_2\right]\times {ℝ}^n\times {ℝ}^n\times {ℝ}^n;ℝ\right)$ and $0<\alpha <1$. Then

$$\delta I_B=0$$

(22)

with

$$q_B\left(t_1\right)=q_{B1}, q_B\left(t_2\right)=q_{B2}, \delta {\dot{q}}_{Bi}=\frac{\mathrm{d}}{\mathrm{d}t}\delta q_{Bi}, \delta B_M^{\alpha }{q}_{Bi}=B_M^{\alpha }\delta q_{Bi}, i=1,2,\cdots ,n$$

(23)

is called the Hamilton principle with the operator $B_M^{\alpha }$, where $q_{B1}=\left(q_{B11},q_{B12},\cdots ,q_{B1n}\right)$, $q_{B2}=\left(q_{B21},q_{B22},\cdots ,q_{B2n}\right)$.

From Equations (6), (22) and (23), we obtain

$$\frac{\partial L_B}{\partial q_{Bi}}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_B}{\partial {\dot{q}}_{Bi}}-A_{M^{*}}^{\alpha }\frac{\partial L_B}{\partial B_M^{\alpha }{q}_{Bi}}=0, i=1,2,\cdots ,n.$$

(24)

Equation (24) is called the Lagrange equation with the operator $B_M^{\alpha }$.

Define the generalized momenta and the Hamiltonian as

$$p_{Bi}=\frac{\partial L_B\left(t,q_B,{\dot{q}}_B,B_M^{\alpha }{q}_B\right)}{\partial {\dot{q}}_{Bi}}, p_{Bi}^{\alpha }=\frac{\partial L_B\left(t,q_B,{\dot{q}}_B,B_M^{\alpha }{q}_B\right)}{\partial B_M^{\alpha }{q}_{Bi}},$$

(25)

$$H_B=p_{Bi}{\dot{q}}_{Bi}+p_{Bi}^{\alpha }\cdot B_M^{\alpha }{q}_{Bi}-L_B\left(t,q_B,{\dot{q}}_B,B_M^{\alpha }{q}_B\right), i=1,2,\cdots ,n.$$

(26)

In this paper, we assume that $B_M^{\alpha }{q}_{Bi}=u_{Bi}\left(t,q_B,{\dot{q}}_B,p_B^{\alpha }\right)$ (or $B_M^{\alpha }{q}_B=u_B\left(t,q_B,{\dot{q}}_B,p_B^{\alpha }\right)$), where $p_B^{\alpha }=\left(p_{B1}^{\alpha },p_{B2}^{\alpha },\cdots ,p_{Bn}^{\alpha }\right)$, $u_B=\left(u_{B1},u_{B2},\cdots ,u_{Bn}\right)$.

Define the elements $H_{Bij}$, $i,j=1,2,\cdots ,n$, of the Hessian matrix $\left[H_{Bij}\right]$ as

$$H_{Bij}=\frac{{\partial }^2{L}_B}{\partial {\dot{q}}_{Bi}\partial {\dot{q}}_{Bj}}, i,j=1,2,\cdots ,n,$$

(27)

then the Lagrangian is called regular if $\det \left[H_{Bij}\right]\ne 0$, and if $\det \left[H_{Bij}\right]=0$, then the Lagrangian $L_B$ is called singular. In this text, we assume that $\det \left[H_{Bij}\right]=0$ and $\mathrm{rank}\left[H_{Bij}\right]=R$, $0\le R<n$. In the sequel, we will discuss two cases, i.e., $1\le R<n$ and $R=0$.

Firstly, when $1\le R<n$, which means that only ${\dot{q}}_{B\sigma }$, $\sigma =1,2,\cdots ,R$, can be determined from Equation (25) while ${\dot{q}}_{B\rho }$, $\rho =R+1,R+2,\cdots ,n$, are random. From Equation (25), we express ${\dot{q}}_{B\sigma }$, $\sigma =1,2,\cdots ,R$, $1\le R<n$, as

$${\dot{q}}_{B\sigma }=f_B^{\sigma }\left(t,q_B,p_B^{\alpha },p_{BE},{\dot{q}}_{BF}\right), (\mathrm{or} {\dot{q}}_{BE}=f_B^E\left(t,q_B,p_B^{\alpha },p_{BE},{\dot{q}}_{BF}\right)),$$

(28)

where $p_{B{\rm E}}=\left(p_{B1},p_{B2},\cdots ,p_{BR}\right)$, ${\dot{q}}_{BE}=\left({\dot{q}}_{B1},{\dot{q}}_{B2},\cdots ,{\dot{q}}_{BR}\right)$, ${\dot{q}}_{BF}=\left({\dot{q}}_{BR+1},{\dot{q}}_{BR+2},\cdots ,{\dot{q}}_{Bn}\right)$, $f_B^E=\left(f_B^1,f_B^2,\cdots ,f_B^R\right)$.

From Equations (25) and (28), we have

$$p_{Bi}=g_{Bi}\left(t,q_B,f_B^E\left(t,q_B,p_B^{\alpha },p_{BE},{\dot{q}}_{BF}\right),{\dot{q}}_{BF},p_B^{\alpha }\right)=g_{Bi}\left(t,q_B,p_B^{\alpha },p_{BE},{\dot{q}}_{BF}\right), i=1,2,\cdots ,n.$$

(29)

For Equation (29), if $i=1,2,\cdots ,R$, then Equation (29) always holds. If $i=R+1,R+2,\cdots ,n$, then from the assumption $\mathrm{rank}\left[H_{Bij}\right]=R$, $1\le R<n$, we have

$$p_{B\rho }=g_{B\rho }\left(t,q_B,p_{BE},p_B^{\alpha }\right), (\mathrm{or} p_{BF}=g_{BF}\left(t,q_B,p_{BE},p_B^{\alpha }\right)),$$

(30)

where $\rho =R+1,R+2,\cdots ,n$, $p_{BF}=\left(p_{BR+1},p_{BR+2},\cdots ,p_{Bn}\right)$, $g_{BF}=\left(g_{BR+1},g_{BR+2},\cdots ,g_{Bn}\right)$, $1\le R<n$. Equation (30) has another form

$${\varphi }_B\left(t,q_B,p_B,p_B^{\alpha }\right)=p_{BF}-g_{BF}\left(t,q_B,p_{BE},p_B^{\alpha }\right)=0,$$

(31)

where ${\varphi }_B=\left({\varphi }_{B1},{\varphi }_{B2},\cdots ,{\varphi }_{Bn-R}\right)$, $1\le R<n$.

Secondly, when $R=0$, which means that no ${\dot{q}}_{Bi}$, $i=1,2,\cdots ,n$, can be determined from Equation (25). Then from Equation (25) and the assumption $\mathrm{rank}\left[H_{Bij}\right]=R$, $R=0$, we have

$$p_{Bi}=g_{Bi}\left(t,q_B,p_B^{\alpha }\right), (\mathrm{or} p_B=g_B\left(t,q_B,p_B^{\alpha }\right)),$$

(32)

where $i=1,2,\cdots ,n$, $p_B=\left(p_{B1},p_{B2},\cdots ,p_{Bn}\right)$, $g_B=\left(g_{B1},g_{B2},\cdots ,g_{Bn}\right)$. Then Equation (32) gives

$${\varphi }_B\left(t,q_B,p_B,p_B^{\alpha }\right)=p_B-g_B\left(t,q_B,p_B^{\alpha }\right)=0,$$

(33)

where ${\varphi }_B=\left({\varphi }_{B1},{\varphi }_{B2},\cdots ,{\varphi }_{Bn}\right)$.

Incorporating Equations (31) and (33), we get

$${\varphi }_B\left(t,q_B,p_B,p_B^{\alpha }\right)=0,$$

(34)

where ${\varphi }_B=\left({\varphi }_{B1},{\varphi }_{B2},\cdots ,{\varphi }_{Bn-R}\right)$, $0\le R<n$. Equation (34) is called the primary constraint with the operator $B_M^{\alpha }$.

Remark 2.

Let${\kappa }_{\alpha }\left(t,\tau \right)={\left(t-\tau \right)}^{\alpha -1}/\Gamma \left(\alpha \right)$, when$M=M_1$,$M=M_2$and$M=M_3$, Equations (24) and (34) give the Lagrange equations and the primary constraints in terms of the left Caputo, the right Caputo and the Riesz-Caputo fractional derivatives, respectively.

We intend to transform the singular Lagrangian systems with the mixed derivatives (Equations (10), (20), (24) and (34)) into the constrained Hamiltonian systems in the following section.

4. Constrained Hamiltonian System and Consistency Condition

4.1. Constrained Hamilton Equation with the Operator $A_M^{\alpha }$

From Equations (11) and (12), we have

$$\delta H_A={\dot{q}}_{Ai}\cdot \delta p_{Ai}+\delta p_{Ai}^{\alpha }\cdot A_M^{\alpha }{q}_{Ai}-\frac{\partial L_A}{\partial q_{Ai}}\delta q_{Ai}, i=1,2,\cdots ,n,$$

(35)

where

$$\frac{\partial L_A}{\partial q_{Ai}}={\dot{p}}_{Ai}+B_{M^{*}}^{\alpha }{p}_{Ai}^{\alpha }-m{\kappa }_{1-\alpha }\left(t_2,t\right)p_{Ai}^{\alpha }\left(t_2\right)+\omega {\kappa }_{1-\alpha }\left(t,t_1\right)p_{Ai}^{\alpha }\left(t_1\right)$$

(36)

From the Hamiltonian $H_A=H_A\left(t,q_A,p_A,p_A^{\alpha }\right)$ we have,

$$\delta H_A=\frac{\partial H_A}{\partial q_{Ai}}\cdot \delta q_{Ai}+\frac{\partial H_A}{\partial p_{Ai}}\cdot \delta p_{Ai}+\frac{\partial H_A}{\partial p_{Ai}^{\alpha }}\cdot \delta p_{Ai}^{\alpha }, i=1,2,\cdots ,n.$$

(37)

Besides, taking isochronous variation of Equation (20), we have

$$\delta {\varphi }_{Aa}=\frac{\partial {\varphi }_{Aa}}{\partial q_{Ai}}\cdot \delta q_{Ai}+\frac{\partial {\varphi }_{Aa}}{\partial p_{Ai}}\cdot \delta p_{Ai}+\frac{\partial {\varphi }_{Aa}}{\partial p_{Ai}^{\alpha }}\cdot \delta p_{Ai}^{\alpha }=0, i=1,2,\cdots ,n, a=1,2,\cdots ,n-R, 0\le R<n.$$

(38)

Introducing the Lagrange multipliers ${\lambda }_{Aa}\left(t\right)$, $a=1,2,\cdots ,n-R$, $0\le R<n$, and from Equations (35)–(38), we have

$$\begin{gathered} {\dot{p}}_{Ai}=-B_{M^{*}}^{\alpha }{p}_{Ai}^{\alpha }-\frac{\partial H_A}{\partial q_{Ai}}+m{\kappa }_{1-\alpha }\left(t_2,t\right)p_{Ai}^{\alpha }\left(t_2\right)-\omega {\kappa }_{1-\alpha }\left(t,t_1\right)p_{Ai}^{\alpha }\left(t_1\right)-{\lambda }_{Aa}\frac{\partial {\varphi }_{Aa}}{\partial q_{Ai}}, \\ {\dot{q}}_{Ai}=\frac{\partial H_A}{\partial p_{Ai}}+{\lambda }_{Aa}\frac{\partial {\varphi }_{Aa}}{\partial p_{Ai}}, A_M^{\alpha }{q}_{Ai}=\frac{\partial H_A}{\partial p_{Ai}^{\alpha }}+{\lambda }_{Aa}\frac{\partial {\varphi }_{Aa}}{\partial p_{Ai}^{\alpha }}, i=1,2,\cdots ,n, a=1,2,\cdots ,n-R, 0\le R<n. \end{gathered}$$

(39)

Equation (39) is called the constrained Hamilton equation with the operator $A_M^{\alpha }$.

