Conserved Quantity and Adiabatic Invariant for Hamiltonian System with Variable Order

S S symmetry

Article Conserved Quantity and Adiabatic Invariant for Hamiltonian System with Variable Order

Chuan-Jing Song * and Yao Cheng School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, China; [email protected] * Correspondence: [email protected]

Received: 26 August 2019; Accepted: 6 October 2019; Published: 10 October 2019

Abstract: Hamiltonian mechanics plays an important role in the development of nonlinear science. This paper aims for a fractional Hamiltonian system of variable order. Several issues are discussed, including diﬀerential equation of motion, Noether symmetry, and perturbation to Noether symmetry. As a result, fractional Hamiltonian mechanics of variable order are established, and conserved quantity and adiabatic invariant are presented. Two applications, fractional isotropic harmonic oscillator model of variable order and fractional Lotka biochemical oscillator model of variable order are given to illustrate the Methods and Results.

Keywords: Hamiltonian system; fractional derivative of variable order; Noether symmetry; conserved quantity; adiabatic invariant

1. Introduction Fractional calculus was firstly proposed because the order of a function’s integral or derivative is a non-integer constant. In fact, fractional calculus started more than 300 years ago when L’Hopital and Leibniz were discussing the meaning of d1/2 y/dx1/2. The development of fractional calculus is much slower than that of integer calculus though many famous mathematicians, such as Fourier, Euler, Riemann, Liouville, Letnikov, Grunwald, etc., have contributed to it (see [1–3] for the history of fractional calculus). In recent decades, because of fractional calculus’s many applications in various fields of science, engineering, biomechanics, economics, and so forth, it has gained more attention (see [4–7] for a review). For example, fractional calculus has been applied to physics [8], quantum mechanics [9–11], field theory [12,13], etc. Among several definitions of the fractional derivatives, which are generally nonlocal operators and are historically applied to study time-dependent or nonlocal processes, Riemann–Liouville fractional derivative and Caputo fractional derivative are the most famous ones. Furthermore, Samko and Ross [14] introduced a generalization of fractional calculus. They considered the order of a function’s integral or derivative as α( , ), where α( , ) is a function · · · · rather than a constant. Because of the nonlocality and memory of fractional order calculus, it is reasonable that the order of a function’s integral or derivative may vary. Afterwards, several works were dedicated to the properties and applications of the fractional operators with variable order [15–20]. For instance, Coimbra [17] examined dynamical behaviors with frictional force using the variable order model. Diaz and Coimbra [18] utilized the differential equation with variable order to reveal some dynamical control behaviors of the nonlinear oscillator. Sun et al. [20] investigated a class of fractional models with variable order. Nowadays, more and more fractional models with variable order have been applied to mechanics, mathematics, and other related subjects (see [21–24] and references therein). In 1996, Riewe started the theory of the fractional calculus of variations by considering that the first-order derivative terms should come from the half-order derivative terms [25,26]. From then on,

Symmetry 2019, 11, 1270; doi:10.3390/sym11101270 www.mdpi.com/journal/symmetry Symmetry 2019, 11, 1270 2 of 23 scholars began to use fractional derivatives to describe the dissipative forces such as frictional forces for the nonconservative systems. Then, fractional calculus of variations was studied by several scholars such as Klimek [27], Agrawal [28,29], Baleanu et al. [30,31], Torres [32], etc. Moreover, the development of the fractional calculus of variations with variable order also made great progress. In [33–35], several results for the fractional calculus of variations with variable order were obtained. Particularly, motivated by references [36–38], a linear combination of the fractional derivative of variable order was introduced. Based on a combined Caputo fractional derivative of variable order (CCVO), the necessary optimality conditions for variational problems were established [39,40], the fractional variational problem of Herglotz type with variable order was studied [41], and two fractional isoperimetric problems and a new variational problem subject to a holonomic constraint were presented [42]. After differential equations of motion are established through the calculus of variations, the next task is to find the solutions to them in dynamics. In fact, the solution to the equations can be given if one can find all of the integrals of the equations. An integral is a conserved quantity; therefore, people try their best to find all of the conserved quantities of a mechanical system. By means of the analysis of forces, Newtonian mechanics gives three conservation laws, i.e., the conservation of momentum, the conservation of mechanical energy, and the conservation of moment of momentum. By means of the analysis of the form of Lagrangian, Lagrangian mechanics gives two conservation laws, i.e., the conservation of generalized energy and the conservation of generalized momentum. The conservation of generalized momentum may be a conservation of momentum, a conservation of moment of momentum, or neither. The physical meaning of the conservation law in Lagrangian mechanics is less clear than that in Newtonian mechanics, whose three conservation laws have very clear physical meaning. However, the conserved quantities deduced by Lagrangian mechanics are more than those deduced by Newtonian mechanics. Since German mathematician Emmy Noether published her famous paper [43], the Noether symmetry method has become a modern method for seeking the conservation law of mechanical systems (see [44] for a review). Similarly, although the physical meaning of the conservation law obtained from the Noether symmetry method is less than that obtained from Lagrangian mechanics, the numbers of the conserved quantities deduced by the Noether symmetry method are more than those deduced by Lagrangian mechanics. Recently, fractional Noether symmetry and fractional conserved quantity are also under strong research. There are two different definitions of fractional conserved quantity. One was introduced by Frederico and Torres [45] by means of a bilinear fractional operator D (D(C) = 0), and the other was introduced by Atanacković[46] by means of the classical definition (dC/dt = 0), where C means a fractional conserved quantity. Fractional Noether symmetry and fractional conserved quantity have been investigated through both definitions. For example, works [47,48] were done based on the former definition, and results [49–53] were obtained on the basis of the latter one. Furthermore, the study of the fractional Noether symmetry and fractional conserved quantity with variable order has also begun. For instance, Odzijewicz, Malinowska, and Torres [54,55] investigated Noether theorem and the second Noether theorem for the fractional Lagrangian system with variable order. Yan and Zhang [56,57] studied Noether symmetry and conserved quantity for the fractional Birkhoffian system in terms of Caputo fractional derivative of variable order (CVO), Riemann–Liouville fractional derivative of variable order (RLVO), Riesz–Caputo fractional derivative of variable order (RCVO), and Riesz–Riemann–Liouville fractional derivative of variable order (RVO). When a dynamical system is disturbed by small forces, the conserved quantity, which refers to the integrability of the system, may also change. Therefore, the research on the perturbation to Noether symmetry and adiabatic invariants is also of great significance for a dynamical system [58,59]. A classical adiabatic invariant means a certain physical quantity that changes more slowly than the parameter that varies very slowly. That is, when a dynamical system is disturbed by small forces and varies very slowly, a slower physical quantity, i.e., adiabatic invariant, will be found. Some important results on the study of perturbation to Noether symmetry and adiabatic invariants for constrained mechanical systems have been obtained [60,61]. Very recently, we studied the perturbation to fractional Symmetry 2019, 11, 1270 3 of 23

Noether symmetry for the Hamiltonian system [53] as well as the Birkhoffian system [62]. Besides, the results on perturbation to Noether symmetry with RLVO for the fractional generalized Birkhoffian system [63] were also presented. The purpose of this paper is to generalize the problem of the calculus of variations, Noether theory, and perturbation to Noether symmetry to the fractional Hamiltonian systems in terms of combined Riemann–Liouville fractional derivatives of variable order (CRLVO) and combined Caputo fractional derivatives of variable order (CCVO), respectively. Here, the CRLVO and the CCVO are the first time to be introduced, and many results obtained before are the special cases of this paper. As the main results, Hamilton equations with the CRLVO and the CCVO are established. Then, Noether symmetry and conserved quantities for the fractional Hamiltonian systems with the CRLVO and the CCVO are presented. Perturbation to Noether symmetry and adiabatic invariants for the disturbed fractional Hamiltonian systems with the CRLVO and the CCVO are also investigated, and two applications are given to illustrate the Methods and Results of this paper. In a word, there are mainly three factors for presenting this paper. Firstly, analytical mechanics. Lagrange was the first man to publish the famous book Analytical Mechanics. Lagrange felt proud when he successfully described mechanics problems mathematically without any diagrams because within the framework of Newtonian mechanics, mechanical phenomena are treated mostly schematically. In fact, the representation by means of analytical mechanics is more general as well as convenient. Secondly, Noether theory. Noether theory not only has mathematical importance but can also be helpful in understanding some inherent physical properties of a dynamical system. Generally, conserved quantity helps reduce the degrees of freedom and make it easier to find the solutions to the differential equations of motion. Thirdly, fractional calculus. Fractional calculus is a useful tool to deal with the dissipative systems, and, as early as 1993, Miller and Ross [64] once said that fractional calculus has been involved in almost every field of science and engineering. This paper is constructed as follows: Combined fractional derivatives of variable order and their properties are given in Section2. Di fferential equations of motion for the Hamiltonian system of variable order are established in Section3. In Sections4 and5, Noether symmetry and conserved quantity, and perturbation to Noether symmetry and adiabatic invariants are investigated. The Results and Methods are illustrated in Section6 with two applications and we finish with Section7 for Conclusions.

