# Noether Symmetry Method for Hamiltonian Mechanics Involving Generalized Operators

Authors: Chuan-Jing Song; Yao Cheng
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

Based on the generalized operators, Hamilton equation, Noether symmetry, and perturbation to Noether symmetry are studied. The main contents are divided into four parts, and every part includes two generalized operators. Firstly, Hamilton equations within generalized operators are established. Secondly, the Noether symmetry method and conserved quantity are studied. Thirdly, perturbation to the Noether symmetry and adiabatic invariant are presented. And finally, two applications are presented to illustrate the methods and results.


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## Body

## 1. Introduction

With the development of discipline and the progress of technology, the dynamics of the constrained mechanical systems was put forward, soAnalytical Mechanics appears. Lagrange is the founder of Analytical Mechanics. Lagrange further studied the motion of the constrained particles after d’Alembert. The Lagrange equation and the d’Alembert-Lagrange principle are the core of Lagrangian mechanics.Hamilton developedAnalytical Mechanics. In his two long papers, “On a General Method in Dynamics” (1834) and “Second Essay on a General Method in Dynamics” (1835), he proposed an integral variational principle and a dynamic equation with generalized coordinate and generalized momentum as independent variables. This principle is called the Hamilton principle. The dynamic equation given by Hamilton is called the canonical equation. The Hamilton principle and the Hamilton canonical equation are the core of Hamiltonian mechanics.Hamilton’s principle is highly general, which can represent the motion law of a holonomic and conservative system by only one functional extreme value. The principle is not only simple and beautiful in form but also rich and profound in connotation. It can be applied to mechanics, optics, electromagnetism, and other fields and can also be applied to approximate calculation [1, 2]. In Ref. [3], the Hamilton principle is applied to the dynamics of the flexible multibody and rigid flexible coupling systems. Hamilton’s principle is extended to the holonomic nonconservative system [4], the high-order system [5], and the nonholonomic system [4]. In addition, the Pfaff-Birkhoff principle [6, 7], generalized Pfaff-Birkhoff principle [8], and Vujanović’s variational principle of nonconservative system [9] are also the generalization of Hamilton’s principle.The Hamilton equation is also of great significance. Firstly, the canonical equation is simpler in form and more symmetrical in structure than the Lagrange equation. It is more convenient for general discussion when solving many complex mechanical problems, such as celestial mechanics and vibration theory. Secondly, the new concepts related to the canonical equations, such as the canonical variables, have many applications in mechanics and physics, such as statistical physics and quantum mechanics. Thirdly, a complete set of integration methods are established for the canonical equations, such as the Poisson theorem, Jacobi method, canonical transformation, and integral invariants [4]. Fourthly, geometric mechanics has been developed due to the symplectic structure of the canonical equations. Besides, Hamiltonian mechanics has also contributed to the formation and the development of the generalized Hamiltonian mechanics [10] and the Birkhoffian mechanics [11]. Finally, considering the perturbation of the Hamilton equation, the KAM theorem appears. The KAM theorem became the beginning of the chaos theory [4].In 1918, the famous paper “Invariante Variations Probleme” by German mathematician Noether revealed the relationship between the conserved quantity of the mechanics system and its internal dynamical symmetry [12]. Reference [13] points out that the application of the mechanics variational principle and its physical significance are based on two theorems: the Hilbert independence theorem and the Noether theorem. The first theorem gives the mathematical argument of the variational principle, and the second theorem reveals its physical significance.The Noether symmetry method is one of the modern integration methods of the Hamiltonian mechanics. For the Hamiltonian system, Noether symmetry is the invariance of the Hamilton action under the infinitesimal transformations. The Noether symmetry method points out that if the infinitesimal generators and the gauge function satisfy the Noether identity, then the conserved quantity of the system can be found [14–20]. The advantage of the Noether theory is that if there is a Noether symmetry, a corresponding conserved quantity can be found and vice versa. The Noether theory can also be used to solve general ordinary differential equations as long as they are expressed as equations of mechanics systems [21].Recent developments in the fields of science, engineering, economics, bioengineering, and applied mathematics have demonstrated that many phenomena in nature are modelled more accurately using fractional derivatives [22, 23]. Fractional Hamiltonian mechanics [24–32] and Noether theorems for them have also been established and investigated [33–36]. The fractional operators mainly referred to the left and right fractional Riemann-Liouville integrals, the left and right fractional Riemann-Liouville differential operators, the left and right fractional Caputo differential operators, the Riesz-Riemann-Liouville differential operator, and the Riesz-Caputo differential operator. However, Agrawal [37] pointed out that the fractional power kernel need not be the only kernel to describe the phenomena of the nature. He defined three new operators which in special cases reduce to the fractional operators listed above. Then, the entire theories of the Hamiltonian mechanics and the corresponding Noether theorem can be redeveloped. And in such a case, the theories of the integer order and the fractional order Hamiltonian mechanics and other results resulted from them would be special cases of the more general new operators.The structure of this paper is organized as follows. Section2 lists the definitions and properties of the generalized operators briefly. Fractional calculus of variations for the Hamiltonian system within generalized operators is studied in Section 3. Based on the generalized operators, Noether symmetry, and conserved quantities, perturbation to Noether symmetry and adiabatic invariants are investigated in Section 4 and Section 5, respectively. Section 6 presents two applications to illustrate the methods and results obtained in this paper. A conclusion is given in Section 7.