For simplicity, we introduce $H_{AT}=H_A+{\lambda }_{Aa}{\varphi }_{Aa}$, $a=1,2,\cdots ,n-R$, $0\le R<n$, then Equation (39) can be written as

$$\begin{gathered} {\dot{p}}_{Ai}=-B_{M^{*}}^{\alpha }{p}_{Ai}^{\alpha }-\frac{\partial H_{AT}}{\partial q_{Ai}}+m{\kappa }_{1-\alpha }\left(t_2,t\right)p_{Ai}^{\alpha }\left(t_2\right)-\omega {\kappa }_{1-\alpha }\left(t,t_1\right)p_{Ai}^{\alpha }\left(t_1\right), \\ {\dot{q}}_{Ai}=\frac{\partial H_{AT}}{\partial p_{Ai}}, A_M^{\alpha }{q}_{Ai}=\frac{\partial H_{AT}}{\partial p_{Ai}^{\alpha }}, i=1,2,\cdots ,n, a=1,2,\cdots ,n-R, 0\le R<n. \end{gathered}$$

(40)

Remark 3.

Let${\kappa }_{\alpha }\left(t,\tau \right)={\left(t-\tau \right)}^{\alpha -1}/\Gamma \left(\alpha \right)$, when$M=M_1$,$M=M_2$and$M=M_3$, Equation (39) (or Equation (40)) gives the constrained Hamilton equations in terms of the left Riemann-Liouville, the right Riemann-Liouville and the Riesz-Riemann-Liouville fractional derivatives, respectively.

4.2. Constrained Hamilton Equation with the Operator $B_M^{\alpha }$

From Equations (25) and (26), we have

$$\delta H_B={\dot{q}}_{Bi}\cdot \delta p_{Bi}+\delta p_{Bi}^{\alpha }\cdot B_M^{\alpha }{q}_{Bi}-\frac{\partial L_B}{\partial q_{Bi}}\delta q_{Bi}, i=1,2,\cdots ,n,$$

(41)

where

$$\frac{\partial L_B}{\partial q_{Bi}}={\dot{p}}_{Bi}+A_{M^{*}}^{\alpha }{p}_{Bi}^{\alpha }.$$

(42)

From the Hamiltonian $H_B=H_B\left(t,q_B,p_B,p_B^{\alpha }\right)$ we have,

$$\delta H_B=\frac{\partial H_B}{\partial q_{Bi}}\cdot \delta q_{Bi}+\frac{\partial H_B}{\partial p_{Bi}}\cdot \delta p_{Bi}+\frac{\partial H_B}{\partial p_{Bi}^{\alpha }}\cdot \delta p_{Bi}^{\alpha }, i=1,2,\cdots ,n.$$

(43)

Besides, taking isochronous variation of Equation (34), we have

$$\delta {\varphi }_{Ba}=\frac{\partial {\varphi }_{Ba}}{\partial q_{Bi}}\cdot \delta q_{Bi}+\frac{\partial {\varphi }_{Ba}}{\partial p_{Bi}}\cdot \delta p_{Bi}+\frac{\partial {\varphi }_{Ba}}{\partial p_{Bi}^{\alpha }}\cdot \delta p_{Bi}^{\alpha }=0, i=1,2,\cdots ,n, a=1,2,\cdots ,n-R, 0\le R<n.$$

(44)

Introducing the Lagrange multipliers ${\lambda }_{Ba}\left(t\right)$, $a=1,2,\cdots ,n-R$, $0\le R<n$, and from Equations (41)–(44), we have

$$\begin{gathered} {\dot{p}}_{Bi}=-A_{M^{*}}^{\alpha }{p}_{Bi}^{\alpha }-\frac{\partial H_B}{\partial q_{Bi}}-{\lambda }_{Ba}\frac{\partial {\varphi }_{Ba}}{\partial q_{Bi}}, {\dot{q}}_{Bi}=\frac{\partial H_B}{\partial p_{Bi}}+{\lambda }_{Ba}\frac{\partial {\varphi }_{Ba}}{\partial p_{Bi}}, \\ {B}_M^{\alpha }{q}_{Bi}=\frac{\partial H_B}{\partial p_{Bi}^{\alpha }}+{\lambda }_{Ba}\frac{\partial {\varphi }_{Ba}}{\partial p_{Bi}^{\alpha }}, i=1,2,\cdots ,n, a=1,2,\cdots ,n-R, 0\le R<n. \end{gathered}$$

(45)

Equation (45) is called the constrained Hamilton equation with the operator $B_M^{\alpha }$.

For simplicity, we introduce $H_{BT}=H_B+{\lambda }_{Ba}{\varphi }_{Ba}$, $a=1,2,\cdots ,n-R$, $0\le R<n$, then Equation (45) can be written as

$$\begin{gathered} {\dot{p}}_{Bi}=-A_{M^{*}}^{\alpha }{p}_{Bi}^{\alpha }-\frac{\partial H_{BT}}{\partial q_{Bi}}, {\dot{q}}_{Bi}=\frac{\partial H_{BT}}{\partial p_{Bi}}, B_M^{\alpha }{q}_{Bi}=\frac{\partial H_{BT}}{\partial p_{Bi}^{\alpha }}, \\ i=1,2,\cdots ,n, a=1,2,\cdots ,n-R, 0\le R<n. \end{gathered}$$

(46)

Remark 4.

Let${\kappa }_{\alpha }\left(t,\tau \right)={\left(t-\tau \right)}^{\alpha -1}/\Gamma \left(\alpha \right)$, when$M=M_1$,$M=M_2$and$M=M_3$, Equation (45) (or Equation (46)) gives the constrained Hamilton equations in terms of the left Caputo, the right Caputo and the Riesz-Caputo fractional derivatives, respectively.

4.3. Consistency Conditions with Generalized Operators

Let $F=F\left(t,q,p,p^{\alpha }\right)$, $G=G\left(t,q,p,p^{\alpha }\right)$, we define the Poisson bracket as

$$\left\{F,G\right\}=\frac{\partial F}{\partial q_i}\frac{\partial G}{\partial p_i}-\frac{\partial F}{\partial p_i}\frac{\partial G}{\partial q_i}, i=1,2,\cdots ,n,$$

(47)

where $q=\left(q_1,q_2,\cdots ,q_n\right)$, $p=\left(p_1,p_2,\cdots ,p_n\right)$, $p^{\alpha }=\left(p_1^{\alpha },p_2^{\alpha },\cdots ,p_n^{\alpha }\right)$.

Then using the Poisson bracket and Equation (40), we obtain

$$\begin{gathered} \left\{{\varphi }_{Aa},H_A\right\}+{\lambda }_{Ab}\left\{{\varphi }_{Aa},{\varphi }_{Ab}\right\}+\frac{\partial {\varphi }_{Aa}}{\partial t}+\frac{\partial {\varphi }_{Aa}}{\partial p_{Ai}^{\alpha }}{\dot{p}}_{Ai}^{\alpha }-\frac{\partial {\varphi }_{Aa}}{\partial p_{Ai}}\left[B_{M^{*}}^{\alpha }{p}_{Ai}^{\alpha }-m{\kappa }_{1-\alpha }\left(t_2,t\right)p_{Ai}^{\alpha }\left(t_2\right)\right. \\ \left.+\omega {\kappa }_{1-\alpha }\left(t,t_1\right)p_{Ai}^{\alpha }\left(t_1\right)\right]=0, i=1,2,\cdots ,n, a,b=1,2,\cdots ,n-R, 0\le R<n. \end{gathered}$$

(48)

Equation (48) is called the consistency condition with the operator $A_M^{\alpha }$.

Similarly, the consistency condition with the operator $B_M^{\alpha }$ has the form

$$\begin{gathered} \left\{{\varphi }_{Ba},H_B\right\}+{\lambda }_{Bb}\left\{{\varphi }_{Ba},{\varphi }_{Bb}\right\}+\frac{\partial {\varphi }_{Ba}}{\partial t}+\frac{\partial {\varphi }_{Ba}}{\partial p_{Bi}^{\alpha }}{\dot{p}}_{Bi}^{\alpha }-\frac{\partial {\varphi }_{Ba}}{\partial p_{Bi}}\cdot A_{M^{*}}^{\alpha }{p}_{Bi}^{\alpha }=0, \\ i=1,2,\cdots ,n, a,b=1,2,\cdots ,n-R, 0\le R<n. \end{gathered}$$

(49)

If $\det \left[\left\{{\varphi }_{Aa},{\varphi }_{Ab}\right\}\right]\ne 0$(resp. $\det \left[\left\{{\varphi }_{Ba},{\varphi }_{Bb}\right\}\right]\ne 0$), $a,b=1,2,\cdots ,n-R$, $0\le R<n$, then all the Lagrange multipliers ${\lambda }_{Aa}$ (resp. ${\lambda }_{Ba}$), $a=1,2,\cdots ,n-R$, $0\le R<n$, can be calculated from Equation (48) (resp. Equation (49)).

If $\det \left[\left\{{\varphi }_{Aa},{\varphi }_{Ab}\right\}\right]=0$ (resp. $\det \left[\left\{{\varphi }_{Ba},{\varphi }_{Bb}\right\}\right]=0$), then the Lagrange multipliers ${\lambda }_{Aa}$ (resp. ${\lambda }_{Ba}$), $a=1,2,\cdots ,n-R$, $0\le R<n$, cannot be calculated completely, and then the new constraint, which is called the secondary constraint, will be deduced. Therefore, the secondary constraint arises from the consistency condition of the primary constraint. Similarly, if the consistency condition of the secondary constraint still cannot give all the Lagrange multipliers, then some new secondary constraints will be established. Anyway, no new secondary constraint will be produced after a finite number of steps for a system with finite degrees of freedom.