2. Combined Fractional Derivative of Variable Order Following the references [54,57], we list the left and right Riemann–Liouville fractional integrals of variable order, the left and right RLVO, and the left and right CVO by

Z t α( , ) 1 α(t,τ) 1 I · · f (t) = (t τ) − f (τ)dτ (1) t1 t ( ( )) t1 Γ α t, τ −

Z t2 α( , ) 1 α(τ,t) 1 tIt · · f (t) = (τ t) − f (τ)dτ (2) 2 t Γ(α(τ, t)) − Z t RL α( , ) d 1 α( , ) d 1 α(t,τ) D · · f (t) = I − · · f (t) = (t τ)− f (τ)dτ (3) t1 t dt t1 t dt Γ(1 α(t, τ)) − t1 − Z t2 RL α( , ) d 1 α( , ) d 1 α(τ,t) D · · f (t) = I − · · f (t) = (τ t)− f (τ)dτ (4) t t2 −dt t t2 −dt Γ(1 α(τ, t)) − t − Z t C α( , ) 1 α(t,τ) d D · · f (t) = (t τ)− f (τ)dτ (5) t1 t Γ(1 α(t, τ)) − dτ t1 − Z t2 C α( , ) 1 α(τ,t) d D · · f (t) = (τ t)− f (τ)dτ (6) t t2 Γ(1 α(τ, t)) − −dτ t − Symmetry 2019, 11, 1270 4 of 23

1 α( , ) where 0 < α( , ) < 1, Γ( ) is the Gamma function, f (t) L1[t1, t2], I − · · f (t) AC[t1, t2], · · · ∈ t1 t ∈ 1 α( , ) I − · · f (t) AC[t1, t2]. In this paper, we assume that α(t, τ) = α(t τ), α(τ, t) = α(τ t). t t2 ∈ − − Based on the deﬁnitions of the RLVO and the CVO, the RVO and the RCVO are deﬁned in [57] as R α( , ) 1 RL α( , ) RL α( , ) D · · f (t) = D · · f (t) D · · f (t) (7) t1 t2 2 t1 t − t t2 RC α( , ) 1 C α( , ) C α( , ) D · · f (t) = D · · f (t) D · · f (t) (8) t1 t2 2 t1 t − t t2 In this paper, CRLVO and CCVO are constructed as

RL α( , ),β( , ) RL α( , ) RL β( , ) Dγ · · · · f (t) = γ D · · f (t) (1 γ) D · · f (t) (9) t1 t − − t t2

C α( , ),β( , ) C α( , ) C β( , ) Dγ · · · · f (t) = γ D · · f (t) (1 γ) D · · f (t) (10) t1 t − − t t2 α( , ) β( , ) where 0 < α( , ) < 1, 0 < β( , ) < 1, D · · and D · · show the arrow of time, and γ is a parameter, · · · · t1 t t t2 0 γ 1, which determines the diﬀerent quantities of information from the past and the future. ≤ ≤ The CRLVO and the CCVO are the most general operators with variable order to some extent. From Equations (9) and (10), the RVO, the RCVO, the left and right Riemann–Liouville fractional derivatives (RL), the left and right Caputo fractional derivatives (C), the Riesz–Riemann–Liouville fractional derivative (R), the Riesz–Caputo fractional derivative (RC), the combined Riemann–Liouville fractional derivative (CRL), the combined Caputo fractional derivative (CC), and the classical integer derivative can be deduced as special cases. We list them as follows: 1 Let β = α, γ = 2 in Equations (9) and (10), so the RVO and the RCVO can be achieved. When α( , ) α, β( , ) β, with two constants 0 < α, β < 1, the left and right RL, the left and · · → · · → right C, the R, the RC, the CRL, and the CC can be obtained as

Z t RL α 1 d α D f (t) = (t τ)− f (τ)dτ (11) t1 t Γ(1 α) dt − − t1

Z t2 RL α 1 d α D f (t) = (τ t)− f (τ)dτ (12) t t2 Γ(1 α) −dt − − t Z t C α 1 α d D f (t) = (t τ)− f (τ)dτ (13) t1 t Γ(1 α) − dτ − t1 Z t2 C α 1 α d D f (t) = (τ t)− f (τ)dτ (14) t t2 Γ(1 α) − −dτ − t 1h i R Dα f (t) = RLDα f (t) RLDα f (t) (15) t1 t2 2 t1 t − t t2 1h i RCDα f (t) = C Dα f (t) CDα f (t) (16) t1 t2 2 t1 t − t t2 RL α,β RL α RL β Dγ f (t) = γ D f (t) (1 γ) D f (t) (17) t1 t − − t t2 C α,β C α C β Dγ f (t) = γ D f (t) (1 γ) D f (t) (18) t1 t − − t t2 Equations (11) to (18) can be found in [65]. When α( , ) 1, we have · · → RL α( , ) C α( , ) RL α( , ) C α( , ) D · · = D · · = d/dt, D · · = D · · = d/dt (19) t1 t t1 t t t2 t t2 − Symmetry 2019, 11, 1270 5 of 23

The formulae of integration by parts for the CVO are

Z t2 Z t2 C α( , ) RL α( , ) 1 α( , ) t g(t) D · · f (t) dt = f (t) D · · g(t) dt + f (t) I − · · g(t) 2 (20) t1 t t t2 t t2 t1 t1 t1

Z t2 Z t2 C α( , ) RL α( , ) 1 α( , ) t g(t) D · · f (t) dt = f (t) D · · g(t) dt f (t) I − · · g(t) 2 (21) t t2 t1 t t1 t t1 t1 t1 − The formulae of integration by parts for the RLVO, RVO, and RCVO are

Z t2 Z t2 RL α( , ) RL α( , ) 1 α( , ) g(t) D · · f (t) dt = f (t) D · · g(t) dt I − · · f (t) g(t) t (22) t1 t t t2 t1 t 1 t1 t1 −

Z t2 Z t2 RL α( , ) RL α( , ) 1 α( , ) g(t) D · · f (t) dt = f (t) D · · g(t) dt I − · · f (t) g(t) t (23) t t2 t1 t t t2 2 t1 t1 − R t2 R α( , ) R t2 R α( , ) 1 1 α( , ) g(t) D · · f (t) dt = f (t) D · · g(t) dt + I − · · f (t) g(t) t t1 t1 t2 − t1 t1 t2 2 t t2 2 (24) 1 1 α( , ) I − · · f (t) g(t) t − 2 t1 t 1

Z t2 Z t2 RC α( , ) R α( , ) R 1 α( , ) t g(t) D · · f (t) dt = f (t) D · · g(t) dt + f (t) I − · · g(t) 2 (25) t1 t2 t1 t2 t1 t2 t1 t1 − t1 For more details about the above Equations (19)–(25), please refer to [54,57,65].

3. Hamilton Equation of Variable Order A uniﬁed Hamilton action and Hamilton principle of variable order will be given ﬁrst, and then the Hamilton action and Hamilton principle in terms of the CRLVO and the CCVO will be discussed, respectively. α( , ),β( , ) Assuming the Lagrangian of variable order is LU t, qj, Dγ · · · · qj , then the generalized momentum of variable order and the Hamiltonian of variable order can be listed as α( , ),β( , ) ∂LU t,qj,Dγ · · · · qj pUi = α( , ),β( , ) , ∂D · · · · q γ i (26) α( , ),β( , ) α( , ),β( , ) HU = p D · · · · q LU t, q , D · · · · q , i, j = 1, 2, , n Ui· γ i − j γ j ··· And the functional

Z t2 α( , ),β( , ) SHU = pUi Dγ · · · · qi HU t, qj, pUj dt (27) t1 · −

1 1 Represents a uniﬁed Hamilton action of variable order, where q , p C , HU is a C function j Uj ∈ with respect to all its arguments. The formula Z t2 α( , ),β( , ) δSHU = δ pUi Dγ · · · · qi HU t, qj, pUj dt = 0 (28) t1 · − with [57] α( , ) α( , ) β( , ) β( , ) δ D · · qi = D · · δqi, δ D · · qi = D · · δqi, δqi t=t = δqi t=t = 0 (29) t1 t t1 t t t2 t t2 1 2 is called a uniﬁed Hamilton principle of variable order. Symmetry 2019, 11, 1270 6 of 23

3.1. Hamilton Equation with the CRLVO When we consider the CRLVO, the Lagrangian of variable order can be denoted by RL α( , ),β( , ) LRL t, qj, Dγ · · · · qj . The generalized momentum of variable order and the Hamiltonian of variable order are RL α( , ),β( , ) ∂LRL t,qj, Dγ · · · · qj pRLi = α( , ),β( , ) , ∂RLD · · · · q γ i (30) RL α( , ),β( , ) RL α( , ),β( , ) HRL = p D · · · · q LRL t, q , D · · · · q RLi· γ i − j γ j Under this condition, Hamilton action of variable order is

Z t2 RL α( , ),β( , ) SHRL = pRLi Dγ · · · · qi HRL t, qj, pRLj dt (31) t1 · −

Hamilton principle of variable order is

Z t2 RL α( , ),β( , ) δSHRL = δ pRLi Dγ · · · · qi HRL t, qj, pRLj dt = 0 (32) t1 · − with [57] RL α( , ) RL α( , ) RL β( , ) RL β( , ) δ D · · qi = D · · δqi, δ D · · qi = D · · δqi, δqi t=t = δqi t=t = 0 (33) t1 t t1 t t t2 t t2 1 2 It follows from Equation (32) that R t2 α( , ),β( , ) = RL · · · · δSHRL δ t pRLi Dγ qi HRL t, qj, pRLj dt 1 · − R t2 α( , ),β( , ) α( , ),β( , ) = RL · · · · + RL · · · · t δpRLi Dγ qi pRLi δ Dγ qi δHRL t, qj, pRLj dt 1 · · − (34) R t2 RL α( , ),β( , ) RL β( , ),α( , ) ∂HRL ∂HRL = δpRLi D · · · · qi δqi D · · · · pRLi δqi δpRLi dt t γ 1 γ ∂qi ∂pRLi 1 · − · − − − R t2 RL α( , ),β( , ) ∂HRL RL β( , ),α( , ) ∂HRL = δpRLi D · · · · qi δqi D · · · · pRLi + dt = 0 t γ ∂pRLi 1 γ ∂qi 1 · − − · − where R t2 α( , ),β( , ) R t2 α( , ) β( , ) RL · · · · = RL · · ( )RL · · t pRLi δ Dγ qidt t pRLi δ γt Dt qi 1 γ t Dt qi dt 1 · 1 · 1 − − 2 R t2 α( , ) β( , ) = RL · · ( ) RL · · t γpRLi t Dt δqi 1 γ pRLi t Dt δqi dt 1 · 1 − − · 2 (35) R t2 α( , ) β( , ) = RL · · ( )RL · · t δqi γt Dt pRLi δqi 1 γ t Dt pRLi dt 1 · 2 − · − 1 R t2 β( , ),α( , ) = RL · · · · t δqi D1 γ pRLi dt − 1 · − From Equations (30) and (34), we have

RL α( , ),β( , ) ∂HRL RL β( , ),α( , ) ∂HRL · · · · = · · · · = Dγ qi , D1 γ pRLi (36) ∂pRLi − − ∂qi

Equation (36) is called the Hamilton equation with the CRLVO.