## 2. Preliminaries

The definitions and properties of the generalized operatorsK, A, and B were studied in detail by Agrawal [37]. Here, we only list their definitions and integration by part formulae.The operatorsK, A, and B have the forms
(1)KMαft=m∫atκαt,τfτdτ+ω∫tbκατ,tfτdτ,α>0,(2)AMαft=DnKMn−αft,n−1<α<n,(3)BMαft=KMn−αDnft,n−1<α<n,where ft is continuous and integrable, a<t<b, M=<a,t,b,m,ω> is a parameter set, m and ω are two real numbers, καt,τ is a kernel which may depend on a parameter α, and n is an integer.Remark 1.
Letκαt,τ=t−τα−1/Γα, then different results will be obtained under different conditions. For example, when M=M1=<a,t,b,1,0>, we have
(4)AMαft=DnKMn−αft=1Γn−αddtn∫att−τn−α−1fτdτ=DaRLtαft,BMαft=KMn−αDnft=1Γn−α∫att−τn−α−1ddτnfτdτ=DaCtαft,i.e., the operator A and the operator B reduce to the left Riemann-Liouville fractional operator and the left Caputo fractional operator, respectively. When M=M2=<a,t,b,0,1>, we have
(5)AMαft=DnKMn−αft=1Γn−αddtn∫tbτ−tn−α−1fτdτ=−1nDtRLbαft,BMαft=KMn−αDnft=1Γn−α∫tbτ−tn−α−1ddτnfτdτ=−1nDtCbαft,i.e., the operator A and the operator B reduce to the right Riemann-Liouville fractional operator and the right Caputo fractional operator, respectively. When M=M3=<a,t,b,1/2,1/2>, we have
(6)AMαft=DnKMn−αft=12Γn−αddtn∫abt−τn−α−1fτdτ=DaRbαft,BMαft=KMn−αDnft=12Γn−α∫abt−τn−α−1ddτnfτdτ=DaRCbαft,i.e., the operator A and the operator B reduce to the Riesz-Riemann-Liouville fractional operator and the Riesz-Caputo fractional operator, respectively.The generalized operatorsK, A, and B satisfy the following integration by part formulae:
(7)∫abgtKMαftdt=∫abftKM∗αgtdt,(8)∫abgtAMαftdt=−1n∫abftBM∗αgtdt+∑j=0n−1−Dn−1−jgtAMα+j−nftt=at=b,(9)∫abgtBMαftdt=−1n∫abftAM∗αgtdt+∑j=0n−1−1jAM∗α+j−ngtDn−1−jftt=at=b,where M∗=<a,t,b,ω,m>, n−1<α<n, and n is an integer.In this text, we setn=1, so 0<α<1. The variational problems within generalized operators A and B are to be studied first.


## 3. Hamilton Equations within Generalized Operators

### 3.1. Hamilton Equation within Generalized OperatorA

LetLA=LAt,qA,AMαqA be the Lagrangian within generalized operator A, qA=qA1,qA2,⋯,qAn, and AMαqA=AMαqA1,AMαqA2,⋯,AMαqAn, then the generalized momentum pA=pA1,pA2,⋯,pAn is defined as pAi=∂LA/∂AMαqAi, i=1,2,⋯n, and the Hamiltonian HA=HAt,qA,pA can be expressed as HA=pAi⋅AMαqAi−LA. Therefore, the Hamilton action within generalized operator A is
(10)SAqA⋅=∫abpAi⋅AMαqAi−HAt,qA,pAdt.Then,(11)δSA=0,with the commutative conditions [38].
(12)δAMαqAi=AMαδqAi,i=1,2,⋯,n,and the boundary conditions
(13)qAa=qAa,qAb=qAb,is called Hamilton principle within generalized operator A, where δ means the isochronous variation.Using Equations (1), (8), and (12), we have
(14)δSA=∫abδpAi⋅AMαqAi+pAi⋅δAMαqAi−∂HA∂qAiδqAi−∂HA∂pAiδpAidt=∫ab−BM∗αpAi+∂HA∂qAi−mpAibκ1−αb,t+ωpAiaκ1−αt,aδqAi+AMαqAi−∂HA∂pAiδpAidt=0.Then, from the HamiltonianHA=pAi⋅AMαqAi−LA, the independence of δqAi, and the arbitrariness of the interval a,b, we get
(15)AMαqAi=∂HA∂pAi,BM∗αpAi=−∂HA∂qAi+mpAibκ1−αb,t−ωpAiaκ1−αt,a.Equation (15) is called the Hamilton equation within generalized operator A.Remark 2.
Letκαt,τ=t−τα−1/Γα; from Equation (15), we can get the Hamilton equations in terms of the left Riemann-Liouville fractional operator, the right Riemann-Liouville fractional operator, and the Riesz-Riemann-Liouville fractional operator by letting M=M1, M=M2, and M=M3, respectively.


### 3.2. Hamilton Equation within Generalized OperatorB

LetLB=LBt,qB,BMαqB be the Lagrangian within the generalized operator B, qB=qB1,qB2,⋯,qBn, and BMαqB=BMαqB1,BMαqB2,⋯,BMαqBn, then the generalized momentum pB=pB1,pB2,⋯,pBn is defined as pBi=∂LB/∂BMαqBi, i=1,2,⋯n, and the Hamiltonian HB=HBt,qB,pB can be expressed as HB=pBi⋅BMαqBi−LB. Therefore, the Hamilton action within the generalized operator B is
(16)SBqB⋅=∫abpBi⋅BMαqBi−HBt,qB,pBdt.Then,(17)δSB=0,with the commutative conditions [38]
(18)δBMαqBi=BMαδqBi,and the boundary conditions
(19)qBa=qBa,qBb=qBb,are called the Hamilton principle within generalized operator B, where δ means the isochronous variation.Using Equations (9), (17), and (19), we have
(20)δSB=∫abδpBi⋅BMαqBi+pBi⋅δBMαqBi−∂HB∂qBiδqBi−∂HB∂pBiδpBidt=∫abBMαqBi−∂HB∂pBiδpBi−AM∗αpBi+∂HB∂qBiδqBidt.Then, from the HamiltonianHB=pBi⋅BMαqBi−LB, the independence of δqBi, and the arbitrariness of the interval a,b, we get
(21)BMαqBi=∂HB∂pBi,AM∗αpBi=−∂HB∂qBi.Equation (21) is called the Hamilton equation within generalized operator B.Remark 3.
Equation (21) is consistent with the result obtained in Ref. [39]. However, the method in Ref. [39] is different.Remark 4.
Letκαt,τ=t−τα−1/Γα; from Equation (21), we can get the Hamilton equations in terms of the left Caputo fractional operator, the right Caputo fractional operator, and the Riesz-Caputo fractional operator by letting M=M1, M=M2, and M=M3, respectively.Remark 5.
In Remark2 and Remark 4, there are six cases in total. If we let α⟶1, then all of them reduce to the classical Hamilton equation, which can be found in Ref. [14].


## 3.1. Hamilton Equation within Generalized OperatorA

LetLA=LAt,qA,AMαqA be the Lagrangian within generalized operator A, qA=qA1,qA2,⋯,qAn, and AMαqA=AMαqA1,AMαqA2,⋯,AMαqAn, then the generalized momentum pA=pA1,pA2,⋯,pAn is defined as pAi=∂LA/∂AMαqAi, i=1,2,⋯n, and the Hamiltonian HA=HAt,qA,pA can be expressed as HA=pAi⋅AMαqAi−LA. Therefore, the Hamilton action within generalized operator A is
(10)SAqA⋅=∫abpAi⋅AMαqAi−HAt,qA,pAdt.Then,(11)δSA=0,with the commutative conditions [38].
(12)δAMαqAi=AMαδqAi,i=1,2,⋯,n,and the boundary conditions
(13)qAa=qAa,qAb=qAb,is called Hamilton principle within generalized operator A, where δ means the isochronous variation.Using Equations (1), (8), and (12), we have
(14)δSA=∫abδpAi⋅AMαqAi+pAi⋅δAMαqAi−∂HA∂qAiδqAi−∂HA∂pAiδpAidt=∫ab−BM∗αpAi+∂HA∂qAi−mpAibκ1−αb,t+ωpAiaκ1−αt,aδqAi+AMαqAi−∂HA∂pAiδpAidt=0.Then, from the HamiltonianHA=pAi⋅AMαqAi−LA, the independence of δqAi, and the arbitrariness of the interval a,b, we get
(15)AMαqAi=∂HA∂pAi,BM∗αpAi=−∂HA∂qAi+mpAibκ1−αb,t−ωpAiaκ1−αt,a.Equation (15) is called the Hamilton equation within generalized operator A.Remark 2.
Letκαt,τ=t−τα−1/Γα; from Equation (15), we can get the Hamilton equations in terms of the left Riemann-Liouville fractional operator, the right Riemann-Liouville fractional operator, and the Riesz-Riemann-Liouville fractional operator by letting M=M1, M=M2, and M=M3, respectively.


