Further Research for Lagrangian Mechanics within Generalized Fractional Operators

Abstract

In this article, the problems of the fractional calculus of variations are discussed based on generalized fractional operators, and the corresponding Lagrange equations are established. Then, the Noether symmetry method and the perturbation to Noether symmetry are analyzed in order to find the integrals of the equations. As a result, the conserved quantities and the adiabatic invariants are obtained. Due to the universality of the generalized fractional operators, the results achieved here can be used to solve other specific problems. Several examples are given to illustrate the universality of the methods and results.

1. Introduction

Fractional calculus, which has many applications [1,2,3,4,5,6,7,8,9], is a hot research topic. Since Riewe [10,11] found that the fractional derivative is a useful tool by which to express dissipative forces, research into the fractional calculus of variations has become popular. The Riemann–Liouville fractional derivative [12,13,14], the Caputo fractional derivative [15,16], the Riesz fractional derivative [17,18], the combined fractional derivative [19,20,21] and so forth [22,23,24,25,26,27] were defined and studied successively by interested scholars. Agrawal [28] introduced three new fractional operators in 2010, which we call generalized fractional operators, on the basis of summarizing the law regarding the definition of the fractional derivative. These generalized fractional operators can be reduced to Riemann–Liouville fractional derivatives, Caputo fractional derivatives, Riesz fractional derivatives, Weyl fractional derivatives, etc. Based on the generalized fractional operators, Agrawal established the Euler–Lagrange equations, developed the concepts of the adjoint differential operators and presented adjoint differential equations. In addition, Agrawal also briefly presented several examples where the kernels were considered different from those given in a preceding paragraph. Of course, the latter kernels were more general. In this paper, we plan to study the fractional calculus of variations in terms of the general kernels and to establish the corresponding differential equations.

For the established differential equations, the next step is to solve them. The symmetry method is a powerful tool in this regard. Symmetry is also called invariance, and there is a potential relationship between symmetry and conservation law. The Noether, Lie and Mei symmetry methods are three commonly used modern integral methods [29,30,31]. The Noether symmetry method was introduced by Noether [32] in 1918. Specifically, Noether symmetry refers to the invariance of the Hamilton action. Noether symmetry always leads to a conserved quantity, which helps reduce the degrees of freedom of the differential equations. Noether revealed the potential relationship between the conserved quantity of a mechanical system and its inherent dynamic symmetry for the first time, and also established the Noether symmetry theory. Many articles have been published on the Noether theorem, such as the Bible of symmetry methods [33], a comprehensive review of Noether’s theorem [34], Noether’s theorem for discrete equations [35], Noether’s theorem for semi-discrete equations [36,37], Noether’s theorem for the fractional Lagrangian system [38,39,40,41,42,43,44,45], Noether’s theorem for the fractional Hamiltonian system [46,47,48,49,50], Noether’s theorem for the fractional Birkhoffian system [51,52,53,54], etc. In this paper, we aim to establish Noether’s theorem within the generalized fractional operators in terms of the general kernels. Due to the generality of the general kernels, many of the results that were obtained previously will be seen to be special cases. Agrawal [28] took the optimal control problem and the fractional optimal control problem as examples by which to explain this generality when he studied the general kernels. Similarly, the Noether theorem obtained here will also become more general.

Furthermore, perturbation to symmetry and adiabatic invariants will also affect the integration of the differential equations of motion [55,56,57,58,59,60,61,62]. Therefore, perturbation to Noether symmetry is also a topic of the paper. This article demonstrates that the results developed here are more general. In addition, some special cases that were obtained under certain conditions agree with those that were obtained using certain other techniques.

This paper is organized as follows: Section 2 lists the definitions and certain basic properties of the generalized fractional operators. Section 3 directs attention to the Euler–Lagrange equation, transversality condition, local and global conserved quantities, perturbation to Noether symmetry and the adiabatic invariant on the basis of the generalized fractional operator $B_M^{\alpha }$. Section 4 focuses on the generalized fractional operator $K_M^{\alpha }$, through which the Noether symmetry and the perturbation to Noether symmetry are also studied. Finally, Section 5 details the conclusion.