3.2. Hamilton Equation with the CCVO When we consider the CCVO, the Lagrangian of variable order can be denoted by C α( , ),β( , ) LC t, qj, Dγ · · · · qj . The generalized momentum of variable order and the Hamiltonian of variable order are C α( , ),β( , ) ∂LC t, qj, Dγ · · · · qj C α( , ),β( , ) C α( , ),β( , ) pCi = , HC = pCi Dγ · · · · qi LC t, qj, Dγ · · · · qj (37) C α( , ),β( , ) · − ∂ Dγ · · · · qi Symmetry 2019, 11, 1270 7 of 23

Under this condition, the Hamilton action of variable order is

Z t2 C α( , ),β( , ) SHC = pCi Dγ · · · · qi HC t, qj, pCj dt (38) t1 · −

The Hamilton principle of variable order is

Z t2 C α( , ),β( , ) δSHC = δ pCi Dγ · · · · qi HC t, qj, pCj dt = 0 (39) t1 · − with [57] C α( , ) C α( , ) C β( , ) C β( , ) δ D · · qi = D · · δqi, δ D · · qi = D · · δqi, δqi t=t = δqi t=t = 0 (40) t1 t t1 t t t2 t t2 1 2 Similarly, from Equations (37) and (39), we can get

C α( , ),β( , ) ∂HC RL β( , ),α( , ) ∂HC · · · · = · · · · = Dγ qi , D1 γ pCi (41) ∂pCi − − ∂qi

Equation (41) is called the Hamilton equation with the CCVO.

Remark 1. When β( , ) = α( , ), γ = 1 , the Hamilton equation in terms of the RVO can be obtained from · · · · 2 Equation (36) as R α( , ) ∂HR R α( , ) ∂HR t Dt · · qi = , t Dt · · pRi = (42) 1 2 ∂pRi 1 2 − ∂qi R α( , ) ∂LR t,qj, D · · qj t1 t2 R α( , ) R α( , ) where pRi = ,HR = pRi D · · qi LR t, qj, D · · qj . R α( , ) t1 t2 t1 t2 ∂ D · · qi · − t1 t2 Similarly, the Hamilton equation in terms of the RCVO can be obtained from Equation (41) as

RC α( , ) ∂HRC R α( , ) ∂HRC t Dt · · qi = , t Dt · · pRCi = (43) 1 2 ∂pRCi 1 2 − ∂qi

RC α( , ) ∂LRC t,qj, D · · qj t1 t2 RC α( , ) RC α( , ) where pRCi = ,HRC = pRCi D · · qi LRC t, qj, D · · qj . RC α( , ) t1 t2 t1 t2 ∂ D · · qi · − t1 t2 Moreover, Equations (42) and (43) can also be obtained through their respective Hamilton principles.

Remark 2. From Equations (30) and (36), the Lagrange equation with the CRLVO can be obtained as RL α( , ),β( , ) RL α( , ),β( , ) ∂LRL t, qj, Dγ · · · · qj ∂LRL t, qj, Dγ · · · · qj RL β( , ),α( , ) D · · · · = 0 (44) 1 γ α( , ),β( , ) ∂qi − − RL ∂ Dγ · · · · qi

It is noted that Equation (44) can be obtained through R t2 α( , ),β( , ) = RL · · · · = δSLRL δ t LRL t, qj, Dγ qj dt 0, 1 (45) RL α( , ) RL α( , ) RL β( , ) RL β( , ) δ D · · qi = D · · δqi, δ D · · qi = D · · δqi, δqi t=t = δqi t=t = 0 t1 t t1 t t t2 t t2 1 2

Similarly, from Equations (37) and (41), the Lagrange equation with the CCVO can be obtained as C α( , ),β( , ) ∂LC t, qj, Dγ · · · · qj ∂LC RL β( , ),α( , ) D · · · · = 0 (46) 1 γ α( , ),β( , ) ∂qi − − C ∂ Dγ · · · · qi Symmetry 2019, 11, 1270 8 of 23

And Equation (46) can also be obtained through R t2 C α( , ),β( , ) C α( , ) C α( , ) δSLC = δ LC t, qj, D · · · · qj dt = 0, δ D · · qi = D · · δqi, t γ t1 t t1 t 1 (47) C β( , ) C β( , ) δ D · · qi = D · · δqi, δqi t=t = δqi t=t = 0 t t2 t t2 1 2

Remark 3. When α( , ) α, β( , ) β, Hamilton equations of variable order, i.e., Equations (36) and (41) · · → · · → reduce to ∂H ∂H RL α,β = RLF RL β,α = RLF Dγ qi , D1 γpRLiF (48) ∂pRLiF − − ∂qi and ∂H ∂H C α,β = CF RL β,α = CF Dγ qi , D1 γpCiF (49) ∂pCiF − − ∂qi Equations (48) and (49) are the Hamilton equations with the CRL and the CC, respectively. Lagrange equations of variable order, i.e., Equations (44) and (46) reduce to

RL α,β RL α,β ∂LRLF t, qj, Dγ qj β,α ∂LRLF t, qj, Dγ qj RLD = 0 (50) 1 γ α,β ∂qi − RL − ∂ Dγ qi and C α,β ∂L β,α ∂LCF t, qj, Dγ qj CF RLD = 0 (51) 1 γ α,β ∂qi − C − ∂ Dγ qi Equations (50) and (51) are the Lagrange equations with the CRL and the CC, respectively. It is noted that Equations (48) to (51) are consistent with the results in [53].

Remark 4. When α( , ), β( , ) 1, Hamilton equations of variable order, i.e., Equations (36) and (41) reduce · · · · → to . ∂H t, qj, pj . ∂H t, qj, pj qi = , pi = (52) ∂pi − ∂qi Lagrange equations of variable order, i.e., Equations (44) and (46) reduce to

. . ∂L t, qj, qj d ∂L t, qj, qj . = 0 (53) ∂qi − dt ∂qi

Equations (52) and (53) are the classical Hamilton equation and the classical Lagrange equation, respectively, which is consistent with the results in [66].

4. Noether Symmetry and Conserved Quantity The classical Noether symmetry means the invariance of the classical Hamilton action under the infinitesimal transformations of time and coordinates. A Noether symmetry can lead to a conserved quantity. Here, we generalize the classical definition of Noether symmetry to fractional form with variable order. Then Noether symmetry and conserved quantity will be discussed in detail with the CRLVO and the CCVO, respectively. Under the infinitesimal transformations t = t + ∆t, qi t = qi(t) + ∆qi, pUi t = pUi(t) + ∆pUi (54) Symmetry 2019, 11, 1270 9 of 23

Whose linear parts are

0 0 t = t + θξ t, qj, pUj , q t = qi(t) + θξ t, qj, pUj , U0 i Ui (55) = ( ) + 0 pUi t pUi t θηUi t, qj, pUj

The uniﬁed Hamilton action of variable order, i.e., Equation (27) changes to be

Z t2 α( , ),β( , ) SHU = pUi Dγ · · · · qi HU t, qj, pUj dt (56) t1 · −

0 0 0 where θ is an infinitesimal parameter, and ξU0, ξUi, and ηUi are infinitesimal generators. Let ∆SHU denote the linear part of SHU SHU, if there exists infinitesimal generators satisfying R t2 d 0 − ∆SHU = dt ∆G dt, then fractional Noether symmetry with variable order can be determined − t1 U 0 = 0 0 by the corresponding infinitesimal transformations, where ∆GU θGU t, qj, pUj , and GU is called a gauge function. For a certain mechanical system, conserved quantity can be deduced from Noether symmetry. A conserved quantity can be seen as a solution to the differential equation of motion of the mechanical system, and help reduce the freedom of the mechanical system. d A quantity IU is called a conserved quantity if and only if dt IU = 0.

4.1. Noether Symmetry and Conserved Quantity with the CRLVO When we consider the CRLVO, the inﬁnitesimal transformations can be expressed as t = t + ∆t, qi t = qi(t) + ∆qi, pRLi t = pRLi(t) + ∆pRLi 0 0 t = t + θRLξ t, qj, pRLj , q t = qi(t) + θRLξ t, qj, pRLj , (57) RL0 i RLi = ( ) + 0 pRLi t pRLi t θRLηRLi t, qj, pRLj .