## 3.2. Hamilton Equation within Generalized OperatorB

LetLB=LBt,qB,BMαqB be the Lagrangian within the generalized operator B, qB=qB1,qB2,⋯,qBn, and BMαqB=BMαqB1,BMαqB2,⋯,BMαqBn, then the generalized momentum pB=pB1,pB2,⋯,pBn is defined as pBi=∂LB/∂BMαqBi, i=1,2,⋯n, and the Hamiltonian HB=HBt,qB,pB can be expressed as HB=pBi⋅BMαqBi−LB. Therefore, the Hamilton action within the generalized operator B is
(16)SBqB⋅=∫abpBi⋅BMαqBi−HBt,qB,pBdt.Then,(17)δSB=0,with the commutative conditions [38]
(18)δBMαqBi=BMαδqBi,and the boundary conditions
(19)qBa=qBa,qBb=qBb,are called the Hamilton principle within generalized operator B, where δ means the isochronous variation.Using Equations (9), (17), and (19), we have
(20)δSB=∫abδpBi⋅BMαqBi+pBi⋅δBMαqBi−∂HB∂qBiδqBi−∂HB∂pBiδpBidt=∫abBMαqBi−∂HB∂pBiδpBi−AM∗αpBi+∂HB∂qBiδqBidt.Then, from the HamiltonianHB=pBi⋅BMαqBi−LB, the independence of δqBi, and the arbitrariness of the interval a,b, we get
(21)BMαqBi=∂HB∂pBi,AM∗αpBi=−∂HB∂qBi.Equation (21) is called the Hamilton equation within generalized operator B.Remark 3.
Equation (21) is consistent with the result obtained in Ref. [39]. However, the method in Ref. [39] is different.Remark 4.
Letκαt,τ=t−τα−1/Γα; from Equation (21), we can get the Hamilton equations in terms of the left Caputo fractional operator, the right Caputo fractional operator, and the Riesz-Caputo fractional operator by letting M=M1, M=M2, and M=M3, respectively.Remark 5.
In Remark2 and Remark 4, there are six cases in total. If we let α⟶1, then all of them reduce to the classical Hamilton equation, which can be found in Ref. [14].


## 4. Noether Theorems within Generalized Operators

Noether symmetry means the invariance of the Hamilton action. Noether symmetry leads to conserved quantity. Therefore, the change of the Hamilton action under the infinitesimal transformations will be studied. There are also two parts: one is the Noether symmetry and conserved quantity in terms of operatorA, and the other is in terms of operator B. We begin with the definition of conserved quantity.Definition 6.
A quantityI is called a conserved quantity if and only if the condition dI/dt=0 holds.


### 4.1. Noether Theorem within Generalized OperatorA

First, we give the infinitesimal transformations within generalized operatorA as
(22)t¯=t+Δt,q¯Ait¯=qAit+ΔqAi,p¯Ait¯=pAit+ΔpAi,i=1,2,⋯,nort¯=t+Δt,q¯At¯=qAt+ΔqA,p¯At¯=pAt+ΔpA.Expanding Equation (22), we have
(23)t¯=t+θAξA00t,qA,pA+οθA,q¯Ait¯=qAit+θAξAi0t,qA,pA+οθA,p¯Ait¯=pAit+θAηAi0t,qA,pA+οθA,i=1,2,⋯,n,where θA is an infinitesimal parameter and ξA00, ξAi0, and ηAi0 are called infinitesimal generators within the generalized operator A.Then, letΔSA be the linear part of S¯A−SA and neglecting the higher order of θA, we get
(24)ΔSA=∫a¯b¯p¯Ai⋅AM¯αq¯Ai−HAt¯,q¯A,p¯Adt¯−∫abpAi⋅AMαqAi−HAt,qA,pAdt=∫abpAi+ΔpAi⋅AMαqAi+AMαδqAi+Δt⋅ddtAMαqAi+ωΔbqAibddtκ1−αb,t−mΔaqAiaddtκ1−αt,a−HAt+Δt,qA+ΔqA,pA+ΔpA−pAiAMαqAi+HAt,qA,pAdt=θA∫abpAi⋅AMαξAi0−q̇AiξA00+pAi⋅ddtAMαqAi−∂HA∂tξA00+pAi⋅AMαqAi−HAξ̇A00−∂HA∂qAiξAi0+ωpAi⋅qAib⋅ξA00b,qAb,pAb⋅ddtκ1−αb,t−mpAi⋅qAia⋅ξA00a,qAa,pAa⋅ddtκ1−αt,adt,where
(25)AM¯αq¯Ai=AMαqAi+AMαδqAi+Δt⋅ddtAMαqAi+ωqAib⋅Δb⋅ddtκ1−αb,t−mqAia⋅Δa⋅ddtκ1−αt,a,M¯=<a¯,t¯,b¯,m,ω>.It follows from Noether symmetry (ΔSA=0) that
(26)pAi⋅AMαξAi0−q̇AiξA00+pAi⋅ddtAMαqAi−∂HA∂tξA00+pAi⋅AMαqAi−HAξ̇A00−∂HA∂qAiξAi0+ωpAi⋅qAib⋅ξA00b,qAb,pAb⋅ddtκ1−αb,t−mpAi⋅qAia⋅ξA00a,qAa,pAa⋅ddtκ1−αt,a=0,i=1,2,⋯,n.Equation (26) is called the Noether identity within the generalized operator A.Finally, the conserved quantities within generalized operators deduced by the Noether symmetry are presented.Theorem 7.
For the Hamiltonian system within generalized operatorA (Equation (15)), if the infinitesimal generators ξA00, ξAi0, and ηAi0 satisfy the Noether identity (Equation (26)), then there exists a conserved quantity
(27)IA0=pAi⋅AMαqAi−HAξA00+∫atpAi⋅AMαξAi0−q̇AiξA00+ξAi0−q̇AiξA00⋅BM∗αpAi−mpAibκ1−αb,τ+ωpAiaκ1−ατ,adτ+ω⋅qAib⋅ξA00b,qAb,pAb∫atpAiτ⋅ddτκ1−αb,τdτ−m⋅qAia⋅ξA00a,qAa,pAa∫atpAiτ⋅ddτκ1−ατ,adτ=const.Proof.
From Equations (15) and (26), we have dIA0/dt=0.If we letΔSA=−∫abd/dtΔGA0dt, ΔGA0=θAGA0t,qA,pA, and GA0 be called the gauge function, then from Equation (24), we have
(28)pAi⋅AMαξAi0−q̇AiξA00+pAi⋅ddtAMαqAi−∂HA∂tξA00+pAi⋅AMαqAi−HAξ̇A00−∂HA∂qAiξAi0+ωpAi⋅qAib⋅ξA00b,qAb,pAb⋅ddtκ1−αb,t−mpAi⋅qAia⋅ξA00a,qAa,pAa⋅ddtκ1−αt,a+ĠA0=0,i=1,2,⋯,n.Equation (28) is called the Noether quasi-identity within the generalized operator A.In this case, we have the following theorem.Theorem 8.
For the Hamiltonian system within the generalized operatorA (Equation (15)), if there exists a gauge function GA0 such that the infinitesimal generators ξA00, ξAi0, and ηAi0 satisfy the Noether quasi-identity (Equation (28)), then there exists a conserved quantity
(29)IAG0=pAi⋅AMαqAi−HAξA00+∫atpAi⋅AMαξAi0−q̇AiξA00+ξAi0−q̇AiξA00⋅BM∗αpAi−mpAibκ1−αb,τ+ωpAiaκ1−ατ,adτ+ω⋅qAib⋅ξA00b,qAb,pAb∫atpAiτ⋅ddτκ1−αb,τdτ+GA0−m⋅qAia⋅ξA00a,qAa,pAa∫atpAiτ⋅ddτκ1−ατ,adτ=const.Proof.
From Equations (15) and (28), we have dIAG0/dt=0.Remark 9.
Letκαt,τ=t−τα−1/Γα; from Equations (26) and (28), Theorem 7, and Theorem 8, we can get Noether identities, Noether quasi-identities, and Noether theorems in terms of the left Riemann-Liouville fractional operator, the right Riemann-Liouville fractional operator, and the Riesz-Riemann-Liouville fractional operator by letting M=M1, M=M2, and M=M3, respectively.