Remark 5.

Let${\kappa }_{\alpha }\left(t,\tau \right)={\left(t-\tau \right)}^{\alpha -1}/\Gamma \left(\alpha \right)$, when$M=M_1$,$M=M_2$and$M=M_3$, Equations (48) and (49) give the consistency conditions in terms of the left Riemann-Liouville, the left Caputo, the right Riemann-Liouville, the right Caputo, the Riesz-Riemann-Liouville and the Riesz-Caputo fractional derivatives, respectively.

5. Noether Symmetry and Conserved Quantity

Noether symmetry is the invariance of the Hamilton action under the infinitesimal transformations of time and coordinates. We begin with Noether symmetry with the operator $A_M^{\alpha }$.

5.1. Noether Symmetry with the Operator $A_M^{\alpha }$

The Hamilton action with the operator $A_M^{\alpha }$ is

$$I_A={\displaystyle {\int }_{t_1}^{t_2}\left[p_{Ai}{\dot{q}}_{Ai}+p_{Ai}^{\alpha }\cdot A_M^{\alpha }{q}_{Ai}-H_A\left(t,q_A,p_A,p_A^{\alpha }\right)\right]}\mathrm{d}t, i=1,2,\cdots ,n.$$

(50)

The infinitesimal transformations are

$$\begin{gathered} \overline{t}=t+\Delta t, {\overline{q}}_{Ai}\left(\overline{t}\right)=q_{Ai}\left(t\right)+\Delta q_{Ai}, {\overline{p}}_{Ai}\left(\overline{t}\right)=p_{Ai}\left(t\right)+\Delta p_{Ai},{\overline{p}}_{Ai}^{\alpha }\left(\overline{t}\right)=p_{Ai}^{\alpha }\left(t\right)+\Delta p_{Ai}^{\alpha }, \\ (\mathrm{or} \overline{t}=t+\Delta t, {\overline{q}}_A\left(\overline{t}\right)=q_A\left(t\right)+\Delta q_A, {\overline{p}}_A\left(\overline{t}\right)=p_A\left(t\right)+\Delta p_A,{\overline{p}}_A^{\alpha }\left(\overline{t}\right)=p_A^{\alpha }\left(t\right)+\Delta p_A^{\alpha }), \end{gathered}$$

(51)

and the expanded forms are

$$\begin{gathered} \overline{t}=t+{\theta }_A{\xi }_{A0}\left(t,q_A,p_A,p_A^{\alpha }\right)+o\left({\theta }_A\right), \\ {\overline{q}}_{Ai}\left(\overline{t}\right)=q_{Ai}\left(t\right)+{\theta }_A{\xi }_{Ai}\left(t,q_A,p_A,p_A^{\alpha }\right)+o\left({\theta }_A\right), \\ {\overline{p}}_{Ai}\left(\overline{t}\right)=p_{Ai}\left(t\right)+{\theta }_A{\eta }_{Ai}\left(t,q_A,p_A,p_A^{\alpha }\right)+o\left({\theta }_A\right), \\ {\overline{p}}_{Ai}^{\alpha }\left(\overline{t}\right)=p_{Ai}^{\alpha }\left(t\right)+{\theta }_A{\eta }_{Ai}^{\alpha }\left(t,q_A,p_A,p_A^{\alpha }\right)+o\left({\theta }_A\right), \end{gathered}$$

(52)

where ${\theta }_A$ is a small parameter, ${\xi }_{A0}$, ${\xi }_{Ai}$, ${\eta }_{Ai}$ and ${\eta }_{Ai}^{\alpha }$, $i=1,2,\cdots ,n$, are the infinitesimal generators of the infinitesimal transformations, $o\left({\theta }_A\right)$ is the higher order of ${\theta }_A$.

Neglecting the higher order of ${\theta }_A$, we have

$$\begin{array}{c}\Delta I_A={\overline{I}}_A-I_A={\displaystyle {\int }_{{\overline{t}}_1}^{{\overline{t}}_2}\left[{\overline{p}}_{Ai}{\dot{\overline{q}}}_{Ai}+{\overline{p}}_{Ai}^{\alpha }\cdot A_{\overline{M}}^{\alpha }{\overline{q}}_{Ai}-H_A\left(\overline{t},{\overline{q}}_A,{\overline{p}}_A,{\overline{p}}_A^{\alpha }\right)\right]}\mathrm{d}\overline{t}-I_A\hfill \\ ={\theta }_A{\displaystyle {\int }_{t_1}^{t_2}\left[{\lambda }_{Aa}\frac{\partial {\varphi }_{Aa}}{\partial p_{Ai}}{\eta }_{Ai}+{\lambda }_{Aa}\frac{\partial {\varphi }_{Aa}}{\partial p_{Ai}^{\alpha }}{\eta }_{Ai}^{\alpha }-\frac{\partial H_A}{\partial q_{Ai}}{\xi }_{Ai}+\left(p_{Ai}^{\alpha }\frac{\mathrm{d}}{\mathrm{d}t}{A}_M^{\alpha }{q}_{Ai}-\frac{\partial H_A}{\partial t}\right){\xi }_{A0}\right.}\hfill \\ +p_{Ai}^{\alpha }{A}_M^{\alpha }\left({\xi }_{Ai}-{\dot{q}}_{Ai}{\xi }_{A0}\right)+\left(p_{Ai}^{\alpha }{A}_M^{\alpha }{q}_{Ai}-H_A\right){\dot{\xi }}_{A0}+\omega p_{Ai}^{\alpha }{q}_{Ai}\left(t_2\right){\xi }_{A0}\left(t_2\right)\frac{\mathrm{d}}{\mathrm{d}t}{\kappa }_{1-\alpha }\left(t_2,t\right)\hfill \\ \left.-m{p}_{Ai}^{\alpha }{q}_{Ai}\left(t_1\right){\xi }_{A0}\left(t_1\right)\frac{\mathrm{d}}{\mathrm{d}t}{\kappa }_{1-\alpha }\left(t,t_1\right)+p_{Ai}{\dot{\xi }}_{Ai}\right]\mathrm{d}t, \hfill \end{array}$$

(53)

where ${\xi }_{A0}\left(t_1\right)={\xi }_{A0}\left(t_1,q_A\left(t_1\right),p_A\left(t_1\right),p_A^{\alpha }\left(t_1\right)\right)$, ${\xi }_{A0}\left(t_2\right)={\xi }_{A0}\left(t_2,q_A\left(t_2\right),p_A\left(t_2\right),p_A^{\alpha }\left(t_2\right)\right)$ and $\overline{M}=<{\overline{t}}_1,\overline{t},{\overline{t}}_2,m,\omega >$.

Noether symmetry requires that $\Delta I_A=0$, that is,

$$\begin{gathered} {\lambda }_{Aa}\frac{\partial {\varphi }_{Aa}}{\partial p_{Ai}}{\eta }_{Ai}+{\lambda }_{Aa}\frac{\partial {\varphi }_{Aa}}{\partial p_{Ai}^{\alpha }}{\eta }_{Ai}^{\alpha }-\frac{\partial H_A}{\partial q_{Ai}}{\xi }_{Ai}+\left(p_{Ai}^{\alpha }\frac{\mathrm{d}}{\mathrm{d}t}{A}_M^{\alpha }{q}_{Ai}-\frac{\partial H_A}{\partial t}\right){\xi }_{A0}+p_{Ai}{\dot{\xi }}_{Ai} \\ +p_{Ai}^{\alpha }{A}_M^{\alpha }\left({\xi }_{Ai}-{\dot{q}}_{Ai}{\xi }_{A0}\right)+\left(p_{Ai}^{\alpha }{A}_M^{\alpha }{q}_{Ai}-H_A\right){\dot{\xi }}_{A0}+\omega p_{Ai}^{\alpha }{q}_{Ai}\left(t_2\right){\xi }_{A0}\left(t_2\right)\frac{\mathrm{d}}{\mathrm{d}t}{\kappa }_{1-\alpha }\left(t_2,t\right) \\ -m{p}_{Ai}^{\alpha }{q}_{Ai}\left(t_1\right){\xi }_{A0}\left(t_1\right)\frac{\mathrm{d}}{\mathrm{d}t}{\kappa }_{1-\alpha }\left(t,t_1\right)=0. \end{gathered}$$

(54)

Equation (54) is called the Noether identity with the operator $A_M^{\alpha }$.

If we let $\Delta I_A=-{\displaystyle {\int }_{t_1}^{t_2}\frac{\mathrm{d}}{\mathrm{d}t}}\left(\Delta G_A\right)\mathrm{d}t$, where $\Delta G_A={\theta }_A{G}_A$, $G_A=G_A\left(t,q_A,p_A,p_A^{\alpha }\right)$ is called a gauge function with the operator $A_M^{\alpha }$, then we obtain

$$\begin{gathered} {\lambda }_{Aa}\frac{\partial {\varphi }_{Aa}}{\partial p_{Ai}}{\eta }_{Ai}+{\lambda }_{Aa}\frac{\partial {\varphi }_{Aa}}{\partial p_{Ai}^{\alpha }}{\eta }_{Ai}^{\alpha }-\frac{\partial H_A}{\partial q_{Ai}}{\xi }_{Ai}+\left(p_{Ai}^{\alpha }\frac{\mathrm{d}}{\mathrm{d}t}{A}_M^{\alpha }{q}_{Ai}-\frac{\partial H_A}{\partial t}\right){\xi }_{A0}+p_{Ai}{\dot{\xi }}_{Ai} \\ +p_{Ai}^{\alpha }{A}_M^{\alpha }\left({\xi }_{Ai}-{\dot{q}}_{Ai}{\xi }_{A0}\right)+\left(p_{Ai}^{\alpha }{A}_M^{\alpha }{q}_{Ai}-H_A\right){\dot{\xi }}_{A0}+\omega p_{Ai}^{\alpha }{q}_{Ai}\left(t_2\right){\xi }_{A0}\left(t_2\right)\frac{\mathrm{d}}{\mathrm{d}t}{\kappa }_{1-\alpha }\left(t_2,t\right) \\ -m{p}_{Ai}^{\alpha }{q}_{Ai}\left(t_1\right){\xi }_{A0}\left(t_1\right)\frac{\mathrm{d}}{\mathrm{d}t}{\kappa }_{1-\alpha }\left(t,t_1\right)+{\dot{G}}_A=0. \end{gathered}$$

(55)

Equation (55) is called the Noether quasi-identity with the operator $A_M^{\alpha }$.

Noether symmetry leads to a conserved quantity. We first present the definition of the conserved quantity.