Under the action of Equation (57), the Hamilton action of variable order, i.e., Equation (31) becomes

Z t2 RL α( , ),β( , ) SHRL = pRLi Dγ · · · · qi HRL t, qj, pRLj dt (58) t1 · −

Then we get

∆SHRL = SHRL SHRL − R t2 RL α( , ),β( , ) d RL α( , ),β( , ) = HRL t + ∆t, qj + ∆qj, pRLj + ∆pRLj + Dγ · · · · qi + ∆t Dγ · · · · qi t1 − dt α(t,t1) β(t2,t) d (t t1)− α( , ),β( , ) d (t2 t)− ( ) − + RL · · · · + ( ) ( ) − γqi t1 ∆t1 dt Γ(1 α(t,t )) Dγ δqi 1 γ qi t2 ∆t2 dt Γ(1 β(t ,t)) − − 1 − − 2 R t2 α( , ),β( , ) ( + ) + d RL · · · · pRLi ∆pRLi 1 dt ∆t dt t pRLi Dγ qi HRL t, qj, pRLj dt (59) · · − 1 · − R t2 RL α( , ),β( , ) d RL α( , ),β( , ) ∂HRL = pRLi Dγ · · · · δqi + pRLi Dγ · · · · qi ∆t t1 · · dt − ∂t α(t,t1) d (t t1)− α( , ),β( , ) d ( ) − + RL · · · · γpRLiqi t1 ∆t1 dt Γ(1 α(t,t )) pRLi Dγ qi HRL dt ∆t − − 1 · − · β(t2,t) d (t2 t)− ∂HRL +(1 γ)pRLiqi(t2)∆t2 − ∆qi dt dt Γ(1 β(t ,t)) ∂qi − − 2 − where α( , ),β( , ) α( , ),β( , ) α( , ),β( , ) α( , ),β( , ) RL · · · · RL · · · · RL · · · · d RL · · · · Dγ qi = Dγ qi + Dγ δqi + ∆t dt Dγ qi α(t,t1) β(t2,t) (60) d (t t1)− d (t2 t)− ( ) − + ( ) ( ) − γqi t1 ∆t1 dt Γ(1 α(t,t )) 1 γ qi t2 ∆t2 dt Γ(1 β(t ,t)) − − 1 − − 2 Symmetry 2019, 11, 1270 10 of 23

R t2 d 0 Let ∆SHRL = ∆G dt, and we obtain − t1 dt RL α( , ),β( , ) . α( , ),β( , ) ∂H p RLD · · · · ξ0 q ξ0 + p d RLD · · · · q RL ξ0 RLi· γ RLi − i RL0 RLi· dt γ i − ∂t RL0 . . α( , ),β( , ) 0 ∂H 0 + RL · · · · RL 0 + pRLi Dγ qi HRL ξRL0 ∂q ξRLi GRL t, qj, pRLj · − · − i (61) α(t,t1) 0 d (t t1)− γpRLiqi(t1)ξ t1, qj(t1), pRLj(t1) − − RL0 dt Γ(1 α(t,t1)) − β(t2,t) d (t2 t)− +( ) ( ) 0 ( ) ( ) − = 1 γ pRLiqi t2 ξRL0 t2, qj t2 , pRLj t2 dt Γ(1 β(t ,t)) 0 − − 2 Equation (61) is called Noether identity with the CRLVO.

0 Theorem 1. For the Hamiltonian system with the CRLVO (Equation (36)), if there exists a gauge function GRL 0 0 such that the inﬁnitesimal generators ξRL0 and ξRLi satisfy Equation (61), then there exists a conserved quantity. R t α( , ),β( , ) . . β( , ),α( , ) = RL · · · · 0 0 + 0 0 RL · · · · + ( IHRL0 t pRLi Dγ ξRLi qiξRL0 ξRLi qiξRL0 D1 γ pRLi dτ pRLi 1 · − − · − · α(τ,t1) α( , ),β( , ) R t d (τ t1)− RL · · · · 0 ( ) 0 ( ) ( ) − Dγ qi HRL ξRL0 t pRLi γqi t1 ξRL0 t1, qj t1 , pRLj t1 dτ Γ(1 α(τ,t )) (62) − · − 1 − 1 β(t2,τ) d (t2 τ)− ( ) ( ) 0 ( ) ( ) − + 0 1 γ qi t2 ξRL0 t2, qj t2 , pRLj t2 dτ Γ(1 β(t ,τ)) dτ GRL − − − 2

Proof. Using Equations (36) and (61), we have ! d . RL α( , ),β( , ) d RL α( , ),β( , ) ∂HRL ∂HRL . ∂HRL . IHRL0 = pRLi Dγ · · · · qi + pRLi Dγ · · · · qi qi pRLi dt · ·dt − ∂t − ∂qi − ∂pRLi

. 0 0 RL α( , ),β( , ) RL α( , ),β( , ) 0 . 0 0 ξ + p D · · · · q HRL ξ + p D · · · · ξ q ξ + ξ · RL0 RLi· γ i − · RL0 RLi· γ RLi − i RL0 RLi

α(t,t1) . 0 RL β( , ),α( , ) 0 d (t t1)− q ξ D · · · · pRLi γpRLiqi(t1)ξ t1, qj(t1), pRLj(t1) − − i RL0 · 1 γ − RL0 dt Γ(1 α(t, t )) − − 1 β(t ,t) 2 . 0 0 d (t2 t)− + (1 γ)pRLiqi(t2)ξ t2, qj(t2), pRLj(t2) − + G − RL0 dt Γ(1 β(t , t)) RL − 2 ! . RL α( , ),β( , ) ∂HRL . ∂HRL . 0 0 . 0 RL β( , ),α( , ) ∂HRL 0 = · · · · + · · · · + pRLi Dγ qi qi pRLi ξRL0 ξRLi qiξRL0 D1 γ pRLi ξRLi · − ∂qi − ∂pRLi · − · − ∂qi ! 0 . 0 RL β( , ),α( , ) ∂HRL = · · · · + = ξRLi qiξRL0 D1 γ pRLi 0 − · − ∂qi

4.2. Noether Symmetry and Conserved Quantity with the CCVO When we consider the CCVO, the inﬁnitesimal transformations can be expressed as t = t + ∆t, qi t = qi(t) + ∆qi, pCi t = pCi(t) + ∆pCi, 0 0 t = t + θCξ t, qj, pCj , q t = qi(t) + θCξ t, qj, pCj , (63) C0 i Ci = ( ) + 0 pCi t pCi t θCηCi t, qj, pCj .

Under the action of Equation (63), the Hamilton action of variable order, i.e., Equation (38) becomes

Z t2 C α( , ),β( , ) SHC = pCi Dγ · · · · qi HC t, qj, pCj dt (64) t1 · − Symmetry 2019, 11, 1270 11 of 23

R t2 d 0 Then let ∆SHC = SHC SHC = ∆G dt, and we obtain − − t1 dt C α( , ),β( , ) . α( , ),β( , ) ∂H p CD · · · · ξ0 q ξ0 + p d CD · · · · q C ξ0 Ci· γ Ci − i C0 Ci· dt γ i − ∂t C0 α(t,t ) . (t t )− 1 . α( , ),β( , ) 0 0 ( ) ( ) − 1 ( ) + C · · · · (65) γpCiξC0 t1, qj t1 , pCj t1 Γ(1 α(t,t )) qi t1 pCi Dγ qi HC ξC0 − − 1 · − · β(t2,t) . 0 0 (t2 t)− . ∂HC 0 +(1 γ)pCiξ t2, qj(t2), pCj(t2) − q (t2) ξ + G t, qj, pCj = 0 C0 Γ(1 β(t ,t)) i ∂qi Ci C − − 2 − Equation (65) is called Noether identity with the CCVO.

0 Theorem 2. For the Hamiltonian system with the CCVO (Equation (41)), if there exists a gauge function GC 0 0 such that the inﬁnitesimal generators ξC0 and ξCi satisfy Equation (65), then there exists a conserved quantity. R t α( , ),β( , ) . . β( , ),α( , ) = C · · · · 0 0 + 0 0 RL · · · · + ( IHC0 t pCi Dγ ξCi qiξC0 ξCi qiξC0 D1 γ pCi dτ pCi 1 · − − · − α(τ,t1) C α( , ),β( , ) 0 R t 0 . (τ t1)− Dγ · · · · qi HC ξ pCi γξ t1, qj(t1), pCj(t1) qi(t1) − (66) · − · C0 − t1 C0 Γ(1 α(τ,t1)) β(t ,τ) − . (t τ)− 2 ( ) ( ) 0 ( ) ( ) 2− + 0 1 γ qi t2 ξC0 t2, qj t2 , pCj t2 Γ(1 β(t ,τ)) dτ GC − − − 2

Proof. The proof is similar to that in Theorem 1 and thus is omitted.

4.3. Some Explanations We will present some special cases in this section on the basis of Theorems 1 and 2, which are the main results of fractional Noether symmetry and fractional conserved quantity with variable order in this paper.

Remark 5. Let β( , ) = α( , ), γ = 1 , and conserved quantities for the Hamiltonian systems with variable · · · · 2 order (Equations (42) and (43)) can be achieved as follows:

Theorem 3. For the Hamiltonian system with the RVO (Equation (42)), if there exists a gauge function 0 0 0 GR t, qj, pRj such that the inﬁnitesimal generators ξR0 t, qj, pRj and ξRi t, qj, pRj satisfy

. 0 R α( , ) 0 . 0 d R α( , ) ∂HR 0 R α( , ) pRi D · · ξ q ξ + pRi D · · qi ξ + pRi D · · qi HR ξR0 ·t1 t2 Ri − i R0 · dt t1 t2 − ∂t R0 ·t1 t2 − · α(t,t1) . 0 0 d (t t1)− ∂HR 0 γpRiqi(t1)ξ t1, qj(t1), pRj(t1) − ξ + GR (67) − R0 dt Γ(1 α(t,t1)) − ∂qi Ri − β(t2,t) d (t2 t)− +( ) ( ) 0 ( ) ( ) − = 1 γ pRiqi t2 ξR0 t2, qj t2 , pRj t2 dt Γ(1 β(t ,t)) 0 − − 2 then there exists a conserved quantity. R t R α( , ) 0 . 0 0 . 0 R α( , ) R α( , ) IHR0 = pRi D · · ξ qiξ + ξ qiξ D · · pRi dτ + pRi D · · qi t1 ·t1 t2 Ri − R0 Ri − R0 ·t1 t2 ·t1 t2 α(τ,t1) R t d (τ t1)− ) 0 ( ) 0 ( ) ( ) − HR ξR0 t pRi γqi t1 ξR0 t1, qj t1 , pRj t1 dτ Γ(1 α(τ,t )) (68) − · − 1 − 1 β(t2,τ) d (t2 τ)− ( ) ( ) 0 ( ) ( ) − + 0 1 γ qi t2 ξR0 t2, qj t2 , pRj t2 dτ Γ(1 β(t ,τ)) dτ GR − − − 2 Symmetry 2019, 11, 1270 12 of 23