### 4.2. Noether Theorem within Generalized OperatorB

The infinitesimal transformations are(30)t¯=t+Δt,q¯Bit¯=qBit+ΔqBi,p¯Bit¯=pBit+ΔpBiort¯=t+Δt,q¯Bt¯=qBt+ΔqB,p¯Bt¯=pBt+ΔpB,and the expanded forms of the infinitesimal transformations are
(31)t¯=t+θBξB00t,qB,pB+οθB,q¯Bit¯=qBit+θBξBi0t,qB,pB+οθB,p¯Bit¯=pBit+θBηBi0t,qB,pB+οθB,where θB is an infinitesimal parameter and ξB00, ξBi0, and ηBi0 are called infinitesimal generators within the generalized operator B.Similarly, letΔSB be the linear part of S¯B−SB and neglecting the higher order of θB, we get
(32)ΔSB=S¯B−SB=∫a¯b¯p¯Bi⋅BM¯αq¯Bi−HBt¯,q¯B,p¯Bdt¯−∫abpBi⋅BMαqBi−HBt,qB,pBdt=∫abpBi+ΔpBi⋅BMαqBi+BMαδqBi+Δt⋅ddtBMαqBi+ωΔbκ1−αb,tq̇Bib−mΔa⋅κ1−αt,aq̇Bia−HBt+Δt,qB+ΔqB,pB+ΔpB−pBiBMαqBi+HBt,qB,pBdt=θB∫abpBi⋅BMαξBi0−q̇BiξB00+pBi⋅ddtBMαqBi−∂HB∂tξB00+pBi⋅BMαqBi−HBξ̇B00−∂HB∂qBiξBi0+ωκ1−αb,tq̇Bib⋅pBiξB00b,qBb,pBb−mκ1−αt,aq̇BiapBiξB00a,qBa,pBadt.where
(33)BM¯αq¯Bi=BMαqBi+BMαδqBi+Δt⋅ddtBMαqBi+ωκ1−αb,tq̇BibΔb−mκ1−αt,aq̇BiaΔa.LettingΔSB=0, we have
(34)pBi⋅BMαξBi0−q̇BiξB00+pBi⋅ddtBMαqBi−∂HB∂tξB00+pBi⋅BMαqBi−HBξ̇B00−∂HB∂qBiξBi0+ωκ1−αb,tq̇Bib⋅pBiξB00b,qBb,pBb−mκ1−αt,aq̇BiapBiξB00a,qBa,pBa=0,i=1,2,⋯,n.Equation (34) is called the Noether identity within the generalized operator B.LetΔSB=−∫abd/dtΔGB0dt, ΔGB0=θBGB0t,qB,pB, and GB0 be called the gauge function, then from Equation (32), we have
(35)pBi⋅BMαξBi0−q̇BiξB00+pBi⋅ddtBMαqBi−∂HB∂tξB00+pBi⋅BMαqBi−HBξ̇B00−∂HB∂qBiξBi0+ωκ1−αb,tq̇Bib⋅pBiξB00b,qBb,pBb−mκ1−αt,aq̇BiapBiξB00a,qBa,pBa+ĠB0=0,i=1,2,⋯,n.Equation (35) is called the Noether quasi-identity within the generalized operator B.Therefore, we have the following theorem.Theorem 10.
For the Hamiltonian system within the generalized operatorB (Equation (21)), if the infinitesimal generators ξB00, ξBi0, and ηBi0 satisfy the Noether identity (Equation (34)), then there exists a conserved quantity
(36)IB0=pBi⋅BMαqBi−HBξB00+∫atpBi⋅BMαξBi0−q̇BiξB00+ξBi0−q̇BiξB00AM∗αpBidτ+ω⋅q̇Bib⋅ξB00b,qBb,pBb∫atpBiτκ1−αb,τdτ−m⋅q̇Bia⋅ξB00a,qBa,pBa∫atpBiτκ1−ατ,adτ=const.Proof.
From Equations (21) and (34), we have dIB0/dt=0.Theorem 11.
For the Hamiltonian system within the generalized operatorB (Equation (21)), if there exists a gauge function GB0 such that the infinitesimal generators ξB00, ξBi0, and ηBi0 satisfy the Noether quasi-identity (Equation (35)), then there exists a conserved quantity
(37)IBG0=pBi⋅BMαqBi−HBξB00+∫atpBi⋅BMαξBi0−q̇BiξB00+ξBi0−q̇BiξB00AM∗αpBidτ+ω⋅q̇Bib⋅ξB00b,qBb,pBb∫atpBiτκ1−αb,τdτ−m⋅q̇Bia⋅ξB00a,qBa,pBa∫atpBiτκ1−ατ,adτ+GB0=const.Proof.
From Equations (21) and (35), we have dIBG0/dt=0.Remark 12.
Letκαt,τ=t−τα−1/Γα; from Equations (34) and (35), Theorem 10, and Theorem 11, we can get Noether identities, Noether quasi-identities, and Noether theorems in terms of the left Caputo fractional operator, the right Caputo fractional operator, and the Riesz-Caputo fractional operator by letting M=M1, M=M2, and M=M3, respectively. Particularly, the results obtained when M=M3 are consistent with the results presented in Ref. [33].Remark 13.
In Remark9 and Remark 12, there are six cases in total. If we let α⟶1, then all of them reduce to the classical Noether identity, the classical Noether quasi-identity, and the classical Noether theorem, which can be found in Ref. [14].