Definition 1.

A quantity$C$is called a conserved quantity if and only if$\mathrm{d}C/\mathrm{d}t=0$holds.

Therefore, we have

Theorem 1.

For the constrained Hamiltonian system with the operator$A_M^{\alpha }$(Equation (39)), if the infinitesimal generators${\xi }_{A0}$,${\xi }_{Ai}$,${\eta }_{Ai}$and${\eta }_{Ai}^{\alpha }$satisfy Equation (54), then there exists a conserved quantity

$$\begin{gathered} C_A=p_{Ai}{\xi }_{Ai}+\left(p_{Ai}^{\alpha }{A}_M^{\alpha }{q}_{Ai}-H_A\right){\xi }_{A0}+{\displaystyle {\int }_{t_1}^t\left\{p_{Ai}^{\alpha }{A}_M^{\alpha }\left({\xi }_{Ai}-{\dot{q}}_{Ai}{\xi }_{A0}\right)+\left({\xi }_{Ai}-{\dot{q}}_{Ai}{\xi }_{A0}\right)\right.} \\ \left.\left[B_{M^{*}}^{\alpha }{p}_{Ai}^{\alpha }-m{\kappa }_{1-\alpha }\left(t_2,\tau \right)p_{Ai}^{\alpha }\left(t_2\right)+\omega {\kappa }_{1-\alpha }\left(\tau ,t_1\right)p_{Ai}^{\alpha }\left(t_1\right)\right]\right\}\mathrm{d}\tau +\omega q_{Ai}\left(t_2\right){\xi }_{A0}\left(t_2\right) \\ \cdot {\displaystyle {\int }_{t_1}^t{p}_{Ai}^{\alpha }\left(\tau \right)\frac{\mathrm{d}}{\mathrm{d}\tau }{\kappa }_{1-\alpha }\left(t_2,\tau \right)}\mathrm{d}\tau -m{q}_{Ai}\left(t_1\right){\xi }_{A0}\left(t_1\right){\displaystyle {\int }_{t_1}^t{p}_{Ai}^{\alpha }\left(\tau \right)\frac{\mathrm{d}}{\mathrm{d}\tau }{\kappa }_{1-\alpha }\left(\tau ,t_1\right)}\mathrm{d}\tau =\mathrm{const}. \end{gathered}$$

(56)

Proof of Theorem 1.

From Equations (20), (39) and (54), we have

$$\begin{array}{c}\mathrm{d}{C}_A/\mathrm{d}t={\dot{p}}_{Ai}{\xi }_{Ai}+p_{Ai}{\dot{\xi }}_{Ai}+\left(p_{Ai}^{\alpha }{A}_M^{\alpha }{q}_{Ai}-H_A\right){\dot{\xi }}_{A0}+p_{Ai}^{\alpha }{A}_M^{\alpha }\left({\xi }_{Ai}-{\dot{q}}_{Ai}{\xi }_{A0}\right)\hfill \\ +\left({\dot{p}}_{Ai}^{\alpha }{A}_M^{\alpha }{q}_{Ai}+p_{Ai}^{\alpha }\cdot \frac{\mathrm{d}}{\mathrm{d}t}{A}_M^{\alpha }{q}_{Ai}-\frac{\partial H_A}{\partial t}-\frac{\partial H_A}{\partial q_{Ai}}{\dot{q}}_{Ai}-\frac{\partial H_A}{\partial p_{Ai}}{\dot{p}}_{Ai}-\frac{\partial H_A}{\partial p_{Ai}^{\alpha }}{\dot{p}}_{Ai}^{\alpha }\right){\xi }_{A0}\hfill \\ +\left({\xi }_{Ai}-{\dot{q}}_{Ai}{\xi }_{A0}\right)\left[B_{M^{*}}^{\alpha }{p}_{Ai}^{\alpha }-m{\kappa }_{1-\alpha }\left(t_2,t\right)p_{Ai}^{\alpha }\left(t_2\right)+\omega {\kappa }_{1-\alpha }\left(t,t_1\right)p_{Ai}^{\alpha }\left(t_1\right)\right]\hfill \\ +\omega q_{Ai}\left(t_2\right){\xi }_{A0}\left(t_2\right)p_{Ai}^{\alpha }\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}{\kappa }_{1-\alpha }\left(t_2,t\right)-m{q}_{Ai}\left(t_1\right){\xi }_{A0}\left(t_1\right)p_{Ai}^{\alpha }\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}{\kappa }_{1-\alpha }\left(t,t_1\right)\hfill \\ =\frac{\partial H_A}{\partial q_{Ai}}{\xi }_{Ai}-{\lambda }_{Aa}\frac{\partial {\varphi }_{Aa}}{\partial p_{Ai}}{\eta }_{Ai}+{\dot{p}}_{Ai}{\xi }_{Ai}+{\xi }_{A0}\left({\dot{p}}_{Ai}^{\alpha }\cdot A_M^{\alpha }{q}_{Ai}-\frac{\partial H_A}{\partial q_{Ai}}{\dot{q}}_{Ai}-\frac{\partial H_A}{\partial p_{Ai}}{\dot{p}}_{Ai}-\frac{\partial H_A}{\partial p_{Ai}^{\alpha }}{\dot{p}}_{Ai}^{\alpha }\right)\hfill \\ +\left({\xi }_{Ai}-{\dot{q}}_{Ai}{\xi }_{A0}\right)\left[B_{M^{*}}^{\alpha }{p}_{Ai}^{\alpha }-m{\kappa }_{1-\alpha }\left(t_2,t\right)p_{Ai}^{\alpha }\left(t_2\right)+\omega {\kappa }_{1-\alpha }\left(t,t_1\right)p_{Ai}^{\alpha }\left(t_1\right)\right]-{\lambda }_{Aa}\frac{\partial {\varphi }_{Aa}}{\partial p_{Ai}^{\alpha }}{\eta }_{Ai}^{\alpha }\hfill \\ ={\dot{p}}_{Ai}{\xi }_{A0}{\lambda }_{Aa}\frac{\partial {\varphi }_{Aa}}{\partial p_{Ai}}-{\lambda }_{Aa}\frac{\partial {\varphi }_{Aa}}{\partial q_{Ai}}\left({\xi }_{Ai}-{\dot{q}}_{Ai}{\xi }_{A0}\right)-{\lambda }_{Aa}\frac{\partial {\varphi }_{Aa}}{\partial p_{Ai}}{\eta }_{Ai}-{\lambda }_{Aa}\frac{\partial {\varphi }_{Aa}}{\partial p_{Ai}^{\alpha }}{\eta }_{Ai}^{\alpha }+{\lambda }_{Aa}\frac{\partial {\varphi }_{Aa}}{\partial p_{Ai}^{\alpha }}{\dot{p}}_{Ai}^{\alpha }{\xi }_{A0}\hfill \\ =-{\lambda }_{Aa}\frac{\partial {\varphi }_{Aa}}{\partial q_{Ai}}\cdot \delta q_{Ai}-{\lambda }_{Aa}\frac{\partial {\varphi }_{Aa}}{\partial p_{Ai}}\cdot \delta p_{Ai}-{\lambda }_{Aa}\frac{\partial {\varphi }_{Aa}}{\partial p_{Ai}^{\alpha }}\cdot \delta p_{Ai}^{\alpha }=-{\lambda }_{Aa}\cdot \delta {\varphi }_{Aa}=0.\hfill \end{array}$$

The proof is completed. □

Theorem 2.

For the constrained Hamiltonian system with the operator$A_M^{\alpha }$(Equation (39)), if there exists a gauge function$G_A$such that the infinitesimal generators${\xi }_{A0}$,${\xi }_{Ai}$,${\eta }_{Ai}$and${\eta }_{Ai}^{\alpha }$satisfy Equation (55), then there exists a conserved quantity

$$\begin{gathered} C_{AG}=p_{Ai}{\xi }_{Ai}+\left(p_{Ai}^{\alpha }{A}_M^{\alpha }{q}_{Ai}-H_A\right){\xi }_{A0}+{\displaystyle {\int }_{t_1}^t\left\{p_{Ai}^{\alpha }{A}_M^{\alpha }\left({\xi }_{Ai}-{\dot{q}}_{Ai}{\xi }_{A0}\right)+\left({\xi }_{Ai}-{\dot{q}}_{Ai}{\xi }_{A0}\right)\right.} \\ \left.\left[B_{M^{*}}^{\alpha }{p}_{Ai}^{\alpha }-m{\kappa }_{1-\alpha }\left(t_2,\tau \right)p_{Ai}^{\alpha }\left(t_2\right)+\omega {\kappa }_{1-\alpha }\left(\tau ,t_1\right)p_{Ai}^{\alpha }\left(t_1\right)\right]\right\}\mathrm{d}\tau +\omega q_{Ai}\left(t_2\right){\xi }_{A0}\left(t_2\right) \\ \cdot {\displaystyle {\int }_{t_1}^t{p}_{Ai}^{\alpha }\left(\tau \right)\frac{\mathrm{d}}{\mathrm{d}\tau }{\kappa }_{1-\alpha }\left(t_2,\tau \right)}\mathrm{d}\tau -m{q}_{Ai}\left(t_1\right){\xi }_{A0}\left(t_1\right){\displaystyle {\int }_{t_1}^t{p}_{Ai}^{\alpha }\left(\tau \right)\frac{\mathrm{d}}{\mathrm{d}\tau }{\kappa }_{1-\alpha }\left(\tau ,t_1\right)}\mathrm{d}\tau +G_A=\mathrm{const}. \end{gathered}$$

(57)

Proof of Theorem 2.

From Equations (20), (39) and (55), we have $\mathrm{d}{C}_{AG}/\mathrm{d}t=0$. □

Remark 6.

Let${\kappa }_{\alpha }\left(t,\tau \right)={\left(t-\tau \right)}^{\alpha -1}/\Gamma \left(\alpha \right)$, when$M=M_1$,$M=M_2$and$M=M_3$, Equation (54), Equation (55), Theorem 1 and Theorem 2 give the Noether identities, Noether quasi-identities and conserved quantities in terms of the left Riemann-Liouville, the right Riemann-Liouville and the Riesz-Riemann-Liouville fractional derivatives, respectively.