2. Preliminaries

Three generalized fractional operators, found in ref. [28], are denoted as $K_M^{\alpha }$, $A_M^{\alpha }$ and $B_M^{\alpha }$. Their definitions and certain necessary properties are listed as follows.

Operators $K_M^{\alpha }$, $A_M^{\alpha }$ and $B_M^{\alpha }$ are defined as

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

(1)

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

(2)

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

(3)

where $f\left(t\right)$ is a function; $a<t<b$, $M=<a,t,b,m,\omega >$ is a parameter set; $m$ and $\omega$ are two real numbers; ${\kappa }_{\alpha }\left(t,\tau \right)$ is a kernel that may depend on a parameter $\alpha$; $n$ is an integer; and $D$ represents the classical integer order derivative.

From their definitions, we determine that these generalized fractional operators are linear. Namely, if $f_1\left(t\right)$ and $f_2\left(t\right)$ are two functions, then $K_M^{\alpha }\left(f_1\left(t\right)+f_2\left(t\right)\right)=K_M^{\alpha }{f}_1\left(t\right)+K_M^{\alpha }{f}_2\left(t\right)$, $A_M^{\alpha }\left(f_1\left(t\right)+f_2\left(t\right)\right)=A_M^{\alpha }{f}_1\left(t\right)+A_M^{\alpha }{f}_2\left(t\right)$, $B_M^{\alpha }\left(f_1\left(t\right)+f_2\left(t\right)\right)=B_M^{\alpha }{f}_1\left(t\right)+B_M^{\alpha }{f}_2\left(t\right)$. Operators $K_M^{\alpha }$, $A_M^{\alpha }$ and $B_M^{\alpha }$ satisfy the following integration by parts formulae:

$${\displaystyle {\int }_a^{b}g\left(t\right)K_M^{\alpha }f\left(t\right)\mathrm{d}t}={\displaystyle {\int }_a^{b}f\left(t\right)K_{M^{*}}^{\alpha }g\left(t\right)\mathrm{d}t},$$

(4)

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

(5)

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

(6)

where $M^{*}=<a,t,b,\omega ,m>$, $f\left(t\right)$ and $g\left(t\right)$ are two functions, and $D$ represents the classical integer order derivative.

In the following sections, the variational problems, the conserved quantities and the adiabatic invariants within the operators  $K_M^{\alpha }$ and $B_M^{\alpha }$ are studied. In this paper, Einstein’s summation convention is used, and we set $0<\alpha <1$.

Before the main text, the exchange relationships between the isochronous variation $\delta$ and the generalized fractional operators $K_M^{\alpha }$, $A_M^{\alpha }$ and $B_M^{\alpha }$ are presented first. By considering two infinitely closed orbits $\gamma$ and $\gamma +\mathrm{d}\gamma$, we denote the generalized coordinates as $q=q\left(t,\gamma \right)$ and $q=q\left(t,\gamma +\mathrm{d}\gamma \right)$, respectively. Then, we have $\delta q=q\left(t,\gamma +\mathrm{d}\gamma \right)-q\left(t,\gamma \right)$. Therefore, we have $\delta B_M^{\alpha }q=B_M^{\alpha }q\left(t,\gamma +\mathrm{d}\gamma \right)-B_M^{\alpha }q\left(t,\gamma \right)=B_M^{\alpha }\left[q\left(t,\gamma +\mathrm{d}\gamma \right)-q\left(t,\gamma \right)\right]=B_M^{\alpha }\delta q$. Similarly, we can obtain $\delta A_M^{\alpha }q=A_M^{\alpha }\delta q$, $\delta K_M^{\alpha }q=K_M^{\alpha }\delta q$.