## 4.1. Noether Theorem within Generalized OperatorA

First, we give the infinitesimal transformations within generalized operatorA as
(22)t¯=t+Δt,q¯Ait¯=qAit+ΔqAi,p¯Ait¯=pAit+ΔpAi,i=1,2,⋯,nort¯=t+Δt,q¯At¯=qAt+ΔqA,p¯At¯=pAt+ΔpA.Expanding Equation (22), we have
(23)t¯=t+θAξA00t,qA,pA+οθA,q¯Ait¯=qAit+θAξAi0t,qA,pA+οθA,p¯Ait¯=pAit+θAηAi0t,qA,pA+οθA,i=1,2,⋯,n,where θA is an infinitesimal parameter and ξA00, ξAi0, and ηAi0 are called infinitesimal generators within the generalized operator A.Then, letΔSA be the linear part of S¯A−SA and neglecting the higher order of θA, we get
(24)ΔSA=∫a¯b¯p¯Ai⋅AM¯αq¯Ai−HAt¯,q¯A,p¯Adt¯−∫abpAi⋅AMαqAi−HAt,qA,pAdt=∫abpAi+ΔpAi⋅AMαqAi+AMαδqAi+Δt⋅ddtAMαqAi+ωΔbqAibddtκ1−αb,t−mΔaqAiaddtκ1−αt,a−HAt+Δt,qA+ΔqA,pA+ΔpA−pAiAMαqAi+HAt,qA,pAdt=θA∫abpAi⋅AMαξAi0−q̇AiξA00+pAi⋅ddtAMαqAi−∂HA∂tξA00+pAi⋅AMαqAi−HAξ̇A00−∂HA∂qAiξAi0+ωpAi⋅qAib⋅ξA00b,qAb,pAb⋅ddtκ1−αb,t−mpAi⋅qAia⋅ξA00a,qAa,pAa⋅ddtκ1−αt,adt,where
(25)AM¯αq¯Ai=AMαqAi+AMαδqAi+Δt⋅ddtAMαqAi+ωqAib⋅Δb⋅ddtκ1−αb,t−mqAia⋅Δa⋅ddtκ1−αt,a,M¯=<a¯,t¯,b¯,m,ω>.It follows from Noether symmetry (ΔSA=0) that
(26)pAi⋅AMαξAi0−q̇AiξA00+pAi⋅ddtAMαqAi−∂HA∂tξA00+pAi⋅AMαqAi−HAξ̇A00−∂HA∂qAiξAi0+ωpAi⋅qAib⋅ξA00b,qAb,pAb⋅ddtκ1−αb,t−mpAi⋅qAia⋅ξA00a,qAa,pAa⋅ddtκ1−αt,a=0,i=1,2,⋯,n.Equation (26) is called the Noether identity within the generalized operator A.Finally, the conserved quantities within generalized operators deduced by the Noether symmetry are presented.Theorem 7.
For the Hamiltonian system within generalized operatorA (Equation (15)), if the infinitesimal generators ξA00, ξAi0, and ηAi0 satisfy the Noether identity (Equation (26)), then there exists a conserved quantity
(27)IA0=pAi⋅AMαqAi−HAξA00+∫atpAi⋅AMαξAi0−q̇AiξA00+ξAi0−q̇AiξA00⋅BM∗αpAi−mpAibκ1−αb,τ+ωpAiaκ1−ατ,adτ+ω⋅qAib⋅ξA00b,qAb,pAb∫atpAiτ⋅ddτκ1−αb,τdτ−m⋅qAia⋅ξA00a,qAa,pAa∫atpAiτ⋅ddτκ1−ατ,adτ=const.Proof.
From Equations (15) and (26), we have dIA0/dt=0.If we letΔSA=−∫abd/dtΔGA0dt, ΔGA0=θAGA0t,qA,pA, and GA0 be called the gauge function, then from Equation (24), we have
(28)pAi⋅AMαξAi0−q̇AiξA00+pAi⋅ddtAMαqAi−∂HA∂tξA00+pAi⋅AMαqAi−HAξ̇A00−∂HA∂qAiξAi0+ωpAi⋅qAib⋅ξA00b,qAb,pAb⋅ddtκ1−αb,t−mpAi⋅qAia⋅ξA00a,qAa,pAa⋅ddtκ1−αt,a+ĠA0=0,i=1,2,⋯,n.Equation (28) is called the Noether quasi-identity within the generalized operator A.In this case, we have the following theorem.Theorem 8.
For the Hamiltonian system within the generalized operatorA (Equation (15)), if there exists a gauge function GA0 such that the infinitesimal generators ξA00, ξAi0, and ηAi0 satisfy the Noether quasi-identity (Equation (28)), then there exists a conserved quantity
(29)IAG0=pAi⋅AMαqAi−HAξA00+∫atpAi⋅AMαξAi0−q̇AiξA00+ξAi0−q̇AiξA00⋅BM∗αpAi−mpAibκ1−αb,τ+ωpAiaκ1−ατ,adτ+ω⋅qAib⋅ξA00b,qAb,pAb∫atpAiτ⋅ddτκ1−αb,τdτ+GA0−m⋅qAia⋅ξA00a,qAa,pAa∫atpAiτ⋅ddτκ1−ατ,adτ=const.Proof.
From Equations (15) and (28), we have dIAG0/dt=0.Remark 9.
Letκαt,τ=t−τα−1/Γα; from Equations (26) and (28), Theorem 7, and Theorem 8, we can get Noether identities, Noether quasi-identities, and Noether theorems in terms of the left Riemann-Liouville fractional operator, the right Riemann-Liouville fractional operator, and the Riesz-Riemann-Liouville fractional operator by letting M=M1, M=M2, and M=M3, respectively.