5.2. Noether Symmetry with the Operator $B_M^{\alpha }$

The Hamilton action with the operator $B_M^{\alpha }$ is

$$I_B={\displaystyle {\int }_{t_1}^{t_2}\left[p_{Bi}{\dot{q}}_{Bi}+p_{Bi}^{\alpha }\cdot B_M^{\alpha }{q}_{Bi}-H_B\left(t,q_B,p_B,p_B^{\alpha }\right)\right]}\mathrm{d}t, i=1,2,\cdots ,n.$$

(58)

The infinitesimal transformations are

$$\begin{gathered} \overline{t}=t+\Delta t, {\overline{q}}_{Bi}\left(\overline{t}\right)=q_{Bi}\left(t\right)+\Delta q_{Bi}, {\overline{p}}_{Bi}\left(\overline{t}\right)=p_{Bi}\left(t\right)+\Delta p_{Bi},{\overline{p}}_{Bi}^{\alpha }\left(\overline{t}\right)=p_{Bi}^{\alpha }\left(t\right)+\Delta p_{Bi}^{\alpha }, \\ (\mathrm{or} \overline{t}=t+\Delta t, {\overline{q}}_B\left(\overline{t}\right)=q_B\left(t\right)+\Delta q_B, {\overline{p}}_B\left(\overline{t}\right)=p_B\left(t\right)+\Delta p_B, {\overline{p}}_B^{\alpha }\left(\overline{t}\right)=p_B^{\alpha }\left(t\right)+\Delta p_B^{\alpha }), \end{gathered}$$

(59)

and the expanded forms are

$$\begin{gathered} \overline{t}=t+{\theta }_B{\xi }_{B0}\left(t,q_B,p_B,p_B^{\alpha }\right)+o\left({\theta }_B\right), {\overline{q}}_{Bi}\left(\overline{t}\right)=q_{Bi}\left(t\right)+{\theta }_B{\xi }_{Bi}\left(t,q_B,p_B,p_B^{\alpha }\right)+o\left({\theta }_B\right), \\ {\overline{p}}_{Bi}\left(\overline{t}\right)=p_{Bi}\left(t\right)+{\theta }_B{\eta }_{Bi}\left(t,q_B,p_B,p_B^{\alpha }\right)+o\left({\theta }_B\right), \\ {\overline{p}}_{Bi}^{\alpha }\left(\overline{t}\right)=p_{Bi}^{\alpha }\left(t\right)+{\theta }_B{\eta }_{Bi}^{\alpha }\left(t,q_B,p_B,p_B^{\alpha }\right)+o\left({\theta }_B\right), \end{gathered}$$

(60)

where ${\theta }_B$ is a small parameter, ${\xi }_{B0}$, ${\xi }_{Bi}$, ${\eta }_{Bi}$ and ${\eta }_{Bi}^{\alpha }$ are the infinitesimal generators of the infinitesimal transformations, $o\left({\theta }_B\right)$ is the higher order of ${\theta }_B$.

Neglecting the higher order of ${\theta }_B$, we have

$$\begin{array}{c}\Delta I_B={\overline{I}}_B-I_B={\displaystyle {\int }_{{\overline{t}}_1}^{{\overline{t}}_2}\left[{\overline{p}}_{Bi}{\dot{\overline{q}}}_{Bi}+{\overline{p}}_{Bi}^{\alpha }\cdot B_{\overline{M}}^{\alpha }{\overline{q}}_{Bi}-H_B\left(\overline{t},{\overline{q}}_B,{\overline{p}}_B,{\overline{p}}_B^{\alpha }\right)\right]}\mathrm{d}\overline{t}-I_B\hfill \\ ={\theta }_B{\displaystyle {\int }_{t_1}^{t_2}\left[{\lambda }_{Ba}\frac{\partial {\varphi }_{Ba}}{\partial p_{Bi}}{\eta }_{Bi}+{\lambda }_{Ba}\frac{\partial {\varphi }_{Ba}}{\partial p_{Bi}^{\alpha }}{\eta }_{Bi}^{\alpha }-\frac{\partial H_B}{\partial q_{Bi}}{\xi }_{Bi}+\left(p_{Bi}^{\alpha }\frac{\mathrm{d}}{\mathrm{d}t}{B}_M^{\alpha }{q}_{Bi}-\frac{\partial H_B}{\partial t}\right){\xi }_{B0}\right.}\hfill \\ +p_{Bi}^{\alpha }{B}_M^{\alpha }\left({\xi }_{Bi}-{\dot{q}}_{Bi}{\xi }_{B0}\right)+\left(p_{Bi}^{\alpha }{B}_M^{\alpha }{q}_{Bi}-H_B\right){\dot{\xi }}_{B0}+\omega p_{Bi}^{\alpha }{\kappa }_{1-\alpha }\left(t_2,t\right){\dot{q}}_{Bi}\left(t_2\right){\xi }_{B0}\left(t_2\right)\hfill \\ \left.-m{p}_{Bi}^{\alpha }{\kappa }_{1-\alpha }\left(t,t_1\right){\dot{q}}_{Bi}\left(t_1\right){\xi }_{B0}\left(t_1\right)+p_{Bi}{\dot{\xi }}_{Bi}\right]\mathrm{d}t,\hfill \end{array}$$

(61)

where ${\xi }_{B0}\left(t_1\right)={\xi }_{B0}\left(t_1,q_B\left(t_1\right),p_B\left(t_1\right),p_B^{\alpha }\left(t_1\right)\right)$, ${\xi }_{B0}\left(t_2\right)={\xi }_{B0}\left(t_2,q_B\left(t_2\right),p_B\left(t_2\right),p_B^{\alpha }\left(t_2\right)\right)$.

Noether symmetry requires that $\Delta I_B=0$, that is,

$$\begin{gathered} {\lambda }_{Ba}\frac{\partial {\varphi }_{Ba}}{\partial p_{Bi}}{\eta }_{Bi}+{\lambda }_{Ba}\frac{\partial {\varphi }_{Ba}}{\partial p_{Bi}^{\alpha }}{\eta }_{Bi}^{\alpha }-\frac{\partial H_B}{\partial q_{Bi}}{\xi }_{Bi}+\left(p_{Bi}^{\alpha }\frac{\mathrm{d}}{\mathrm{d}t}{B}_M^{\alpha }{q}_{Bi}-\frac{\partial H_B}{\partial t}\right){\xi }_{B0}+p_{Bi}{\dot{\xi }}_{Bi} \\ +p_{Bi}^{\alpha }{B}_M^{\alpha }\left({\xi }_{Bi}-{\dot{q}}_{Bi}{\xi }_{B0}\right)+\left(p_{Bi}^{\alpha }{B}_M^{\alpha }{q}_{Bi}-H_B\right){\dot{\xi }}_{B0}+\omega p_{Bi}^{\alpha }{\kappa }_{1-\alpha }\left(t_2,t\right){\dot{q}}_{Bi}\left(t_2\right){\xi }_{B0}\left(t_2\right) \\ -m{p}_{Bi}^{\alpha }{\kappa }_{1-\alpha }\left(t,t_1\right){\dot{q}}_{Bi}\left(t_1\right){\xi }_{B0}\left(t_1\right)=0. \end{gathered}$$

(62)

Equation (62) is called the Noether identity with the operator $B_M^{\alpha }$.

If we let $\Delta I_B=-{\displaystyle {\int }_{t_1}^{t_2}\frac{\mathrm{d}}{\mathrm{d}t}}\left(\Delta G_B\right)\mathrm{d}t$, where $\Delta G_B={\theta }_B{G}_B$, $G_B=G_B\left(t,q_B,p_B,p_B^{\alpha }\right)$ is called a gauge function with the operator $B_M^{\alpha }$, then we obtain

$$\begin{gathered} {\lambda }_{Ba}\frac{\partial {\varphi }_{Ba}}{\partial p_{Bi}}{\eta }_{Bi}+{\lambda }_{Ba}\frac{\partial {\varphi }_{Ba}}{\partial p_{Bi}^{\alpha }}{\eta }_{Bi}^{\alpha }-\frac{\partial H_B}{\partial q_{Bi}}{\xi }_{Bi}+\left(p_{Bi}^{\alpha }\frac{\mathrm{d}}{\mathrm{d}t}{B}_M^{\alpha }{q}_{Bi}-\frac{\partial H_B}{\partial t}\right){\xi }_{B0}+p_{Bi}{\dot{\xi }}_{Bi} \\ +p_{Bi}^{\alpha }{B}_M^{\alpha }\left({\xi }_{Bi}-{\dot{q}}_{Bi}{\xi }_{B0}\right)+\left(p_{Bi}^{\alpha }{B}_M^{\alpha }{q}_{Bi}-H_B\right){\dot{\xi }}_{B0}+\omega p_{Bi}^{\alpha }{\kappa }_{1-\alpha }\left(t_2,t\right){\dot{q}}_{Bi}\left(t_2\right){\xi }_{B0}\left(t_2\right) \\ -m{p}_{Bi}^{\alpha }{\kappa }_{1-\alpha }\left(t,t_1\right){\dot{q}}_{Bi}\left(t_1\right){\xi }_{B0}\left(t_1\right)+{\dot{G}}_B=0. \end{gathered}$$

(63)

Equation (63) is called the Noether quasi-identity with the operator $B_M^{\alpha }$. Therefore, we have

Theorem 3 .

For the constrained Hamiltonian system with the operator$B_M^{\alpha }$(Equation (45)), if the infinitesimal generators${\xi }_{B0}$,${\xi }_{Bi}$,${\eta }_{Bi}$and${\eta }_{Bi}^{\alpha }$satisfy Equation (62), then there exists a conserved quantity

$$\begin{gathered} C_B=p_{Bi}{\xi }_{Bi}+\left(p_{Bi}^{\alpha }{B}_M^{\alpha }{q}_{Bi}-H_B\right){\xi }_{B0}+{\displaystyle {\int }_{t_1}^t\left[p_{Bi}^{\alpha }{B}_M^{\alpha }\left({\xi }_{Bi}-{\dot{q}}_{Bi}{\xi }_{B0}\right)+\left({\xi }_{Bi}-{\dot{q}}_{Bi}{\xi }_{B0}\right)A_{M^{*}}^{\alpha }{p}_{Bi}^{\alpha }\right]}\mathrm{d}\tau \\ +\omega {\dot{q}}_{Bi}\left(t_2\right){\xi }_{B0}\left(t_2\right)\cdot {\displaystyle {\int }_{t_1}^t{p}_{Bi}^{\alpha }\left(\tau \right){\kappa }_{1-\alpha }\left(t_2,\tau \right)}\mathrm{d}\tau -m{\dot{q}}_{Bi}\left(t_1\right){\xi }_{B0}\left(t_1\right){\displaystyle {\int }_{t_1}^t{p}_{Bi}^{\alpha }\left(\tau \right){\kappa }_{1-\alpha }\left(\tau ,t_1\right)}\mathrm{d}\tau =\mathrm{const}. \end{gathered}$$

(64)

Proof of Theorem 3.

From Equations (34), (45) and (62), we have $\mathrm{d}{C}_B/\mathrm{d}t=0$. □

Theorem 4.