3. Noether Theorem within the Generalized Fractional Operator of $B_M^{\alpha }$

3.1. Euler–Lagrange Equation and Transversality Condition

Hamilton action on the basis of the operator $B_M^{\alpha }$ gives

$$I_B\left[\mathit{q}\left(\cdot \right)\right]={\displaystyle {\int }_a^b{L}_B\left(t,\mathit{q},\dot{\mathit{q}},\mathit{z}\right)\mathrm{d}t}$$

(7)

where $\mathit{z}=\mathit{h}\left(t\right)+{\mathit{B}}_{\mathit{M}}^{\mathbf{\alpha }}\mathit{q}\left(t\right)$, $\mathit{q}=\left(q_1,q_2,\cdots ,q_n\right)$, $\dot{\mathit{q}}=\left({\dot{q}}_1,{\dot{q}}_2,\cdots ,{\dot{q}}_n\right)$, $\mathit{h}=\left(h_1,h_2,\cdots ,h_n\right)$, $h_i$ is an arbitrary function of time, $i=1,2,\cdots ,n$, ${\mathit{B}}_{\mathit{M}}^{\mathbf{\alpha }}\mathit{q}=\left(B_M^{\alpha }{q}_1,B_M^{\alpha }{q}_2,\cdots ,B_M^{\alpha }{q}_n\right)$, $\mathit{z}=\left(z_1,z_2,\cdots ,z_n\right)$. Then

$$\delta I_B=0$$

(8)

with the conditions

$$\delta B_M^{\alpha }{q}_i=B_M^{\alpha }\left(\delta q_i\right)$$

(9)

and

$$\mathit{q}\left(a\right)={\mathit{q}}_{\mathit{a}}, \mathit{q}\left(b\right)={\mathit{q}}_{\mathit{b}}$$

(10)

is the Hamilton principle, where ${\mathit{q}}_{\mathit{a}}=\left(q_{a1},q_{a2},\cdots ,q_{an}\right)$, ${\mathit{q}}_{\mathit{b}}=\left(q_{b1},q_{b2},\cdots ,q_{bn}\right)$, $q_{ai}$ and $q_{bj}$ are constants, $i,j=1,2,\cdots ,n$.

Through using Equation (9) and the integration by parts formulae, we have, from Equation (8),

$$\begin{array}{cc}\delta I_B\hfill & ={\displaystyle {\int }_a^b\left(\frac{\partial L_B}{\partial q_i}\delta q_i+\frac{\partial L_B}{\partial {\dot{q}}_i}\delta {\dot{q}}_i+\frac{\partial L_B}{\partial z_i}\delta z_i\right)\mathrm{d}t}\hfill \\ & ={\displaystyle {\int }_a^b\left(\frac{\partial L_B}{\partial q_i}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_B}{\partial {\dot{q}}_i}-A_{M^{*}}^{\alpha }\frac{\partial L_B}{\partial z_i}\right)\cdot \delta q_i\mathrm{d}t}+{\left.\frac{\partial L_B}{\partial {\dot{q}}_i}\cdot \delta q_i\right|}_{t=a}^{t=b}+{\left.K_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}\cdot \delta q_i\right|}_{t=a}^{t=b}=0.\hfill \end{array}$$

(11)

From Equation (10), the independence of $\delta q_i$ and the arbitrariness of the interval $\left[a,b\right]$ follows, such that

$$\frac{\partial L_B}{\partial q_i}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_B}{\partial {\dot{q}}_i}-A_{M^{*}}^{\alpha }\frac{\partial L_B}{\partial z_i}=0.$$

(12)

Equation (12) is the Euler–Lagrange equation in terms of the generalized fractional operator $B_M^{\alpha }$.

In addition, the transversality conditions within the generalized fractional operator $B_M^{\alpha }$ can also be obtained from Equation (11). When $\mathit{q}\left(a\right)$ and $\mathit{q}\left(b\right)$ are not specified, we have

$${\left.\left\{\frac{\partial L_B}{\partial {\dot{q}}_i}+K_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}\right\}\right|}_{t=a}=0$$

(13)

and

$${\left.\left\{\frac{\partial L_B}{\partial {\dot{q}}_i}+K_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}\right\}\right|}_{t=b}=0,$$

(14)

respectively.

3.2. Conserved Quantity

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

Firstly, two special conserved quantities, obtained only from the Euler–Lagrange equation within the generalized fractional operator $B_M^{\alpha }$ (Equation (12)), are presented.