## 4.2. Noether Theorem within Generalized OperatorB

The infinitesimal transformations are(30)t¯=t+Δt,q¯Bit¯=qBit+ΔqBi,p¯Bit¯=pBit+ΔpBiort¯=t+Δt,q¯Bt¯=qBt+ΔqB,p¯Bt¯=pBt+ΔpB,and the expanded forms of the infinitesimal transformations are
(31)t¯=t+θBξB00t,qB,pB+οθB,q¯Bit¯=qBit+θBξBi0t,qB,pB+οθB,p¯Bit¯=pBit+θBηBi0t,qB,pB+οθB,where θB is an infinitesimal parameter and ξB00, ξBi0, and ηBi0 are called infinitesimal generators within the generalized operator B.Similarly, letΔSB be the linear part of S¯B−SB and neglecting the higher order of θB, we get
(32)ΔSB=S¯B−SB=∫a¯b¯p¯Bi⋅BM¯αq¯Bi−HBt¯,q¯B,p¯Bdt¯−∫abpBi⋅BMαqBi−HBt,qB,pBdt=∫abpBi+ΔpBi⋅BMαqBi+BMαδqBi+Δt⋅ddtBMαqBi+ωΔbκ1−αb,tq̇Bib−mΔa⋅κ1−αt,aq̇Bia−HBt+Δt,qB+ΔqB,pB+ΔpB−pBiBMαqBi+HBt,qB,pBdt=θB∫abpBi⋅BMαξBi0−q̇BiξB00+pBi⋅ddtBMαqBi−∂HB∂tξB00+pBi⋅BMαqBi−HBξ̇B00−∂HB∂qBiξBi0+ωκ1−αb,tq̇Bib⋅pBiξB00b,qBb,pBb−mκ1−αt,aq̇BiapBiξB00a,qBa,pBadt.where
(33)BM¯αq¯Bi=BMαqBi+BMαδqBi+Δt⋅ddtBMαqBi+ωκ1−αb,tq̇BibΔb−mκ1−αt,aq̇BiaΔa.LettingΔSB=0, we have
(34)pBi⋅BMαξBi0−q̇BiξB00+pBi⋅ddtBMαqBi−∂HB∂tξB00+pBi⋅BMαqBi−HBξ̇B00−∂HB∂qBiξBi0+ωκ1−αb,tq̇Bib⋅pBiξB00b,qBb,pBb−mκ1−αt,aq̇BiapBiξB00a,qBa,pBa=0,i=1,2,⋯,n.Equation (34) is called the Noether identity within the generalized operator B.LetΔSB=−∫abd/dtΔGB0dt, ΔGB0=θBGB0t,qB,pB, and GB0 be called the gauge function, then from Equation (32), we have
(35)pBi⋅BMαξBi0−q̇BiξB00+pBi⋅ddtBMαqBi−∂HB∂tξB00+pBi⋅BMαqBi−HBξ̇B00−∂HB∂qBiξBi0+ωκ1−αb,tq̇Bib⋅pBiξB00b,qBb,pBb−mκ1−αt,aq̇BiapBiξB00a,qBa,pBa+ĠB0=0,i=1,2,⋯,n.Equation (35) is called the Noether quasi-identity within the generalized operator B.Therefore, we have the following theorem.Theorem 10.
For the Hamiltonian system within the generalized operatorB (Equation (21)), if the infinitesimal generators ξB00, ξBi0, and ηBi0 satisfy the Noether identity (Equation (34)), then there exists a conserved quantity
(36)IB0=pBi⋅BMαqBi−HBξB00+∫atpBi⋅BMαξBi0−q̇BiξB00+ξBi0−q̇BiξB00AM∗αpBidτ+ω⋅q̇Bib⋅ξB00b,qBb,pBb∫atpBiτκ1−αb,τdτ−m⋅q̇Bia⋅ξB00a,qBa,pBa∫atpBiτκ1−ατ,adτ=const.Proof.
From Equations (21) and (34), we have dIB0/dt=0.Theorem 11.
For the Hamiltonian system within the generalized operatorB (Equation (21)), if there exists a gauge function GB0 such that the infinitesimal generators ξB00, ξBi0, and ηBi0 satisfy the Noether quasi-identity (Equation (35)), then there exists a conserved quantity
(37)IBG0=pBi⋅BMαqBi−HBξB00+∫atpBi⋅BMαξBi0−q̇BiξB00+ξBi0−q̇BiξB00AM∗αpBidτ+ω⋅q̇Bib⋅ξB00b,qBb,pBb∫atpBiτκ1−αb,τdτ−m⋅q̇Bia⋅ξB00a,qBa,pBa∫atpBiτκ1−ατ,adτ+GB0=const.Proof.
From Equations (21) and (35), we have dIBG0/dt=0.Remark 12.
Letκαt,τ=t−τα−1/Γα; from Equations (34) and (35), Theorem 10, and Theorem 11, we can get Noether identities, Noether quasi-identities, and Noether theorems in terms of the left Caputo fractional operator, the right Caputo fractional operator, and the Riesz-Caputo fractional operator by letting M=M1, M=M2, and M=M3, respectively. Particularly, the results obtained when M=M3 are consistent with the results presented in Ref. [33].Remark 13.
In Remark9 and Remark 12, there are six cases in total. If we let α⟶1, then all of them reduce to the classical Noether identity, the classical Noether quasi-identity, and the classical Noether theorem, which can be found in Ref. [14].