For the constrained Hamiltonian system with the operator$B_M^{\alpha }$(Equation (45)), if there exists a gauge function$G_B$such that the infinitesimal generators${\xi }_{B0}$,${\xi }_{Bi}$,${\eta }_{Bi}$and${\eta }_{Bi}^{\alpha }$satisfy Equation (63), then there exists a conserved quantity

$$\begin{gathered} C_{BG}=p_{Bi}{\xi }_{Bi}+{\displaystyle {\int }_{t_1}^t\left[p_{Bi}^{\alpha }{B}_M^{\alpha }\left({\xi }_{Bi}-{\dot{q}}_{Bi}{\xi }_{B0}\right)+\left({\xi }_{Bi}-{\dot{q}}_{Bi}{\xi }_{B0}\right)A_{M^{*}}^{\alpha }{p}_{Bi}^{\alpha }\right]}\mathrm{d}\tau \\ +\omega {\dot{q}}_{Bi}\left(t_2\right){\xi }_{B0}\left(t_2\right)\cdot {\displaystyle {\int }_{t_1}^t{p}_{Bi}^{\alpha }\left(\tau \right){\kappa }_{1-\alpha }\left(t_2,\tau \right)}\mathrm{d}\tau +\left(p_{Bi}^{\alpha }{B}_M^{\alpha }{q}_{Bi}-H_B\right){\xi }_{B0} \\ -m{\dot{q}}_{Bi}\left(t_1\right){\xi }_{B0}\left(t_1\right){\displaystyle {\int }_{t_1}^t{p}_{Bi}^{\alpha }\left(\tau \right){\kappa }_{1-\alpha }\left(\tau ,t_1\right)}\mathrm{d}\tau +G_B=\mathrm{const}. \end{gathered}$$

(65)

Proof of Theorem 4.

From Equations (34), (45) and (63), we have $\mathrm{d}{C}_{BG}/\mathrm{d}t=0$. □

Remark 7.

Let${\kappa }_{\alpha }\left(t,\tau \right)={\left(t-\tau \right)}^{\alpha -1}/\Gamma \left(\alpha \right)$, when$M=M_1$,$M=M_2$and$M=M_3$, Equation (62), Equation (63), Theorem 3 and Theorem 4 give the Noether identities, the Noether quasi-identities and the conserved quantities in terms of the left Caputo, the right Caputo and the Riesz-Caputo fractional derivatives, respectively.

Remark 8.

When the gauge function$G_A=0$(resp.$G_B=0$), Theorem 2 (resp. Theorem 4) reduces to Theorem 1 (resp. Theorem 3). Hence, the Noether-quasi symmetry is more general than the Noether symmetry.

6. Lie Symmetry and Conserved Quantity

Lie symmetry means an invariance of the differential equations under the infinitesimal transformations of time and coordinates. Lie symmetry can also lead to a conserved quantity under certain conditions.

6.1. Lie Symmetry with the Operator $A_M^{\alpha }$

We rewrite the constrained Hamilton equation with the operator $A_M^{\alpha }$ (Equation (39)) as

$$\begin{gathered} {\dot{p}}_{Ai}=-B_{M^{*}}^{\alpha }{p}_{Ai}^{\alpha }+f_{Ai}\left(t,q_A,p_A,p_A^{\alpha }\right), {\dot{q}}_{Ai}=S_{Ai}\left(t,q_A,p_A,p_A^{\alpha }\right), \\ {A}_M^{\alpha }{q}_{Ai}=h_{Ai}\left(t,q_A,p_A,p_A^{\alpha }\right), i=1,2,\cdots ,n. \end{gathered}$$

(66)

Then under the condition ${\kappa }_{1-\alpha }\left(t,t\right)=0$, we have

$$\begin{array}{c}{\dot{\overline{q}}}_{Ai}-S_{Ai}\left(\overline{t},{\overline{q}}_A,{\overline{p}}_A,{\overline{p}}_A^{\alpha }\right)=\frac{\mathrm{d}{\overline{q}}_{Ai}}{\mathrm{d}\overline{t}}-S_{Ai}\left(t+\Delta t,q_A+\Delta q_A,p_A+\Delta p_A,p_A^{\alpha }+\Delta p_A^{\alpha }\right)\hfill \\ =\frac{\mathrm{d}{q}_{Ai}+\mathrm{d}\Delta q_{Ai}}{\mathrm{d}t+\mathrm{d}\Delta t}-S_{Ai}\left(t+\Delta t,q_A+\Delta q_A,p_A+\Delta p_A,p_A^{\alpha }+\Delta p_A^{\alpha }\right)\hfill \\ =\frac{\left(\frac{\mathrm{d}{q}_{Ai}}{\mathrm{d}t}+\frac{\mathrm{d}\Delta q_{Ai}}{\mathrm{d}t}\right)\left(1-\frac{\mathrm{d}\Delta t}{\mathrm{d}t}\right)}{\left(1+\frac{\mathrm{d}\Delta t}{\mathrm{d}t}\right)\left(1-\frac{\mathrm{d}\Delta t}{\mathrm{d}t}\right)}-S_{Ai}\left(t+\Delta t,q_A+\Delta q_A,p_A+\Delta p_A,p_A^{\alpha }+\Delta p_A^{\alpha }\right)\hfill \\ ={\dot{q}}_{Ai}+\frac{\mathrm{d}\Delta q_{Ai}}{\mathrm{d}t}-{\dot{q}}_{Ai}\frac{\mathrm{d}\Delta t}{\mathrm{d}t}-S_{Ai}\left(t,q_A,p_A,p_A^{\alpha }\right)-\frac{\partial S_{Ai}}{\partial t}\cdot \Delta t-\frac{\partial S_{Ai}}{\partial q_{Ak}}\cdot \Delta q_{Ak}-\frac{\partial S_{Ai}}{\partial p_{Ak}}\cdot \Delta p_{Ak}-\frac{\partial S_{Ai}}{\partial p_{Ak}^{\alpha }}\cdot \Delta p_{Ak}^{\alpha }\hfill \\ ={\dot{q}}_{Ai}-S_{Ai}\left(t,q_A,p_A,p_A^{\alpha }\right)+{\theta }_A\left[{\dot{\xi }}_{Ai}-{\dot{q}}_{Ai}{\dot{\xi }}_{A0}-X_A^{\left(0\right)}\left(S_{Ai}\right)\right],\hfill \end{array}$$

(67)

where $X_A^{\left(0\right)}={\xi }_{A0}\frac{\partial }{\partial t}+{\xi }_{Ak}\frac{\partial }{\partial q_{Ak}}+{\eta }_{Ak}\frac{\partial }{\partial p_{Ak}}+{\eta }_{Ak}^{\alpha }\frac{\partial }{\partial p_{Ak}^{\alpha }}$, $k=1,2,\cdots ,n$.

$$\begin{array}{c}{A}_M^{\alpha }{\overline{q}}_{Ai}\left(\overline{t}\right)-h_{Ai}\left(\overline{t},{\overline{q}}_A,{\overline{p}}_A,{\overline{p}}_A^{\alpha }\right)=A_M^{\alpha }{\overline{q}}_{Ai}\left(\overline{t}\right)-h_{Ai}\left(t+\Delta t,q_A+\Delta q_A,p_A+\Delta p_A,p_A^{\alpha }+\Delta p_A^{\alpha }\right)\\ =A_M^{\alpha }{q}_{Ai}+A_M^{\alpha }\delta q_{Ai}+\Delta t\cdot \frac{\mathrm{d}}{\mathrm{d}t}{A}_M^{\alpha }{q}_{Ai}+\frac{\mathrm{d}}{\mathrm{d}t}\left[-m{\kappa }_{1-\alpha }\left(t,t_1\right)\Delta t_1{q}_{Ai}\left(t_1\right)+\omega {\kappa }_{1-\alpha }\left(t_2,t\right)\Delta t_2{q}_{Ai}\left(t_2\right)\right]\hfill \\ -h_{Ai}\left(t,q_A,p_A,p_A^{\alpha }\right)-\frac{\partial h_{Ai}}{\partial t}\Delta t-\frac{\partial h_{Ai}}{\partial q_{Ak}}\Delta q_{Ak}-\frac{\partial h_{Ai}}{\partial p_{Ak}}\Delta p_{Ak}-\frac{\partial h_{Ai}}{\partial p_{Ak}^{\alpha }}\Delta p_{Ak}^{\alpha }\hfill \\ =A_M^{\alpha }{q}_{Ai}-h_{Ai}\left(t,q_A,p_A,p_A^{\alpha }\right)+{\theta }_A\left\{A_M^{\alpha }\left({\xi }_{Ai}-{\dot{q}}_{Ai}{\xi }_{A0}\right)+{\xi }_{A0}\cdot \frac{\mathrm{d}}{\mathrm{d}t}{A}_M^{\alpha }{q}_{Ai}\right.\hfill \\ \left.+\frac{\mathrm{d}}{\mathrm{d}t}\left[-m{\kappa }_{1-\alpha }\left(t,t_1\right){\xi }_{A0}\left(t_1\right)q_{Ai}\left(t_1\right)+\omega {\kappa }_{1-\alpha }\left(t_2,t\right){\xi }_{A0}\left(t_2\right)q_{Ai}\left(t_2\right)\right]-X_A^{\left(0\right)}\left(h_{Ai}\right)\right\}, \hfill \end{array}$$