When $\mathit{q}$ is not explicitly contained in $L_B$, i.e., $L_B\ne L_B\left(\mathit{q}\right)$, Equation (12) provides $0=-\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L_B}{\partial {\dot{q}}_i}+K_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}\right)$, hence,

$$\frac{\partial L_B}{\partial {\dot{q}}_i}+K_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}=\mathrm{const}, i=1,2,\cdots ,n.$$

(15)

When the time $t$ is not explicitly contained in $L_B$, i.e., $L_B\ne L_B\left(t\right)$, then $z_i=B_M^{\alpha }{q}_i\left(t\right)$. In addition, when the condition $\frac{\mathrm{d}}{\mathrm{d}\tau }{\kappa }_{\alpha }\left(t,\tau \right)=-\frac{\mathrm{d}}{\mathrm{d}t}{\kappa }_{\alpha }\left(t,\tau \right)$ holds, from the definition of the operator $B_M^{\alpha }$, we have

$$\frac{\mathrm{d}}{\mathrm{d}t}{B}_M^{\alpha }{q}_i=B_M^{\alpha }{\dot{q}}_i+m{\dot{q}}_i\left(a\right){\kappa }_{1-\alpha }\left(t,a\right)-\omega {\dot{q}}_i\left(b\right){\kappa }_{1-\alpha }\left(b,t\right).$$

(16)

In this case, for any interval $\left[a,b\right]$, if ${\dot{q}}_i\left(a\right)={\dot{q}}_i\left(b\right)=0$ holds, then we have

$${\displaystyle {\int }_a^b\frac{\mathrm{d}}{\mathrm{d}t}\left[L_B-{\dot{q}}_i\left(\frac{\partial L_B}{\partial {\dot{q}}_i}+K_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}\right)\right]\mathrm{d}t}={\displaystyle {\int }_a^b\left[\frac{\partial L_B}{\partial z_i}\cdot \left(\frac{\mathrm{d}}{\mathrm{d}t}{B}_M^{\alpha }{q}_i-B_M^{\alpha }{\dot{q}}_i\right)\right]\mathrm{d}t}=0.$$

(17)

Therefore,

$$L_B-{\dot{q}}_i\left(\frac{\partial L_B}{\partial {\dot{q}}_i}+K_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}\right)=\mathrm{const}, i=1,2,\cdots ,n.$$

(18)

We call Equations (15) and (18) local, conserved quantities.

Secondly, we then study the Noether symmetry and conserved quantity, the latter of which we call the global conserved quantity. Each Noether symmetry leads to the establishment of a global conserved quantity.

The infinitesimal transformations have the forms

$$\overline{t}=t+\Delta t, {\overline{q}}_i\left(\overline{t}\right)=q_i\left(t\right)+\Delta q_i (\mathrm{or} \overline{\mathit{q}}\left(\overline{t}\right)=\mathit{q}\left(t\right)+\Delta \mathit{q}),$$

(19)

whose expansions are

$$\overline{t}=t+\theta {\xi }_0^0\left(t,\mathit{q},\dot{\mathit{q}},\mathit{z}\right)+o\left(\theta \right), {\overline{q}}_i\left(\overline{t}\right)=q_i\left(t\right)+\theta {\xi }_i^0\left(t,\mathit{q},\dot{\mathit{q}},\mathit{z}\right)+o\left(\theta \right),$$

(20)

where $\theta$ is an infinitesimal parameter, and ${\xi }_0^0$ and ${\xi }_i^0$ are called infinitesimal generators of the infinitesimal transformations, $i=1,2,\cdots ,n$.

Before studying Noether symmetry, we present two necessary results,

$$\Delta {\dot{q}}_i=\delta {\dot{q}}_i+{\ddot{q}}_i\Delta t=\frac{\mathrm{d}}{\mathrm{d}t}\Delta q_i-{\dot{q}}_i\frac{\mathrm{d}}{\mathrm{d}t}\Delta t,$$