## 5. Adiabatic Invariants within Generalized Operators

This section begins with the definition of the adiabatic invariant.Definition 14.
A quantityIz is called an adiabatic invariant if Iz contains a parameter ε, whose highest power is z, and also satisfies that dIz/dt is in proportion to εz+1.When the systems (Equations (15) and (21)) are disturbed by small forces, the conserved quantities may also change.Assuming that the Hamiltonian system within the generalized operatorA (Equation (15)), the gauge function GA and the infinitesimal generators ξA0, ξAi, and ηAi are disturbed as
(38)BM∗αpAi=−∂HA∂qAi+mpAibκ1−αb,t−ωpAiaκ1−αt,a−εAWAit,qA,pA,AMαqAi=∂HA∂pAi,i=1,2,⋯,n,(39)GA=GA0+εAGA1+εA2GA2+⋯=εAsGAs,ξA0=ξA00+εAξA01+εA2ξA02+⋯=εAsξA0s,ξAi=ξAi0+εAξAi1+εA2ξAi2+⋯=εAsξAis,ηAi=ηAi0+εAηAi1+εA2ηAi2+⋯=εAsηAis,s=0,1,2,⋯,then we have the following theorem.Theorem 15.
For the disturbed Hamiltonian system within the generalized operatorA (Equation (38)), if there exists a gauge function GAs such that the infinitesimal generators ξA0s, ξAis, and ηAis satisfy
(40)pAi⋅AMαξAis−q̇AiξA0s+pAi⋅ddtAMαqAi−∂HA∂tξA0s+pAi⋅AMαqAi−HAξ̇A0s−∂HA∂qAiξAis+ωpAi⋅qAib⋅ξA0sb,qAb,pAb⋅ddtκ1−αb,t−WAiξAis−1−q̇AiξA0s−1−mpAi⋅qAia⋅ξA0sa,qAa,pAa⋅ddtκ1−αt,a+ĠAs=0,where ξAis−1=ξA0s−1=0 when s=0, then there exists an adiabatic invariant
(41)IAGz=∑s=0zεAspAi⋅AMαqAi−HAξA0s+∫atpAi⋅AMαξAis−q̇AiξA0s+ξAis−q̇AiξA0s⋅BM∗αpAi−mpAibκ1−αb,τ+ωpAiaκ1−ατ,adτ+ω⋅qAib⋅ξA0sb,qAb,pAb∫atpAiτ⋅ddτκ1−αb,τdτ+GAs−m⋅qAia⋅ξA0sa,qAa,pAa∫atpAiτ⋅ddτκ1−ατ,adτ.Proof.
From Equations (38) and (40), we have d/dtIAGz=−εAz+1WAiξAiz−q̇AiξA0z.If the Hamiltonian system within the generalized operatorB (Equation (21)), the gauge function GB and the infinitesimal generators ξB0, ξBi, and ηBi are disturbed as
(42)BMαqBi=∂HB∂pBi,AM∗αpBi=−∂HB∂qBi−εBWBit,qB,pB,i=1,2,⋯,n,(43)GB=GB0+εBGB1+εB2GB2+⋯=εBsGBs,ξB0=ξB00+εBξB01+εB2ξB02+⋯=εBsξB0s,(44)ξBi=ξBi0+εBξBi1+εB2ξBi2+⋯=εBsξBis,ηBi=ηBi0+εBηBi1+εB2ηBi2+⋯=εBsηBis,(45)s=0,1,2,⋯,then we have the following theorem.Theorem 16.
For the disturbed Hamiltonian system within the generalized operatorB (Equation (42)), if there exists a gauge function GBs such that the infinitesimal generators ξB0s, ξBis, and ηBis satisfy
(46)pBi⋅BMαξBis−q̇BiξB0s+pBi⋅ddtBMαqBi−∂HB∂tξB0s+pBi⋅BMαqBi−HBξ̇B0s−∂HB∂qBiξBis+ωκ1−αb,tq̇Bib⋅pBiξB0sb,qBb,pBb−WBiξBis−1−q̇BiξB0s−1−mκ1−αt,aq̇BiapBiξB0sa,qBa,pBa+ĠBs=0,
whereξBis−1=ξB0s−1=0 when s=0, then there exists an adiabatic invariant
(47)IBGz=∑s=0zεBspBi⋅BMαqBi−HBξB0s+∫atpBi⋅BMαξBis−q̇BiξB0s+ξBis−q̇BiξB0s⋅AM∗αpBidτ+ω⋅q̇Bib⋅ξB0sb,qBb,pBb∫atpBiτκ1−αb,τdτ−m⋅q̇Bia⋅ξB0sa,qBa,pBa∫atpBiτκ1−ατ,adτ+GBs.Proof.
From Equations (42) and (46), we have d/dtIBGz=−εBz+1WBiξBiz−q̇BiξB0z.Remark 17.
Lettingκαt,τ=t−τα−1/Γα, we can get the adiabatic invariants in terms of the left Riemann-Liouville fractional operator, the right Riemann-Liouville fractional operator, and the Riesz-Riemann-Liouville fractional operator from Theorem 15 by letting M=M1, M=M2, and M=M3, respectively. Similarly, we can get the adiabatic invariants in terms of the left Caputo fractional operator, the right Caputo fractional operator, and the Riesz-Caputo fractional operator by letting M=M1, M=M2, and M=M3, respectively. Particularly, the adiabatic invariant within the Riesz-Caputo fractional operator is consistent with the result obtained in Ref. [33].Remark 18.
In Remark17, there are six cases in total. If we let α⟶1, then all of them reduce to the classical adiabatic invariant, which can be found in Ref. [33].Remark 19.
The adiabatic invariants of Theorem15 and Theorem 16 reduce to conserved quantities of Theorem 8 and Theorem 11 when z=0, respectively.In the following text, we give two applications to illustrate the results and methods.


## 6. Applications

### 6.1. Application 1

We try to find the conserved quantity and adiabatic invariant for the two-dimensional isotropic harmonic oscillator within the generalized operatorA, whose Lagrangian has the form
(48)LA=12αAAMαqA12+AMαqA22−12βAqA12+qA22,where αA and βA are constants.From Equation (48), we have
(49)pA1=αAAMαqA1,pA2=αAAMαqA2,HA=pA122αA+pA222αA+12βAqA12+qA22.Then, Equation (15) gives
(50)AMαqA1=pA1αA,AMαqA2=pA2αA,BM∗αpA1=−βAqA1+mpA1bκ1−αb,t−ωpA1aκ1−αt,a,BM∗αpA2=−βAqA2+mpA2bκ1−αb,t−ωpA2aκ1−αt,a.And under the conditiond/dtκt,τ=−d/dτκt,τ, we can verify that
(51)ξA00=1,ξA10=ξA20=0,GA0=0,satisfy the Noether quasi-identity (Equation (28)). Therefore, from Theorem 8, we have
(52)IAG0=pA1⋅AMαqA1+pA2⋅AMαqA2−HA−∫atpA1⋅ddτAMαqA1+q̇A1⋅BM∗αpA1−mpA1bκ1−αb,τ+ωpA1aκ1−ατ,a+pA2⋅ddτAMαqA2+q̇A2BM∗αpA2−mpA2bκ1−αb,τ+ωpA2aκ1−ατ,adτ=const.When the system is disturbed by−εAWA1t,qA,pA=−εAqA2 and −εAWA2t,qA,pA=−εAqA1, then we can find that
(53)ξA01=1,ξA11=ξA21=0,GA1=−qA1qA2,is a solution to Equation (40). Therefore, from Theorem 15, we obtain
(54)IAG1=IAG0+εAIAG0−qA1qA2.Specially, letκαt,τ=t−τα−1/Γα and M=M1 (or M=M2 or M=M3); further, letting α⟶1, we have
(55)IAG0C=−HA=const,IAG1C=−HA−εAHA+qA1qA2.