Theorem 4. For the Hamiltonian system with the RCVO (Equation (43)), if there exists a gauge function 0 0 0 GRC t, qj, pRCj such that the inﬁnitesimal generators ξRC0 t, qj, pRCj and ξRCi t, qj, pRCj satisfy RC α( , ) 0 . 0 d RC α( , ) ∂HRC 0 pRCi D · · ξ q ξ + pRCi D · · qi ξ ·t1 t2 RCi − i RC0 · dt t1 t2 − ∂t RC0 α(t,t1) . 0 0 (t t1)− . RC α( , ) γpRCiξ t , qj(t ), pRCj(t ) − q (t ) + pRCi D · · qi HRC ξ (69) RC0 1 1 1 Γ(1 α(t,t )) i 1 t1 t2 RC0 − − 1 · − · β(t2,t) . 0 0 (t2 t)− . ∂HRC 0 +(1 γ)pRCiξ t2, qj(t2), pRCj(t2) − q (t2) ξ + G = 0 RC0 Γ(1 β(t ,t)) i ∂qi RCi RC − − 2 − then there exists a conserved quantity. R t RC α( , ) 0 . 0 0 . 0 R α( , ) IHRC0 = pRCi D · · ξ qiξ + ξ qiξ D · · pRCi dτ + (pRCi t1 ·t1 t2 RCi − RC0 RCi − RC0 ·t1 t2 α(τ,t1) RC α( , ) 0 R t 0 . (τ t1)− D · · qi HRC ξ pRCi γξ t1, qj(t1), pRCj(t1) qi(t1) − (70) ·t1 t2 − · RC0 − t1 RC0 Γ(1 α(τ,t1)) β(t ,τ) − . (t τ)− 2 ( ) ( ) 0 ( ) ( ) 2− + 0 1 γ qi t2 ξRC0 t2, qj t2 , pRCj t2 Γ(1 β(t ,τ)) dτ GRC − − − 2

Remark 6. Conserved quantities for the Lagrangian systems with variable order (i.e. Equations (44) and (46)) can be achieved as follows:

Theorem 5. For the Lagrangian system with the CRLVO (Equation (44)), if there exists a gauge function α( , ),β( , ) 0 RL · · · · 0 0 GLRL t, qj, Dγ qj such that the inﬁnitesimal generators ξLRL0 t, qj and ξLRLi t, qj satisfy

α( , ),β( , ) ∂L t,q ,RLD · · · · q . . RL j γ j α( , ),β( , ) . 0 0 ∂L ∂L RL · · · · 0 0 + + + RL 0 + RL α( , ),β( , ) Dγ ξLRLi qiξLRL0 LRL ξRL0 GLRL ∂q ξLRLi ∂t ∂RLD · · · · q · − · i γ i ! ( ) ( ) ( ) α(t,t1) ∂LRL d RL α , ,β , 0 γqi t1 ∂LRL 0 d (t t1)− (71) + ( ) ( ) D · · · · qi ξ ( ) ( ) ξ t1, qj(t1) − RL α , ,β , dt γ LRL0 RL α , ,β , LRL0 dt Γ(1 α(t,t1)) ∂ Dγ · · · · qi · − ∂ Dγ · · · · qi − β(t2,t) (1 γ)qi(t2)∂LRL 0 d (t2 t)− + − ( ) ( ) ξ t2, qj(t2) − = 0 RL α , ,β , LRL0 dt Γ(1 β(t2,t)) ∂ Dγ · · · · qi − then there exists a conserved quantity. " R t 0 . 0 RL β( , ),α( , ) ∂LRL ∂LRL ILRL0 = ξ q ξ D · · · · ( ) ( ) + ( ) ( ) t1 LRLi i LRL0 1 γ RL α , ,β , RL α , ,β , − − ∂ Dγ · · · · qi ∂ Dγ · · · · qi h RL α( , ),β( , ) 0 . 0 0 R t ∂LRL 0 D · · · · ξ q ξ dτ + LRL ξ ( ) ( ) γqi(t1)ξ t1, qj(t1) (72) γ LRLi i LRL0 LRL0 t1 RL α , ,β , LRL0 · − · − ∂ Dγ · · · · qi α(τ,t1) β(t2,τ) d (τ t1)− d (t2 τ)− − ( ) ( ) 0 ( ) − + 0 dτ Γ(1 α(τ,t )) 1 γ qi t2 ξLRL0 t2, qj t2 dτ Γ(1 β(t ,τ)) dτ GLRL · − 1 − − − 2

Theorem 6. For the Lagrangian system with the CCVO (Equation (46)), if there exists a gauge function α( , ),β( , ) 0 C · · · · 0 0 GLC t, qj, Dγ qj such that the inﬁnitesimal generators ξLC0 t, qj and ξLCi t, qj satisfy

α( , ),β( , ) ∂L t,q ,CD · · · · q C j γ j α( , ),β( , ) . ∂L α( , ),β( , ) CD · · · · ξ0 q ξ0 + C d CD · · · · q C α( , ),β( , ) γ LCi i LC0 C α( , ),β( , ) dt γ i ∂ Dγ · · · · qi · − ∂ Dγ · · · · qi · β(t2,t) ∂LC 0 ∂LC 0 (t2 t)− . ∂LC 0 + ξ + (1 γ) ( ) ( ) ξ t2, qj(t2) − q (t2) + ξ (73) ∂t LC0 C α , ,β , LC0 Γ(1 β(t2,t)) i ∂qi LCi − ∂ Dγ · · · · qi − α(t,t1) . 0 . 0 ∂LC 0 (t t1)− . γ ( ) ( ) ξ t1, qj(t1) − q (t1) + LC ξ + G = 0 C α , ,β , LC0 Γ(1 α(t,t1)) i LC0 LC − ∂ Dγ · · · · qi − · Symmetry 2019, 11, 1270 13 of 23 then there exists a conserved quantity. " R t ∂L α( , ),β( , ) . . = C C · · · · 0 0 + 0 0 ILC0 t α( , ),β( , ) Dγ ξLCi qiξLC0 ξLCi qiξLC0 1 ∂CD · · · · q · − − γ # i R t . α(τ,t1) RL β( , ),α( , ) ∂LC ∂LC 0 (τ t1)− (74) D · · · · ( ) ( ) dτ ( ) ( ) γξ t1, qj(t1) q (t1) − 1 γ C α , ,β , t1 C α , ,β , LC0 i Γ(1 α(τ,t1)) · − ∂ Dγ · · · · qi − ∂ Dγ · · · · qi − β(t ,τ) . (t τ)− 2 ( ) ( ) 0 ( ) 2− + 0 + 0 1 γ qi t2 ξLC0 t2, qj t2 Γ(1 β(t ,τ)) dτ LC ξLC0 GLC − − − 2 ·

Remark 7. When α( , ) α , β( , ) β, conserved quantities for the Hamiltonian systems with the CRL · · → · · → and the CC (Equations (48) and (49)) and the Lagrangian systems with the CRL and the CC (Equations (50) and (51)) can be deduced as follows:

Theorem 7. For the Hamiltonian system with the CRL (Equation (48)), if there exists a gauge function 0 0 0 GRLF t, qj, pRLjF such that the inﬁnitesimal generators ξRL0F t, qj, pRLjF and ξRLiF t, qj, pRLjF satisfy

. RL α,β 0 0 + d RL α,β ∂HRLF 0 + RL α,β pRLiF Dγ ξRLiF qiξRL0F pRLiF dt Dγ qi ∂t ξRL0F pRLiF Dγ qi · . 0 − . 0 · − · ∂HRLF 0 γpRLiF 0 d α HRLF) ξ ξ + G qi(t1)ξ t1, qj(t1), pRLjF(t1) (t t1)− (75) RL0F ∂qi RLiF RLF Γ(1 α) RL0F dt − · − − − − (1 γ)pRLiF 0 d β + − ( ) ( ) ( ) ( )− = Γ(1 β) qi t2 ξRL0F t2, qj t2 , pRLjF t2 dt t2 t 0 − − then there exists a conserved quantity. R t RL α,β 0 . 0 0 . 0 RL β,α IHRL0F = pRLiF Dγ ξ qiξ + ξ qiξ D pRLiF dτ + (pRLiF t1 · RLiF − RL0F RLiF − RL0F · 1 γ − RL α,β 0 R t γqi(t1) 0 d α Dγ qi HRLF ξ pRLiF ξ t1, qj(t1), pRLjF(t1) (τ t1)− (76) · − · RL0F − t1 Γ(1 α) RL0F dτ − − (1 γ)qi(t2) 0 d β 0 − ( ) ( ) ( )− + Γ(1 β) ξRL0F t2, qj t2 , pRLjF t2 dτ t2 τ dτ GRLF − − −

Theorem 8. For the Hamiltonian system with the CC (Equation (49)), if there exists a gauge function 0 0 0 GCF t, qj, pCjF such that the inﬁnitesimal generators ξC0F t, qj, pCjF and ξCiF t, qj, pCjF satisfy

. C α,β 0 0 + d C α,β ∂HCF 0 pCiF Dγ ξCiF qiξC0F pCiF dt Dγ qi ∂t ξC0F · − α · − . 0 0 (t t1)− . C α,β γpCiFξ t , qj(t ), pCjF(t ) − q (t ) + pCiF D qi HCF ξ (77) − C0F 1 1 1 Γ(1 α) i 1 · γ − · C0F − β . 0 0 (t2 t)− . ∂HCF 0 +(1 γ)pCiFξ t2, qj(t2), pCjF(t2) − q (t2) ξ + G = 0 C0F Γ(1 β) i ∂qi CiF CF − − − then there exists a conserved quantity. R t C α,β 0 . 0 0 . 0 RL β,α IHC0F = pCiF Dγ ξ qiξ + ξ qiξ D pCiF dτ + (pCiF t1 · CiF − C0F CiF − C0F · 1 γ − α C α,β 0 R t 0 . (τ t1)− Dγ qi HCF ξ pCiF γξ t1, qj(t1), pCjF(t1) qi(t1) − (78) · − · C0F − t1 C0F Γ(1 α) β − . (t τ)− ( ) ( ) 0 ( ) ( ) 2− + 0 1 γ qi t2 ξC0F t2, qj t2 , pCjF t2 Γ(1 β) dτ GCF − − − Symmetry 2019, 11, 1270 14 of 23

Theorem 9. For the Lagrangian system with the CRL (Equation (50)), if there exists a gauge function 0 RL α,β 0 0 GLRLF t, qj, Dγ qj such that the inﬁnitesimal generators ξLRL0F t, qj and ξLRLiF t, qj satisfy