(21)

and

$$\begin{gathered} \Delta z_i={\overline{z}}_i\left(\overline{t}\right)-z_i\left(t\right)={\overline{h}}_i\left(\overline{t}\right)+B_{\overline{M}}^{\alpha }{\overline{q}}_i\left(\overline{t}\right)-h_i\left(t\right)-B_M^{\alpha }{q}_i\left(t\right) \\ ={\dot{h}}_i\left(t\right)\Delta t+B_M^{\alpha }\delta q_i+\Delta t\cdot \frac{\mathrm{d}}{\mathrm{d}t}{B}_M^{\alpha }{q}_i+\omega {\kappa }_{1-\alpha }\left(b,t\right){\dot{q}}_i\left(b\right)\Delta b-m{\kappa }_{1-\alpha }\left(t,a\right){\dot{q}}_i\left(a\right)\Delta a, \end{gathered}$$

(22)

where $\overline{M}=<\overline{a},\overline{t},\overline{b},m,\omega >$. Then, under the condition $\dot{\mathit{q}}\left(a\right)=\dot{\mathit{q}}\left(b\right)=0$, let $\Delta I_B$ be the linear part of ${\overline{I}}_B-I_B$. Moreover, when neglecting the higher order of $\theta$, we obtain

$$\begin{array}{cc}& ={\displaystyle {\int }_a^b\left(\frac{\partial L_B}{\partial t}\Delta t+\frac{\partial L_B}{\partial q_i}\Delta q_i+\frac{\partial L_B}{\partial {\dot{q}}_i}\Delta {\dot{q}}_i+\frac{\partial L_B}{\partial z_i}\Delta z_i+L_B\cdot \frac{\mathrm{d}}{\mathrm{d}t}\Delta t\right)}\mathrm{d}t\hfill \\ & =\theta {\displaystyle {\int }_a^b\left[\frac{\partial L_B}{\partial t}{\xi }_0^0+\frac{\partial L_B}{\partial q_i}{\xi }_i^0+\frac{\partial L_B}{\partial {\dot{q}}_i}\left({\dot{\xi }}_i^0-{\dot{q}}_i{\dot{\xi }}_0^0\right)+\frac{\partial L_B}{\partial z_i}\cdot \left({\dot{h}}_i\left(t\right)+\frac{\mathrm{d}}{\mathrm{d}t}{B}_M^{\alpha }{q}_i\right){\xi }_0^0\right.}\hfill \\ & \hspace{1em}\left.+L_B\cdot {\dot{\xi }}_0^0+K_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}\cdot \left({\dot{\xi }}_i^0-{\dot{q}}_i{\dot{\xi }}_0^0-{\ddot{q}}_i{\xi }_0^0\right)\right]\mathrm{d}t.\hfill \end{array}$$

(23)

Noether symmetry requires $\Delta I_B=0$. Hence, we obtain

Theorem 1. 

For the Lagrangian system (Equation (12)), if ${\xi }_0^0$ and ${\xi }_i^0$ satisfy the following Noether identity

$$\begin{gathered} \frac{\partial L_B}{\partial t}{\xi }_0^0+\frac{\partial L_B}{\partial q_i}{\xi }_i^0+\frac{\partial L_B}{\partial {\dot{q}}_i}\left({\dot{\xi }}_i^0-{\dot{q}}_i{\dot{\xi }}_0^0\right)+\frac{\partial L_B}{\partial z_i}\cdot \left({\dot{h}}_i\left(t\right)+\frac{\mathrm{d}}{\mathrm{d}t}{B}_M^{\alpha }{q}_i\right){\xi }_0^0 \\ +L_B\cdot {\dot{\xi }}_0^0+K_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}\cdot \left({\dot{\xi }}_i^0-{\dot{q}}_i{\dot{\xi }}_0^0-{\ddot{q}}_i{\xi }_0^0\right)=0, i=1,2,\cdots ,n, \end{gathered}$$

(24)

then there exists a conserved quantity

$$C_{B0}=L_B{\xi }_0^0+\left(\frac{\partial L_B}{\partial {\dot{q}}_i}+K_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}\right)\cdot \left({\xi }_i^0-{\dot{q}}_i{\xi }_0^0\right), i=1,2,\cdots ,n.$$

(25)

Proof. 