(68)

where

$$\begin{array}{l}{\xi }_{A0}\left(t_1\right)={\xi }_{A0}\left(t_1,q_A\left(t_1\right),p_A\left(t_1\right),p_A^{\alpha }\left(t_1\right)\right), {\xi }_{A0}\left(t_2\right)={\xi }_{A0}\left(t_2,q_A\left(t_2\right),p_A\left(t_2\right),p_A^{\alpha }\left(t_2\right)\right),\\ A_M^{\alpha }{\overline{q}}_{Ai}\left(\overline{t}\right)=A_M^{\alpha }{q}_{Ai}+\Delta t\cdot \frac{\mathrm{d}}{\mathrm{d}t}{A}_M^{\alpha }{q}_{Ai}+\frac{\mathrm{d}}{\mathrm{d}t}\left[-m{\kappa }_{1-\alpha }\left(t,t_1\right)\Delta t_1{q}_{Ai}\left(t_1\right)+\omega {\kappa }_{1-\alpha }\left(t_2,t\right)\Delta t_2{q}_{Ai}\left(t_2\right)\right]\hfill \\ +A_M^{\alpha }\delta q_{Ai}.\hfill \\ {\dot{\overline{p}}}_{Ai}+B_{M^{*}}^{\alpha }{\overline{p}}_{Ai}^{\alpha }\left(\overline{t}\right)-f_{Ai}\left(\overline{t},{\overline{q}}_A,{\overline{p}}_A,{\overline{p}}_A^{\alpha }\right)=\frac{\mathrm{d}{p}_{Ai}+\mathrm{d}\Delta p_{Ai}}{\mathrm{d}t+\mathrm{d}\Delta t}+B_{M^{*}}^{\alpha }{\overline{p}}_{Ai}^{\alpha }\left(\overline{t}\right)-f_{Ai}\left(\overline{t},{\overline{q}}_A,{\overline{p}}_A,{\overline{p}}_A^{\alpha }\right)\hfill \\ =\frac{\left(\frac{\mathrm{d}{p}_{Ai}}{\mathrm{d}t}+\frac{\mathrm{d}\Delta p_{Ai}}{\mathrm{d}t}\right)\left(1-\frac{\mathrm{d}\Delta t}{\mathrm{d}t}\right)}{\left(1+\frac{\mathrm{d}\Delta t}{\mathrm{d}t}\right)\left(1-\frac{\mathrm{d}\Delta t}{\mathrm{d}t}\right)}+B_{M^{*}}^{\alpha }{\overline{p}}_{Ai}^{\alpha }-f_{Ai}\left(t+\Delta t,q_A+\Delta q_A,p_A+\Delta p_A,p_A^{\alpha }+\Delta p_A^{\alpha }\right)\hfill \\ ={\dot{p}}_{Ai}+B_{M^{*}}^{\alpha }{p}_{Ai}^{\alpha }-f_{Ai}\left(t,q_A,p_A,p_A^{\alpha }\right)+{\theta }_A\left[{\dot{\eta }}_{Ai}-{\dot{p}}_{Ai}{\dot{\xi }}_{A0}+B_{M^{*}}^{\alpha }\left({\eta }_{Ai}^{\alpha }-{\dot{p}}_{Ai}^{\alpha }{\xi }_{A0}\right)+{\xi }_{A0}\cdot \frac{\mathrm{d}}{\mathrm{d}t}{B}_{M^{*}}^{\alpha }{p}_{Ai}^{\alpha }\right.\hfill \\ \left.-\omega {\kappa }_{1-\alpha }\left(t,t_1\right){\dot{p}}_{Ai}^{\alpha }\left(t_1\right)\cdot {\xi }_{A0}\left(t_1\right)+m{\kappa }_{1-\alpha }\left(t_2,t\right){\dot{p}}_{Ai}^{\alpha }\left(t_2\right)\cdot {\xi }_{A0}\left(t_2\right)-X_A^{\left(0\right)}\left(f_{Ai}\right)\right], \hfill \end{array}$$

(69)

where

$$B_{M^{*}}^{\alpha }{\overline{p}}_{Ai}^{\alpha }=B_{M^{*}}^{\alpha }{p}_{Ai}^{\alpha }+B_{M^{*}}^{\alpha }\delta p_{Ai}^{\alpha }+\Delta t\cdot \frac{\mathrm{d}}{\mathrm{d}t}{B}_{M^{*}}^{\alpha }{p}_{Ai}^{\alpha }-\omega {\kappa }_{1-\alpha }\left(t,t_1\right){\dot{p}}_{Ai}^{\alpha }\left(t_1\right)\cdot \Delta t_1+m{\kappa }_{1-\alpha }\left(t_2,t\right){\dot{p}}_{Ai}^{\alpha }\left(t_2\right)\cdot \Delta t_2.$$

Lie symmetry requires that

$$\begin{gathered} {\dot{\xi }}_{Ai}-{\dot{q}}_{Ai}{\dot{\xi }}_{A0}-X_A^{\left(0\right)}\left(S_{Ai}\right)=0, \\ {A}_M^{\alpha }\left({\xi }_{Ai}-{\dot{q}}_{Ai}{\xi }_{A0}\right)+\frac{\mathrm{d}}{\mathrm{d}t}\left[-m{\kappa }_{1-\alpha }\left(t,t_1\right){\xi }_{A0}\left(t_1\right)q_{Ai}\left(t_1\right)+\omega {\kappa }_{1-\alpha }\left(t_2,t\right){\xi }_{A0}\left(t_2\right)q_{Ai}\left(t_2\right)\right] \\ +{\xi }_{A0}\cdot \frac{\mathrm{d}}{\mathrm{d}t}{A}_M^{\alpha }{q}_{Ai}-X_A^{\left(0\right)}\left(h_{Ai}\right)=0, \\ {\dot{\eta }}_{Ai}-{\dot{p}}_{Ai}{\dot{\xi }}_{A0}+B_{M^{*}}^{\alpha }\left({\eta }_{Ai}^{\alpha }-{\dot{p}}_{Ai}^{\alpha }{\xi }_{A0}\right)+{\xi }_{A0}\cdot \frac{\mathrm{d}}{\mathrm{d}t}{B}_{M^{*}}^{\alpha }{p}_{Ai}^{\alpha }-\omega {\kappa }_{1-\alpha }\left(t,t_1\right){\dot{p}}_{Ai}^{\alpha }\left(t_1\right) \\ \cdot {\xi }_{A0}\left(t_1\right)+m{\kappa }_{1-\alpha }\left(t_2,t\right){\dot{p}}_{Ai}^{\alpha }\left(t_2\right)\cdot {\xi }_{A0}\left(t_2\right)-X_A^{\left(0\right)}\left(f_{Ai}\right)=0 \end{gathered}$$

(70)

Equation (70) is called the determined equation for the constrained Hamiltonian system with the operator $A_M^{\alpha }$ (Equation (39)). Lie symmetry also leads to a conserved quantity under certain conditions. Therefore, we have

Theorem 5.

For the constrained Hamiltonian system with the operator$A_M^{\alpha }$(Equation (39)), if the infinitesimal generators${\xi }_{A0}$,${\xi }_{Ai}$,${\eta }_{Ai}$and${\eta }_{Ai}^{\alpha }$satisfy Equation (54) and the determined equation (Equation (70)), then there exists a conserved quantity Equation (56).

Proof of Theorem 5.

From Equations (20), (39) and (54), we can get the intended result. □

6.2. Lie Symmetry with the Operator $B_M^{\alpha }$

We rewrite the constrained Hamilton equation with the operator $B_M^{\alpha }$ (Equation (45)) as

$$\begin{gathered} {\dot{p}}_{Bi}=-A_{M^{*}}^{\alpha }{p}_{Bi}^{\alpha }+f_{Bi}\left(t,q_B,p_B,p_B^{\alpha }\right), {\dot{q}}_{Bi}=S_{Bi}\left(t,q_B,p_B,p_B^{\alpha }\right), \\ {B}_M^{\alpha }{q}_{Bi}=h_{Bi}\left(t,q_B,p_B,p_B^{\alpha }\right), i=1,2,\cdots ,n \end{gathered}$$

(71)

Then under the condition ${\kappa }_{1-\alpha }\left(t,t\right)=0$, we have

$$\begin{gathered} {\dot{\overline{q}}}_{Bi}-S_{Bi}\left(\overline{t},{\overline{q}}_B,{\overline{p}}_B,{\overline{p}}_B^{\alpha }\right)=\frac{\mathrm{d}{\overline{q}}_{Bi}}{\mathrm{d}\overline{t}}-S_{Bi}\left(t+\Delta t,q_B+\Delta q_B,p_B+\Delta p_B,p_B^{\alpha }+\Delta p_B^{\alpha }\right) \\ =\frac{\mathrm{d}{q}_{Bi}+\mathrm{d}\Delta q_{Bi}}{\mathrm{d}t+\mathrm{d}\Delta t}-S_{Bi}\left(t+\Delta t,q_B+\Delta q_B,p_B+\Delta p_B,p_B^{\alpha }+\Delta p_B^{\alpha }\right) \\ ={\dot{q}}_{Bi}-S_{Bi}\left(t,q_B,p_B,p_B^{\alpha }\right)+{\theta }_B\left[{\dot{\xi }}_{Bi}-{\dot{q}}_{Bi}{\dot{\xi }}_{B0}-X_B^{\left(0\right)}\left(S_{Bi}\right)\right], \end{gathered}$$

(72)

where

$$\begin{gathered} X_B^{\left(0\right)}={\xi }_{B0}\frac{\partial }{\partial t}+{\xi }_{Bk}\frac{\partial }{\partial q_{Bk}}+{\eta }_{Bk}\frac{\partial }{\partial p_{Bk}}+{\eta }_{Bk}^{\alpha }\frac{\partial }{\partial p_{Bk}^{\alpha }}, k=1,2,\cdots ,n. \\ {B}_M^{\alpha }{\overline{q}}_{Bi}\left(\overline{t}\right)-h_{Bi}\left(\overline{t},{\overline{q}}_B,{\overline{p}}_B,{\overline{p}}_B^{\alpha }\right)=B_M^{\alpha }{\overline{q}}_{Bi}\left(\overline{t}\right)-h_{Bi}\left(t+\Delta t,q_B+\Delta q_B,p_B+\Delta p_B,p_B^{\alpha }+\Delta p_B^{\alpha }\right) \\ =B_M^{\alpha }{q}_{Bi}-h_{Bi}\left(t,q_B,p_B,p_B^{\alpha }\right)+{\theta }_B\left[B_M^{\alpha }\left({\xi }_{Bi}-{\dot{q}}_{Bi}{\xi }_{B0}\right)+{\xi }_{B0}\cdot \frac{\mathrm{d}}{\mathrm{d}t}{B}_M^{\alpha }{q}_{Bi}\right. \\ \left.-m{\kappa }_{1-\alpha }\left(t,t_1\right){\xi }_{B0}\left(t_1\right){\dot{q}}_{Bi}\left(t_1\right)+\omega {\kappa }_{1-\alpha }\left(t_2,t\right){\xi }_{B0}\left(t_2\right){\dot{q}}_{Bi}\left(t_2\right)-X_B^{\left(0\right)}\left(h_{Bi}\right)\right], \end{gathered}$$