### 6.2. Application 2

The Lotka biochemical oscillator model’s Hamiltonian has the form(56)HB=αB1pB1−αB2qB1+βB1exppB1−βB2expqB1.We try to find its conserved quantity and adiabatic invariant based on generalized operatorB.Equation (21) gives the Hamilton equation within the generalized operator B(57)BMαqB1=αB1+βB1exppB1,BMαqB2=0,AM∗αpB1=αB2+βB2expqB1,AM∗αpB2=0.Then, under the conditiond/dtκt,τ=−d/dτκt,τ, it is easy to verify that
(58)ξB00=1,ξB10=0,GB0=0,is a solution to the Noether quasi-identity (Equation (35)). Therefore, from Theorem 11, we have
(59)IBG0=pB1⋅BMαqB1−HB−∫atq̇B1AM∗αpB1+pB1⋅ddτBMαqB1dτ=const.When the system is disturbed by−εBWB1t,qB,pB=−εB2qB1+1 and −εBWB2t,qB,pB=0, then we can find that
(60)ξB01=1,ξB11=0,GB1=−qB12−qB1,is a solution to Equation (46). Therefore, from Theorem 16, we have
(61)IBG1=pB1⋅BMαqB1−HB−∫atq̇B1AM∗αpB1+pB1⋅ddτBMαqB1dτ+εBpB1⋅BMαqB1−HB−∫atq̇B1AM∗αpB1+pB1⋅ddτBMαqB1dτ−qB12−qB1.Specially, letκαt,τ=t−τα−1/Γα and M=M1 (or M=M2 or M=M3); further, letting α⟶1, we have
(62)IBG0C=−HB=const,IBG1C=−HB−εBHB+qB12+qB1.


## 6.1. Application 1

We try to find the conserved quantity and adiabatic invariant for the two-dimensional isotropic harmonic oscillator within the generalized operatorA, whose Lagrangian has the form
(48)LA=12αAAMαqA12+AMαqA22−12βAqA12+qA22,where αA and βA are constants.From Equation (48), we have
(49)pA1=αAAMαqA1,pA2=αAAMαqA2,HA=pA122αA+pA222αA+12βAqA12+qA22.Then, Equation (15) gives
(50)AMαqA1=pA1αA,AMαqA2=pA2αA,BM∗αpA1=−βAqA1+mpA1bκ1−αb,t−ωpA1aκ1−αt,a,BM∗αpA2=−βAqA2+mpA2bκ1−αb,t−ωpA2aκ1−αt,a.And under the conditiond/dtκt,τ=−d/dτκt,τ, we can verify that
(51)ξA00=1,ξA10=ξA20=0,GA0=0,satisfy the Noether quasi-identity (Equation (28)). Therefore, from Theorem 8, we have
(52)IAG0=pA1⋅AMαqA1+pA2⋅AMαqA2−HA−∫atpA1⋅ddτAMαqA1+q̇A1⋅BM∗αpA1−mpA1bκ1−αb,τ+ωpA1aκ1−ατ,a+pA2⋅ddτAMαqA2+q̇A2BM∗αpA2−mpA2bκ1−αb,τ+ωpA2aκ1−ατ,adτ=const.When the system is disturbed by−εAWA1t,qA,pA=−εAqA2 and −εAWA2t,qA,pA=−εAqA1, then we can find that
(53)ξA01=1,ξA11=ξA21=0,GA1=−qA1qA2,is a solution to Equation (40). Therefore, from Theorem 15, we obtain
(54)IAG1=IAG0+εAIAG0−qA1qA2.Specially, letκαt,τ=t−τα−1/Γα and M=M1 (or M=M2 or M=M3); further, letting α⟶1, we have
(55)IAG0C=−HA=const,IAG1C=−HA−εAHA+qA1qA2.


## 6.2. Application 2

The Lotka biochemical oscillator model’s Hamiltonian has the form(56)HB=αB1pB1−αB2qB1+βB1exppB1−βB2expqB1.We try to find its conserved quantity and adiabatic invariant based on generalized operatorB.Equation (21) gives the Hamilton equation within the generalized operator B(57)BMαqB1=αB1+βB1exppB1,BMαqB2=0,AM∗αpB1=αB2+βB2expqB1,AM∗αpB2=0.Then, under the conditiond/dtκt,τ=−d/dτκt,τ, it is easy to verify that
(58)ξB00=1,ξB10=0,GB0=0,is a solution to the Noether quasi-identity (Equation (35)). Therefore, from Theorem 11, we have
(59)IBG0=pB1⋅BMαqB1−HB−∫atq̇B1AM∗αpB1+pB1⋅ddτBMαqB1dτ=const.When the system is disturbed by−εBWB1t,qB,pB=−εB2qB1+1 and −εBWB2t,qB,pB=0, then we can find that
(60)ξB01=1,ξB11=0,GB1=−qB12−qB1,is a solution to Equation (46). Therefore, from Theorem 16, we have
(61)IBG1=pB1⋅BMαqB1−HB−∫atq̇B1AM∗αpB1+pB1⋅ddτBMαqB1dτ+εBpB1⋅BMαqB1−HB−∫atq̇B1AM∗αpB1+pB1⋅ddτBMαqB1dτ−qB12−qB1.Specially, letκαt,τ=t−τα−1/Γα and M=M1 (or M=M2 or M=M3); further, letting α⟶1, we have
(62)IBG0C=−HB=const,IBG1C=−HB−εBHB+qB12+qB1.


## 7. Results and Discussion

Hamilton equations, Noether theorems, and adiabatic invariants are obtained for the Hamiltonian systems on the basis of the generalized operators. The Hamilton equation in terms of the generalized operatorA (Equation (15)), the Hamilton equation in terms of the generalized operator B (Equation (21)), the Noether symmetries and conserved quantities in terms of the generalized operator A (Theorem 7 and Theorem 8), the Noether symmetries and conserved quantities in terms of the generalized operator B (Theorem 10 and Theorem 11), the perturbation to Noether symmetries and adiabatic invariants in terms of the generalized fractional operators (Theorem 15 and Theorem 16) are all new work.However, we still need to make efforts to finish the follow-up work successfully. For example, MATLAB is a useful tool to simulate the results of the applications to directly illustrate the achievements obtained in this paper. Therefore, the appliance of MATLAB is necessary. Besides, except for the Noether symmetry method, the Lie symmetry method, and the Mei symmetry method are also significant methods to be studied for the Hamiltonian systems within generalized operators. Time scale calculus is recently proposed with tremendous potential for applications. The Hamiltonian mechanics within mixed time scale and generalized operators is a new research direction, which also deserves to be done.



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