By using Equations (12) and (24), we obtain

$$\begin{gathered} \frac{\mathrm{d}}{\mathrm{d}t}{C}_{B0}={\xi }_0\left(\frac{\partial L_B}{\partial t}+\frac{\partial L_B}{\partial q_i}\cdot {\dot{q}}_i+\frac{\partial L_B}{\partial {\dot{q}}_i}\cdot {\ddot{q}}_i+\frac{\partial L_B}{\partial z_i}\cdot {\dot{z}}_i\right)+\left({\xi }_i-{\dot{q}}_i{\xi }_0\right)\left(\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_B}{\partial {\dot{q}}_i}+A_{M^{*}}^{\alpha }\frac{\partial L_B}{\partial z_i}\right) \\ +L_B{\dot{\xi }}_0+\left(\frac{\partial L_B}{\partial {\dot{q}}_i}+K_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}\right)\cdot \left({\dot{\xi }}_i-{\dot{q}}_i{\dot{\xi }}_0-{\ddot{q}}_i{\xi }_0\right)=\left({\xi }_i-{\dot{q}}_i{\xi }_0\right)\left(-\frac{\partial L_B}{\partial q_i}+\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_B}{\partial {\dot{q}}_i}+A_{M^{*}}^{\alpha }\frac{\partial L_B}{\partial z_i}\right)=0. \end{gathered}$$

Therefore, this proof is completed. □

If $\Delta I_B=-{\displaystyle {\int }_a^b\frac{\mathrm{d}}{\mathrm{d}t}\Delta G^0\mathrm{d}t}$, $\Delta G^0=\theta G^0\left(t,\mathit{q},\dot{\mathit{q}},\mathit{z}\right)+o\left(\theta \right)$, we obtain

Theorem 2. 

For the Lagrangian system (Equation (12)), if ${\xi }_0^0$, ${\xi }_i^0$ and a function $G^0$ satisfy the following Noether quasi-identity

$$\begin{gathered} \frac{\partial L_B}{\partial t}{\xi }_0^0+\frac{\partial L_B}{\partial q_i}{\xi }_i^0+\frac{\partial L_B}{\partial {\dot{q}}_i}\left({\dot{\xi }}_i^0-{\dot{q}}_i{\dot{\xi }}_0^0\right)+\frac{\partial L_B}{\partial z_i}\cdot \left({\dot{h}}_i\left(t\right)+\frac{\mathrm{d}}{\mathrm{d}t}{B}_M^{\alpha }{q}_i\right){\xi }_0^0 \\ +L_B\cdot {\dot{\xi }}_0^0+K_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}\cdot \left({\dot{\xi }}_i^0-{\dot{q}}_i{\dot{\xi }}_0^0-{\ddot{q}}_i{\xi }_0^0\right)+{\dot{G}}^0=0, i=1,2,\cdots ,n, \end{gathered}$$

(26)

then there exists a conserved quantity

$$C_{BG0}=L_B{\xi }_0^0+\left(\frac{\partial L_B}{\partial {\dot{q}}_i}+K_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}\right)\cdot \left({\xi }_i^0-{\dot{q}}_i{\xi }_0^0\right)+G^0.$$

(27)

Proof. 

From Equations (12) and (26), we can obtain $\mathrm{d}{C}_{BG0}/\mathrm{d}t=0$. □

3.3. Perturbation to Noether Symmetry and Adiabatic Invariant

When the system (Equation (12)) is disturbed, the original conserved quantity may change. Perturbation to Noether symmetry and the adiabatic invariant are studied in this section.

A quantity $C_z$ is called an adiabatic invariant if it contains a small parameter $\epsilon$, whose highest power is $z$ and which satisfies that $\mathrm{d}{C}_z/\mathrm{d}t$ is in proportion to ${\epsilon }^{z+1}$.

We assume that the Lagrangian system within the generalized fractional operator $B_M^{\alpha }$ is disturbed as

$$\frac{\partial L_B}{\partial q_i}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_B}{\partial {\dot{q}}_i}-A_{M^{*}}^{\alpha }\frac{\partial L_B}{\partial z_i}-\epsilon W_i\left(t,\mathit{q},\dot{\mathit{q}},\mathit{z}\right)=0.$$

(28)