RL α,β . ∂LRLF t,qj, Dγ qj α,β . 0 RLD ξ0 q ξ0 + L ξ + ∂LRLF ξ0 + ∂LRLF RL α,β γ LRLiF i LRL0F RLF RL0F ∂q LRLiF ∂t ∂ D qi · − · i γ ! ∂LRLF d RL α,β 0 γqi(t1) ∂LRLF 0 d α + D q ξ ξ t , q (t ) (t t )− (79) RL α,β dt γ i LRL0F Γ(1 α) RL α,β LRL0F 1 j 1 dt 1 ∂ Dγ qi · − − ∂ Dγ qi − . 0 (1 γ)qi(t2) ∂LRLF 0 d β + − ξ t , q (t ) (t t)− + G = 0 Γ(1 β) RL α,β LRL0F 2 j 2 dt 2 LRLF − ∂ Dγ qi − then there exists a conserved quantity. " R t . β,α I = ξ0 q ξ0 RLD ∂LRLF + ∂LRLF LRL0F t LRLiF i LRL0F 1 γ RL α,β RL α,β 1 − − ∂ Dγ qi ∂ Dγ qi α,β . i R t h RLD ξ0 q ξ0 dτ + L ξ0 ∂LRLF γq (t )ξ0 t , q (t ) (80) γ LRLiF i LRL0F RLF LRL0F t RL α,β i 1 LRL0F 1 j 1 · − · − 1 ∂ Dγ qi α β d (τ t1)− d (t2 τ)− − ( ) ( ) 0 ( ) − + 0 dτ Γ(1 α) 1 γ qi t2 ξLRL0F t2, qj t2 dτ Γ(1 β) dτ GLRLF · − − − −

Theorem 10. For the Lagrangian system with the CC (Equation (51)), if there exists a gauge function 0 C α,β 0 0 GLCF t, qj, Dγ qj such that the inﬁnitesimal generators ξLC0F t, qj and ξLCiF t, qj satisfy

C α,β ∂LCF t,qj, Dγ qj α,β . α,β CD ξ0 q ξ0 + ∂LCF d CD q C α,β γ LCiF i LC0F C α,β dt γ i ∂ Dγ qi · − ∂ Dγ qi · β ∂LCF 0 ∂LCF 0 (t2 t)− . ∂LCF 0 + ξ + (1 γ) , ξ t2, qj(t2) − q (t2) + ξ (81) ∂t LC0F C α β LC0F Γ(1 β) i ∂qi LCiF − ∂ Dγ qi − α . . ∂L (t t )− . 0 0 γ CF ξ0 t , q (t ) − 1 q (t ) + L ξ + G = 0 C α,β LC0F 1 j 1 Γ(1 α) i 1 CF LC0F LCF − ∂ Dγ qi − · then there exists a conserved quantity. " R t α,β . . I = ∂LCF CD ξ0 q ξ0 + ξ0 q ξ0 LC0F t C α,β γ LCiF i LC0F LCiF i LC0F 1 ∂ Dγ qi · − − # α β,α ∂L R t ∂L . (τ t )− RLD CF dτ CF γξ0 t , q (t ) q (t ) − 1 (82) 1 γ C α,β t C α,β LC0F 1 j 1 i 1 Γ(1 α) · − ∂ Dγ qi − 1 ∂ Dγ qi − β . (t τ)− ( ) ( ) 0 ( ) 2− + 0 + 0 1 γ qi t2 ξLC0F t2, qj t2 Γ(1 β) dτ LCF ξLC0F GLCF − − − ·

Remark 8. The results of Theorems 7–10 are consistent with those in [53].

Remark 9. When α( , ), β( , ) 1, conserved quantities for the Hamiltonian system (52) and the Lagrangian · · · · → system (53) can be obtained as follows:

0 Theorem 11. For the classical Hamiltonian system (Equation (52)), if there exists a gauge function G t, qj, pj 0 0 such that the inﬁnitesimal generators ξ0 t, qj, pj and ξi t, qj, pj satisfy

. 0 . 0 . 0 ∂H 0 ∂H 0 piξi ξ0 Hξ0 ξi + G = 0 (83) − ∂t − − ∂qi then there exists a conserved quantity. I = p ξ0 Hξ0 + G0 (84) 0 i i − 0 Symmetry 2019, 11, 1270 15 of 23

0 . Theorem 12. For the classical Lagrangian system (Equation (53)), if there exists a gauge function GL t, qj, qj 0 0 such that the inﬁnitesimal generators ξL0 t, qj and ξLi t, qj satisfy

. ∂L t, qj, q . 0 . 0 . 0 . 0 j . ∂L 0 ∂L 0 . ξLi qiξL0 + L ξL0 + GL + ξLi + ξL0 = 0 (85) ∂qi · − · ∂qi ∂t then there exists a conserved quantity.

∂L 0 . 0 0 0 IL0 = . ξLi qiξL0 + L ξL0 + GL (86) ∂qi − ·

Remark 10. The results of Theorems 11 and 12 are consistent with those in [66].

5. Perturbation to Noether Symmetry and Adiabatic Invariants Noether symmetry is one of the methods of finding the solutions to the differential equations of motion. However, if the mechanical system is disturbed by small forces, the original conserved quantity may also change. Perturbation to Noether symmetry and adiabatic invariants will be studied in this section because they also have a close relationship with the integration of the mechanical system. Supposing the mechanical system is disturbed by εWUi, the way of disturbing the infinitesimal generators and the gauge function is set as

0 1 2 2 0 1 2 2 GU = G + εG + ε G + , ξU = ξ + εξ + ε ξ + U U U ··· 0 U0 U0 U0 ··· (87) ξ = ξ0 + εξ1 + ε2ξ2 + , η = η0 + εη1 + ε2η2 + Ui Ui Ui Ui ··· Ui Ui Ui Ui ··· where GU is the gauge function of the disturbed system, and ξU0, ξUi, and ηUi are the inﬁnitesimal generators of the disturbed system. If a quantity Iz, which has a parameter ε and the highest power of ε as z, satisﬁes the rule that z+1 dIz/dt is in direct proportion to ε , then the quantity Iz is called an adiabatic invariant. Then we have:

Theorem 13. For the disturbed Hamiltonian system in terms of the CRLVO

RL α( , ),β( , ) ∂HRL RL β( , ),α( , ) ∂HRL · · · · = · · · · = Dγ qi , D1 γ pRLi εWRLi t, qj, pRLj (88) ∂pRLi − − ∂qi −

m m If there exists a gauge function GRL t, qj, pRLj such that the inﬁnitesimal generators ξRL0 t, qj, pRLj and m ξRLi t, qj, pRLj satisfy α( , ),β( , ) . α( , ),β( , ) ∂H α( , ),β( , ) p RLD · · · · ξm q ξm + p d RLD · · · · q RL ξm + p RLD · · · · q RLi· γ RLi − i RL0 RLi· dt γ i − ∂t RL0 RLi· γ i . m . m α(t,t1) ∂HRL m m d (t t1)− HRL) ξRL0 ξ + GRL γpRLiqi(t1)ξ t1, qj(t1), pRLj(t1) − (89) − · − ∂qi RLi − RL0 dt Γ(1 α(t,t1)) β(t2,t) − d (t2 t)− . +( ) ( ) m ( ) ( ) − m 1 m 1 = 1 γ pRLiqi t2 ξRL0 t2, qj t2 , pRLj t2 dt Γ(1 β(t ,t)) WRLi ξRLi− qiξRL−0 0 − − 2 − − Symmetry 2019, 11, 1270 16 of 23 then there exists an adiabatic invariant.

z P R t α( , ),β( , ) . . β( , ),α( , ) = m RL · · · · m m + m m RL · · · · + ( IHRLz ε t pRLi Dγ ξRLi qiξRL0 ξRLi qiξRL0 D1 γ pRLi dτ pRLi m=0 1 · − − · − α(τ,t1) α( , ),β( , ) R t d (τ t1)− RL · · · · m ( ) m ( ) ( ) − Dγ qi HRL ξRL0 t pRLi γqi t1 ξRL0 t1, qj t1 , pRLj t1 dτ Γ(1 α(τ,t )) · − · − 1 − 1 β(t2,τ) d (t2 τ)− ( ) ( ) m ( ) ( ) − + m 1 γ qi t2 ξRL0 t2, qj t2 , pRLj t2 dτ Γ(1 β(t ,τ)) dτ GRL − − − 2 (90) m 1 = m 1 = = where ξRLi− ξRL−0 0 when m 0.