(73)

where

$$\begin{gathered} {\xi }_{B0}\left(t_1\right)={\xi }_{B0}\left(t_1,q_B\left(t_1\right),p_B\left(t_1\right),p_B^{\alpha }\left(t_1\right)\right), \\ {\xi }_{B0}\left(t_2\right)={\xi }_{B0}\left(t_2,q_B\left(t_2\right),p_B\left(t_2\right),p_B^{\alpha }\left(t_2\right)\right). \\ {\dot{\overline{p}}}_{Bi}+A_{M^{*}}^{\alpha }{\overline{p}}_{Bi}^{\alpha }\left(\overline{t}\right)-f_{Bi}\left(\overline{t},{\overline{q}}_B,{\overline{p}}_B,{\overline{p}}_B^{\alpha }\right)=\frac{\mathrm{d}{p}_{Bi}+\mathrm{d}\Delta p_{Bi}}{\mathrm{d}t+\mathrm{d}\Delta t}+A_{M^{*}}^{\alpha }{\overline{p}}_{Bi}^{\alpha }\left(\overline{t}\right)-f_{Bi}\left(\overline{t},{\overline{q}}_B,{\overline{p}}_B,{\overline{p}}_B^{\alpha }\right) \\ ={\dot{p}}_{Bi}+A_{M^{*}}^{\alpha }{p}_{Bi}^{\alpha }-f_{Bi}\left(t,q_B,p_B,p_B^{\alpha }\right)+{\theta }_B\left\{{\dot{\eta }}_{Bi}-{\dot{p}}_{Bi}{\dot{\xi }}_{B0}+A_{M^{*}}^{\alpha }\left({\eta }_{Bi}^{\alpha }-{\dot{p}}_{Bi}^{\alpha }{\xi }_{B0}\right)+{\xi }_{B0}\cdot \frac{\mathrm{d}}{\mathrm{d}t}{A}_{M^{*}}^{\alpha }{p}_{Bi}^{\alpha }\right. \\ \left.+\frac{\mathrm{d}}{\mathrm{d}t}\left[-\omega {\kappa }_{1-\alpha }\left(t,t_1\right)p_{Bi}^{\alpha }\left(t_1\right)\cdot {\xi }_{B0}\left(t_1\right)+m{\kappa }_{1-\alpha }\left(t_2,t\right)p_{Bi}^{\alpha }\left(t_2\right)\cdot {\xi }_{B0}\left(t_2\right)\right]-X^{\left(0\right)}\left(f_{Bi}\right)\right\}. \end{gathered}$$

(74)

Lie symmetry requires that

$$\begin{gathered} {\dot{\xi }}_{Bi}-{\dot{q}}_{Bi}{\dot{\xi }}_{B0}-X_B^{\left(0\right)}\left(S_{Bi}\right)=0, \\ {B}_M^{\alpha }\left({\xi }_{Bi}-{\dot{q}}_{Bi}{\xi }_{B0}\right)+{\xi }_{B0}\cdot \frac{\mathrm{d}}{\mathrm{d}t}{B}_M^{\alpha }{q}_{Bi}-m{\kappa }_{1-\alpha }\left(t,t_1\right){\xi }_{B0}\left(t_1\right){\dot{q}}_{Bi}\left(t_1\right) \\ +\omega {\kappa }_{1-\alpha }\left(t_2,t\right){\xi }_{B0}\left(t_2\right){\dot{q}}_{Bi}\left(t_2\right)-X_B^{\left(0\right)}\left(h_{Bi}\right)=0, \\ {\dot{\eta }}_{Bi}-{\dot{p}}_{Bi}{\dot{\xi }}_{B0}+A_{M^{*}}^{\alpha }\left({\eta }_{Bi}^{\alpha }-{\dot{p}}_{Bi}^{\alpha }{\xi }_{B0}\right)+{\xi }_{B0}\cdot \frac{\mathrm{d}}{\mathrm{d}t}{A}_{M^{*}}^{\alpha }{p}_{Bi}^{\alpha }-X^{\left(0\right)}\left(f_{Bi}\right) \\ +\frac{\mathrm{d}}{\mathrm{d}t}\left[-\omega {\kappa }_{1-\alpha }\left(t,t_1\right)p_{Bi}^{\alpha }\left(t_1\right)\cdot {\xi }_{B0}\left(t_1\right)+m{\kappa }_{1-\alpha }\left(t_2,t\right)p_{Bi}^{\alpha }\left(t_2\right)\cdot {\xi }_{B0}\left(t_2\right)\right]=0. \end{gathered}$$

(75)

Equation (75) is called the determined equation for the constrained Hamiltonian system with the operator $B_M^{\alpha }$ (Equation (45)). Lie symmetry also leads to a conserved quantity under certain conditions. Therefore, we have

Theorem 6.

For the constrained Hamiltonian system with the operator$B_M^{\alpha }$(Equation (45)), if the infinitesimal generators${\xi }_{B0}$,${\xi }_{Bi}$,${\eta }_{Bi}$and${\eta }_{Bi}^{\alpha }$satisfy Equation (62) and the determined equation (Equation (75), then there exists a conserved quantity Equation (64).

Proof of Theorem 6.

From Equations (34), (45) and (62), we can get the intended result. □

7. An Example

Try to find the conserved quantity for the following singular system with the operator $A_M^{\alpha }$, whose Lagrangian is

$$L_A={\dot{q}}_{A1}{q}_{A2}-q_{A1}{\dot{q}}_{A2}+{\left(q_{A1}\right)}^2+{\left(q_{A2}\right)}^2+\frac{1}{2}\left[{\left(A_M^{\alpha }{q}_{A1}\right)}^2+{\left(A_M^{\alpha }{q}_{A2}\right)}^2\right].$$

(76)

From Equations (11) and (12), we have

$$\begin{gathered} p_{A1}=\frac{\partial L_A}{\partial {\dot{q}}_{A1}}=q_{A2}, p_{A2}=\frac{\partial L_A}{\partial {\dot{q}}_{A2}}=-q_{A1}, p_{A1}^{\alpha }=\frac{\partial L_A}{\partial A_M^{\alpha }{q}_{A1}}=A_M^{\alpha }{q}_{A1}, p_{A2}^{\alpha }=\frac{\partial L_A}{\partial A_M^{\alpha }{q}_{A2}}=A_M^{\alpha }{q}_{A2}, \\ {H}_A=p_{A1}{\dot{q}}_{A1}+p_{A2}{\dot{q}}_{A2}+p_{A1}^{\alpha }{A}_M^{\alpha }{q}_{A1}+p_{A2}^{\alpha }{A}_M^{\alpha }{q}_{A2}-L_A=\frac{1}{2}\left[{\left(A_M^{\alpha }{q}_{A1}\right)}^2+{\left(A_M^{\alpha }{q}_{A2}\right)}^2\right] \\ -{\left(q_{A1}\right)}^2-{\left(q_{A2}\right)}^2=\frac{1}{2}\left[{\left(p_{A1}^{\alpha }\right)}^2+{\left(p_{A2}^{\alpha }\right)}^2\right]-{\left(q_{A1}\right)}^2-{\left(q_{A2}\right)}^2. \end{gathered}$$

(77)

Therefore, the Hamiltonian and the two primary constraints have the form

$$H_A=\frac{1}{2}\left[{\left(p_{A1}^{\alpha }\right)}^2+{\left(p_{A2}^{\alpha }\right)}^2\right]-q_{A1}^2-q_{A2}^2,$$

(78)

$${\varphi }_{A1}=p_{A1}-q_{A2}=0, {\varphi }_{A2}=p_{A2}+q_{A1}=0.$$

(79)

Then from Equation (48), the two Lagrange multipliers ${\lambda }_{A1}$ and ${\lambda }_{A2}$ can be calculated as

$$\begin{gathered} {\lambda }_{A1}=-q_{A2}+\frac{1}{2}\left[B_{M^{*}}^{\alpha }{p}_{A2}^{\alpha }-m{p}_{A2}^{\alpha }\left(t_2\right){\kappa }_{1-\alpha }\left(t_2,t\right)+\omega p_{A2}^{\alpha }\left(t_1\right){\kappa }_{1-\alpha }\left(t,t_1\right)\right], \\ {\lambda }_{A2}=q_{A1}-\frac{1}{2}\left[B_{M^{*}}^{\alpha }{p}_{A1}^{\alpha }-m{p}_{A1}^{\alpha }\left(t_2\right){\kappa }_{1-\alpha }\left(t_2,t\right)+\omega p_{A1}^{\alpha }\left(t_1\right){\kappa }_{1-\alpha }\left(t,t_1\right)\right]. \end{gathered}$$

(80)

Therefore, the constrained Hamilton equation with the operator $A_M^{\alpha }$ can be obtained. And we can also verify that under the condition $\frac{\mathrm{d}}{\mathrm{d}t}{\kappa }_{\alpha }\left(t,\tau \right)=-\frac{\mathrm{d}}{\mathrm{d}\tau }{\kappa }_{\alpha }\left(t,\tau \right)$,

$${\xi }_{A0}=1, {\xi }_{A1}={\xi }_{A2}=0, {\eta }_{A1}={\eta }_{A2}=0, {\eta }_{A1}^{\alpha }={\eta }_{A2}^{\alpha }=0, G_A=0$$

(81)

is a solution to the Noether quasi-identity (Equation (55)). Then Theorem 2 gives

$$\begin{gathered} C_{AG}=\frac{1}{2}\left[{\left(p_{A1}^{\alpha }\right)}^2+{\left(p_{A2}^{\alpha }\right)}^2\right]+q_{A1}^2+q_{A2}^2-{\displaystyle {\int }_{t_1}^t\left\{p_{A1}^{\alpha }\frac{\mathrm{d}}{\mathrm{d}t}{A}_M^{\alpha }{q}_{A1}+p_{A2}^{\alpha }\frac{\mathrm{d}}{\mathrm{d}t}{A}_M^{\alpha }{q}_{A2}\right.} \\ +{\dot{q}}_{A1}\left[B_{M^{*}}^{\alpha }{p}_{A1}^{\alpha }-m{\kappa }_{1-\alpha }\left(t_2,\tau \right)p_{A1}^{\alpha }\left(t_2\right)+\omega {\kappa }_{1-\alpha }\left(\tau ,t_1\right)p_{A1}^{\alpha }\left(t_1\right)\right] \\ \left.+{\dot{q}}_{A2}\left[B_{M^{*}}^{\alpha }{p}_{A2}^{\alpha }-m{\kappa }_{1-\alpha }\left(t_2,\tau \right)p_{A2}^{\alpha }\left(t_2\right)+\omega {\kappa }_{1-\alpha }\left(\tau ,t_1\right)p_{A2}^{\alpha }\left(t_1\right)\right]\right\}\mathrm{d}\tau =\mathrm{const}. \end{gathered}$$

(82)

Specially, let ${\kappa }_{\alpha }\left(t,\tau \right)={\left(t-\tau \right)}^{\alpha -1}/\Gamma \left(\alpha \right)$, when $M=M_1$ (or $M=M_2$ or $M=M_3$) and $\alpha \to 1$, we have $C_{AG}=-H_A=\mathrm{const}$.

8. Results and Discussion

Based on the mixed integer order derivative and generalized operators, the singular Lagrange equations, primary constraints, constrained Hamilton equations, consistency conditions and conserved quantities were investigated. All are new works. In fact, Lie symmetry can lead to the Noether type conserved quantity as well as the Hojman conserved quantity. Here, we only presented the Noether type conserved quantity simply. Next, Lie symmetry and the Hojman conserved quantity, Mei symmetry and the Mei type conserved quantity and the relationships among the three symmetry methods will be studied. Singular systems on time scales is also a hot topic that needs to be investigated further.