Proof. Using Equations (88) and (89), we have

Xz " d m . RL α( , ),β( , ) d RL α( , ),β( , ) ∂HRL . ∂HRL . IHRLz = ε p D · · · · q + p D · · · · q q p dt RLi· γ i RLi·dt γ i − ∂q i − ∂p RLi m=0 i RLi ! . m ∂HRL m RL α( , ),β( , ) RL α( , ),β( , ) m . m ξ + p D · · · · q HRL ξ + p D · · · · ξ q ξ − ∂t · RL0 RLi· γ i − · RL0 RLi· γ RLi − i RL0

α(t,t1) m . m RL β( , ),α( , ) m d (t t1)− + ξ q ξ D · · · · pRLi γpRLiqi(t1)ξ t1, qj(t1), pRLj(t1) − RLi − i RL0 · 1 γ − RL0 dt Γ(1 α(t, t )) − − 1 β(t ,t)  ( ) 2 . m m d t2 t −  +(1 γ)pRLiqi(t2)ξ t2, qj(t2), pRLj(t2) − + GRL − RL0 dt Γ(1 β(t , t))  − 2 Xz " ! # m m . m RL β( , ),α( , ) ∂HRL m 1 . m 1 = ε ξ q ξ D · · · · p + + W ξ − q ξ − RLi − i RL0 · 1 γ RLi ∂q RLi RLi − i RL0 m=0 − i Xz mh m . m m 1 . m 1i z+1 z . z = ε εW ξ q ξ + W ξ − q ξ − = ε W ξ q ξ − RLi RLi − i RL0 RLi RLi − i RL0 − RLi RLi − i RL0 m=0

Theorem 14. For the disturbed Hamiltonian system in terms of the CCVO

C α( , ),β( , ) ∂HC RL β( , ),α( , ) ∂HC · · · · = · · · · = Dγ qi , D1 γ pCi εWCi t, qj, pCj (91) ∂pCi − − ∂qi −

m m If there exists a gauge function GC t, qj, pCj such that the inﬁnitesimal generators ξC0 t, qj, pCj and m ξCi t, qj, pCj satisfy α( , ),β( , ) . α( , ),β( , ) ∂H . p CD · · · · ξm q ξm + p d CD · · · · q C ξm W ξm 1 q ξm 1 Ci· γ Ci − i C0 Ci· dt γ i − ∂t C0 − Ci Ci− − i C0− α(t,t ) . (t t )− 1 . α( , ),β( , ) m m ( ) ( ) − 1 ( ) + C · · · · (92) γpCiξC0 t1, qj t1 , pCj t1 Γ(1 α(t,t )) qi t1 pCi Dγ qi HC ξC0 − − 1 · − · β(t2,t) . m m (t2 t)− . ∂HC m +(1 γ)pCiξ t2, qj(t2), pCj(t2) − q (t2) ξ + G = 0 C0 Γ(1 β(t ,t)) i ∂qi Ci C − − 2 − Then there exists an adiabatic invariant.

z P R t α( , ),β( , ) . . β( , ),α( , ) = m C · · · · m m + m m RL · · · · + ( IHCz ε t pCi Dγ ξCi qiξC0 ξCi qiξC0 D1 γ pCi dτ pCi m=0 1 · − − · − R . α(τ,t1) C α( , ),β( , ) m t m (τ t1)− (93) Dγ · · · · qi HC ξ pCi γξ t1, qj(t1), pCj(t1) qi(t1) − · − · C0 − t1 C0 Γ(1 α(τ,t1)) β(t ,τ) − . (t τ)− 2 ( ) ( ) m ( ) ( ) 2− + m 1 γ qi t2 ξC0 t2, qj t2 , pCj t2 Γ(1 β(t ,τ)) dτ GC − − − 2 Symmetry 2019, 11, 1270 17 of 23

m 1 = m 1 = = where ξCi− ξC0− 0 when m 0.

Theorems 13 and 14 are the main results of this paper for the disturbed Hamiltonian systems with variable order. Based on the two theorems, adiabatic invariants for the disturbed Lagrangian systems with variable order can also been obtained.

Theorem 15. For the disturbed Lagrangian system in terms of the CRLVO RL α( , ),β( , ) ∂LRL t, qj, Dγ · · · · qj RL β( , ),α( , ) ∂LRL RL α( , ),β( , ) D · · · · = εW t, q , D · · · · q (94) 1 γ α( , ),β( , ) LRLi j γ j ∂qi − − RL ∂ Dγ · · · · qi α( , ),β( , ) m RL · · · · m If there exists a gauge function GLRL t, qj, Dγ qj such that the inﬁnitesimal generators ξLRL0 t, qj m and ξLRLi t, qj satisfy

. . ∂L α( , ),β( , ) . m m ∂L ∂L RL RL · · · · m m + + + RL m + RL α( , ),β( , ) Dγ ξLRLi qiξLRL0 LRL ξRL0 GLRL ∂q ξLRLi ∂t ∂RLD · · · · q · − · i γ i ! α(t,t1) ∂LRL d RL α( , ),β( , ) m γqi(t1)∂LRL m d (t t1)− + ( ) ( ) D · · · · qi ξ ( ) ( ) ξ t1, qj(t1) − (95) RL α , ,β , dt γ LRL0 RL α , ,β , LRL0 dt Γ(1 α(t,t1)) ∂ Dγ · · · · qi · − ∂ Dγ · · · · qi − β(t2,t) (1 γ)qi(t2)∂LRL m d (t2 t)− m 1 . m 1 + − ( ) ( ) ξ t2, qj(t2) − WLRLi ξ − q ξ − = 0 RL α , ,β , LRL0 dt Γ(1 β(t2,t)) LRLi i LRL0 ∂ Dγ · · · · qi − − −

Then there exists an adiabatic invariant. ( " z P m R t m . m RL β( , ),α( , ) ∂LRL ∂LRL ILRLz = ε ξ q ξ D · · · · ( ) ( ) + ( ) ( ) t1 LRLi i LRL0 1 γ RL α , ,β , RL α , ,β , m=0 − − ∂ Dγ · · · · qi ∂ Dγ · · · · qi h RL α( , ),β( , ) m . m m R t ∂LRL m D · · · · ξ q ξ dτ + LRL ξ ( ) ( ) γqi(t1)ξ t1, qj(t1) (96) γ LRLi i LRL0 LRL0 t1 RL α , ,β , LRL0 · − · − ∂ Dγ · · · · qi α(τ,t1) β(t2,τ) d (τ t1)− d (t2 τ)− − ( ) ( ) m ( ) − + m dτ Γ(1 α(τ,t )) 1 γ qi t2 ξLRL0 t2, qj t2 dτ Γ(1 β(t ,τ)) dτ GLRL · − 1 − − − 2 m 1 = m 1 = = where ξLRLi− ξLRL− 0 0 when m 0.

Theorem 16. For the disturbed Lagrangian system in terms of the CCVO C α( , ),β( , ) ∂LC t, qj, Dγ · · · · qj ∂LC RL β( , ),α( , ) C α( , ),β( , ) D · · · · = εW t, q , D · · · · q (97) 1 γ α( , ),β( , ) LCi j γ j ∂qi − − C ∂ Dγ · · · · qi α( , ),β( , ) m C · · · · m If there exists a gauge function GLC t, qj, Dγ qj such that the inﬁnitesimal generators ξLC0 t, qj and m ξLCi t, qj satisfy ! ∂L α( , ),β( , ) . ∂L α( , ),β( , ) ∂L C CD · · · · ξm q ξm + C d CD · · · · q + C ξm C α( , ),β( , ) γ LCi i LC0 C α( , ),β( , ) dt γ i ∂t LC0 ∂ Dγ · · · · qi · − ∂ Dγ · · · · qi · β(t2,t) . m . m ∂LC m (t2 t)− . ∂LC m +(1 γ) ( ) ( ) ξ t2, qj(t2) − q (t2) + ξ + LC ξ + G (98) C α , ,β , LC0 Γ(1 β(t2,t)) i ∂qi LCi LC0 LC − ∂ Dγ · · · · qi − · α(t,t1) ∂LC m (t t1)− . m 1 . m 1 γ ( ) ( ) ξ t1, qj(t1) − q (t1) WLCi ξ − q ξ − = 0 C α , ,β , LC0 Γ(1 α(t,t1)) i LCi i LC0 − ∂ Dγ · · · · qi − − − Symmetry 2019, 11, 1270 18 of 23

Then there exists an adiabatic invariant.

z ( " P R t ∂L α( , ),β( , ) . . = m C C · · · · m m + m m ILCz ε t α( , ),β( , ) Dγ ξLCi qiξLC0 ξLCi qiξLC0 1 ∂CD · · · · q · − − m=0 γ# i R t . α(τ,t1) RL β( , ),α( , ) ∂LC ∂LC m (τ t1)− (99) D · · · · ( ) ( ) dτ ( ) ( ) γξ t1, qj(t1) q (t1) − 1 γ C α , ,β , t1 C α , ,β , LC0 i Γ(1 α(τ,t1)) · − ∂ Dγ · · · · qi − ∂ Dγ · · · · qi − β(t ,τ) . (t τ)− 2 ( ) ( ) m ( ) 2− + m + m 1 γ qi t2 ξLC0 t2, qj t2 Γ(1 β(t ,τ)) dτ LC ξLC0 GLC − − − 2 · m 1 = m 1 = = where ξLCi− ξLC−0 0 when m 0.

Moreover, adiabatic invariants for the disturbed Hamiltonian systems with the RVO and the RCVO, the disturbed Hamiltonian systems with the CRL and the CC, the disturbed Lagrangian systems with the CRL and the CC, the disturbed classical Hamiltonian system, and the disturbed classical Lagrangian system can also be deduced. We only present them in the forms of remarks rather than describe them in detail.

Remark 11. Let β( , ) = α( , ), γ = 1 , so adiabatic invariants for the disturbed Hamiltonian systems in terms · · · · 2 of the RVO and the RCVO can be achieved.

Remark 12. Let α( , ) α, β( , ) β, so adiabatic invariants for the disturbed Hamiltonian systems with · · → · · → the CRL and the CC as well as the disturbed Lagrangian systems with the CRL and the CC can be deduced.

Remark 13. Let α( , ), β( , ) 1, so adiabatic invariants for the disturbed classical Hamiltonian system and · · · · → the disturbed classical Lagrangian system can be deduced.

6. Applications In this section, we present two applications to illustrate the Methods and Results. Application 1. The diﬀerential equation of the Lotka biochemical oscillator [67] of variable order is

RL α( , ),β( , ) RL β( , ),α( , ) · · · · = + · · · · = + Dγ x1 α1 β1 exp x2, D1 γ x2 α2 β2 exp x1 (100) − where α1, α2, β1, and β2 are constants.

Let x1 = q, x2 = pRL, so the Hamiltonian can be expressed as

HRL = α pRL α q + β exp pRL β exp q (101) 1 − 2 1 − 2 From Equation (61), we have

. 0 0 d RL α( , ),β( , ) RL α( , ),β( , ) pRL ξ D · · · · q + pRL D · · · · q α pRL + α q β exp pRL + β exp q ξ · RL0· dt γ · γ − 1 2 − 1 2 · RL0 α(t,t1) . 0 0 d (t t1)− RL α( , ),β( , ) 0 . 0 pRL γq(t1)ξ t1, qj(t1), pRLj(t1) − + pRL Dγ · · · · ξ qξ + GRL (102) − · RL0 dt Γ(1 α(t,t1)) · RL − RL0 − β(t2,t) d (t2 t)− +( + ) 0 + ( ) ( ) 0 ( ) ( ) − = α2 β2 exp q ξRL pRL 1 γ q t2 ξRL0 t2, qj t2 , pRL j t2 dt Γ(1 β(t ,t)) 0 · − − 2 It is obvious that ξ0 = 1, ξ0 = 0, G0 = 0 (103) RL0 − RL RL satisﬁes Equation (102), where [57]

α(t,t1) β(t2,t) d RL α( , ),β( , ) RL α( , ),β( , ) . d (t t1)− d (t2 t)− D · · · · q = D · · · · q + γq(t1) − (1 γ)q(t2) − (104) dt γ γ dt Γ(1 α(t, t )) − − dt Γ(1 β(t , t)) − 1 − 2 Symmetry 2019, 11, 1270 19 of 23

Then, a conserved quantity can be obtained from Theorem 1 as

Z t d RL α( , ),β( , ) . RL β( , ),α( , ) RL α( , ),β( , ) = · · · · + · · · · · · · · IHRL0 pRL Dγ q q D1 γ pRL dτ pRL Dγ q HRL (105) t1 ·dτ − − · −

When the Lotka biochemical oscillator model is disturbed as

RL α( , ),β( , ) RL β( , ),α( , ) · · · · = + · · · · = + ( + ) Dγ q α1 β1 exp pRL, D1 γ pRL α2 β2 exp q ε 2q 1 (106) − − We can get the following solution to Equation (89)

1 1 1 2 ξRL0 = 1, ξRL = 0, GRL = q + q (107)

Then, the ﬁrst order adiabatic invariant can be obtained from Theorem 13 as R t α( , ),β( , ) . β( , ),α( , ) α( , ),β( , ) = d RL · · · · + RL · · · · RL · · · · + IHRL1 t pRL dτ Dγ q q D1 γ pRL dτ pRL Dγ q α1pRL α2q 1 · − − · − 2 RL α( , ),β( , ) β1 exp pRL + β2 exp q) + ε q + q + pRL Dγ · · · · q α1pRL + α2q β1 exp pRL + β2 exp q (108) − · − − R t α( , ),β( , ) . β( , ),α( , ) d RL · · · · + RL · · · · t pRL dτ Dγ q q D1 γ pRL dτ − 1 · − When α, β 1, the classical diﬀerential equation of the Lotka biochemical oscillator, the classical → Hamiltonian, the classical conserved quantity, and the classical adiabatic invariant can be obtained, which are consistent with the results in [66]. Application 2. The Lagrangian of the two-dimensional isotropic harmonic oscillator of variable order is

" 2 2# 1 C α( , ),β( , ) C α( , ),β( , ) 1 h 2 2i L = m D · · · · q + D · · · · q k (q ) + (q ) (109) C 2 γ 1 γ 2 − 2 1 2 where k and m are constants. Firstly, the generalized momentum and the Hamiltonian of variable order can be obtained from Equation (37): C α( , ),β( , ) ∂LC t, qj, Dγ · · · · qj C α( , ),β( , ) pC1 = = m Dγ · · · · q1 C α( , ),β( , ) ∂ Dγ · · · · q1 C α( , ),β( , ) ∂LC t, qj, Dγ · · · · qj C α( , ),β( , ) pC2 = = m Dγ · · · · q2 C α( , ),β( , ) ∂ Dγ · · · · q2 C α( , ),β( , ) C α( , ),β( , ) C α( , ),β( , ) HC = pC1 Dγ · · · · q1 + pC2 Dγ · · · · q2 LC t, qj, Dγ · · · · qj · · − p p p 2 p 2 h i = p C1 + p C2 1 m C1 + C2 + 1 k (q )2 + (q )2 (110) C1· m C2· m − 2 m m 2 1 2 2 2 (pC1) (pC2) 1 h 2 2i = 2m + 2m + 2 k (q1) + (q2) From Equation (41), we have

C α( , ),β( , ) pC1 C α( , ),β( , ) pC2 Dγ · · · · q1 = m , Dγ · · · · q2 = m β( , ),α( , ) β( , ),α( , ) (111) RL · · · · = RL · · · · = D1 γ pC1 kq1, D1 γ pC2 kq2 − − − − And from Equation (65), we ﬁnd that

ξ0 = 1, ξ0 = ξ0 = 0, G0 = 0 (112) C0 − C1 C2 C Symmetry 2019, 11, 1270 20 of 23 is a proper solution. A conserved quantity can be obtained from Theorem 2 as

2 2 C α( , ),β( , ) C α( , ),β( , ) (pC1) (pC2) 1 2 2 IHC0 = pC1 Dγ · · · · q1 + pC2 Dγ · · · · q2 2m 2m 2 k (q1) + (q2) − · · − − − R t α( , ),β( , ) α( , ),β( , ) . β( , ),α( , ) + d C · · · · + d C · · · · + RL · · · · (113) t pC1 dτ Dγ q1 pC2 dτ Dγ q2 q1 D1 γ pC1 1 · · · − . β( , ),α( , ) + RL · · · · q2 D1 γ pC2 dτ · − When the isotropic harmonic oscillator model of variable order is disturbed as

α( , ),β( , ) p α( , ),β( , ) p β( , ),α( , ) CD · · · · q = C1 , CD · · · · q = C2 , RLD · · · · p = kq εq γ 1 m γ 2 m 1 γ C1 − 1 − 2 β( , ),α( , ) − (114) RL · · · · = D1 γ pC2 kq2 εq1 − − − we obtain 1 1 1 1 ξC0 = 1, ξC1 = ξC2 = 0, GC = q1q2 (115) which satisﬁes Equation (92). Then, the ﬁrst order adiabatic invariant can be obtained from Theorem 14 as

2 2 C α( , ),β( , ) C α( , ),β( , ) (pC1) (pC2) 1 2 2 IHC1 = pC1 Dγ · · · · q1 + pC2 Dγ · · · · q2 2m 2m 2 k (q1) + (q2) − · · − − − R t α( , ),β( , ) α( , ),β( , ) . β( , ),α( , ) . + d C · · · · + d C · · · · + RL · · · · + t pC1 dτ Dγ q1 pC2 dτ Dγ q2 q1 D1 γ pC1 q2 1 · · · − β( , ),α( , ) R t α( , ),β( , ) α( , ),β( , ) . RL · · · · d C · · · · + d C · · · · + D1 γ pC2 dτ ε t pC1 dτ Dγ q1 pC2 dτ Dγ q2 q1 (116) · − − 1 · · β( , ),α( , ) . β( , ),α( , ) α( , ),β( , ) α( , ),β( , ) RLD · · · · p + q RLD · · · · p dτ p CD · · · · q + p CD · · · · q · 1 γ C1 2· 1 γ C2 − C1· γ 1 C2· γ 2 − 2 − 2 (p ) (p ) C1 C2 1 k (q )2 + (q )2 q q − 2m − 2m − 2 1 2 − 1 2 When α, β 1, the classical Lagrangian, the classical Hamiltonian of the two-dimensional → isotropic harmonic oscillator, the classical conserved quantity, and the classical adiabatic invariant can be obtained, which is consistent with the results in [66].

7. Conclusions Equations of motion, Noether symmetry and conserved quantities, and perturbation to Noether symmetry and adiabatic invariants are investigated here. Hamilton equations with the CRLVO (Equation (36)), CCVO (Equation (41)), RVO (Equation (42)), RCVO (Equation (43)), CRL (Equation (48)), and CC (Equation (49)) are established. Then, Noether symmetry and conserved quantities are studied for those six Hamiltonian systems (Theorem 1, Theorem 2, Theorem 3, Theorem 4, Theorem 7, and Theorem 8). Meanwhile, Lagrange equations with the CRLVO (Equation (44)), CCVO (Equation (46)), CRL (Equation (50)), and CC (Equation (51)) are presented. Then Noether symmetry and conserved quantities are investigated for those four Lagrangian systems (Theorem 5, Theorem 6, Theorem 9, and Theorem 10). As for perturbation to Noether symmetry and adiabatic invariants, only adiabatic invariants with the CRLVO and CCVO for the Hamiltonian systems and the Lagrangian systems are studied and described in detail (Theorem 13, Theorem 14, Theorem 15, and Theorem 16). Among the results obtained in this paper, Equations (36), (41)–(44), and (46), Theorems 1–6, and Theorems 13–16 are new. Equations (48)–(53) and Theorems 7–12, which are deduced from the main results of this paper, are consistent with the existing results. It is generally known that there are three kinds of symmetry in analytical mechanics, i.e., Noether symmetry, Lie symmetry, and Mei symmetry. Noether symmetry plays an important role in finding the solutions to the differential equations of motion for mechanical systems because conserved quantities can be deduced from it. In fact, Lie symmetry and Mei symmetry are also able to deduce conserved quantities under certain conditions. The Lie symmetry means the invariance of the differential equations Symmetry 2019, 11, 1270 21 of 23 of motion under the infinitesimal transformations of time and coordinates. The Mei symmetry means the invariance under which the transformed dynamical functions still satisfy the original differential equations of motion. The conserved quantities deduced directly by the Lie symmetry and the Mei symmetry are called the Hojman conserved quantity and the Mei conserved quantity, respectively. However, only Noether symmetry is considered here, so Lie symmetry, Mei symmetry, perturbation to Lie symmetry, and perturbation to Mei symmetry are important and valuable aspects waiting to be studied.

Author Contributions: Writing—original draft, C.-J.S.; Writing—review and editing, Y.C. Funding: This research was funded by the National Natural Science Foundation of China (grant numbers 11972241, 11802193, 11801396, 11572212), the Natural Science Foundation of Jiangsu Province (grant number BK20170374), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (grant numbers 18KJB130005, 17KJB110016), the Jiangsu Government Scholarship for Overseas Studies, the Science Research Foundation of Suzhou University of Science and Technology (grant number 331812137), and the Natural Science Foundation of Suzhou University of Science and Technology. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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