The infinitesimal generators ${\xi }_0\left(t,\mathit{q},\dot{\mathit{q}},\mathit{z}\right)$ and ${\xi }_i\left(t,\mathit{q},\dot{\mathit{q}},\mathit{z}\right)$ of the infinitesimal transformations and the function $G\left(t,\mathit{q},\dot{\mathit{q}},\mathit{z}\right)$ for the perturbed Lagrangian system within the generalized fractional operator $B_M^{\alpha }$ (Equation (28)) have the forms

$$\begin{gathered} {\xi }_0={\xi }_0^0+\epsilon {\xi }_0^1+{\epsilon }^2{\xi }_0^2+\cdots ={\epsilon }^s{\xi }_0^s, {\xi }_i={\xi }_i^0+\epsilon {\xi }_i^1+{\epsilon }^2{\xi }_i^2+\cdots ={\epsilon }^s{\xi }_i^s, \\ G=G^0+\epsilon G^1+{\epsilon }^2{G}^2+\cdots ={\epsilon }^s{G}^s, s=0,1,2\cdots . \end{gathered}$$

(29)

Then, we obtain

Theorem 3. 

For the perturbed Lagrangian system (Equation (28)), if ${\xi }_0^s$, ${\xi }_i^s$ and a function $G^s$ satisfy

$$\begin{gathered} \frac{\partial L_B}{\partial t}{\xi }_0^s+\frac{\partial L_B}{\partial q_i}{\xi }_i^s+\frac{\partial L_B}{\partial {\dot{q}}_i}\left({\dot{\xi }}_i^s-{\dot{q}}_i{\dot{\xi }}_0^s\right)+\frac{\partial L_B}{\partial z_i}\cdot \left({\dot{h}}_i\left(t\right)+\frac{\mathrm{d}}{\mathrm{d}t}{B}_M^{\alpha }{q}_i\right){\xi }_0^s \\ +L_B\cdot {\dot{\xi }}_0^s+K_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}\cdot \left({\dot{\xi }}_i^s-{\dot{q}}_i{\dot{\xi }}_0^s-{\ddot{q}}_i{\xi }_0^s\right)-W_i\left({\xi }_i^{s-1}-{\dot{q}}_i{\xi }_0^{s-1}\right)+{\dot{G}}^s=0, i=1,2,\cdots ,n, \end{gathered}$$

(30)

then the following adiabatic invariant

$$C_{BGz}={\displaystyle \sum _{s=0}^z{\epsilon }^s\left[L_B{\xi }_0^s+\left(\frac{\partial L_B}{\partial {\dot{q}}_i}+K_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}\right)\cdot \left({\xi }_i^s-{\dot{q}}_i{\xi }_0^s\right)+G^s\right]}, i=1,2,\cdots ,n$$

(31)

exists, where ${\xi }_i^{s-1}={\xi }_0^{s-1}=0$ when $s=0$.

Proof. 

Using Equations (28) and (30), we obtain

$$\begin{array}{cc}\frac{\mathrm{d}{C}_{BGz}}{\mathrm{d}t}\hfill & ={\displaystyle \sum _{s=0}^z{\epsilon }^s\left[L_B{\dot{\xi }}_0^s+{\xi }_0^s\left(\frac{\partial L_B}{\partial t}+\frac{\partial L_B}{\partial q_i}{\dot{q}}_i+\frac{\partial L_B}{\partial {\dot{q}}_i}{\ddot{q}}_i+\frac{\partial L_B}{\partial z_i}{\dot{z}}_i\right)\right.}\hfill \\ & \left.+\left(\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_B}{\partial {\dot{q}}_i}+A_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}\right)\cdot \left({\xi }_i^s-{\dot{q}}_i{\xi }_0^s\right)+\left(\frac{\partial L_B}{\partial {\dot{q}}_i}+K_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}\right)\cdot \left({\dot{\xi }}_i^s-{\ddot{q}}_i{\xi }_0^s-{\dot{q}}_i{\dot{\xi }}_0^s\right)+{\dot{G}}^s\right]\hfill \\ & ={\displaystyle \sum _{s=0}^z{\epsilon }^s\left[W_i\left({\xi }_i^{s-1}-{\dot{q}}_i{\xi }_0^{s-1}\right)+\left(\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_B}{\partial {\dot{q}}_i}+A_{M^{*}}^{\alpha }\frac{\partial L_B}{\partial z_i}-\frac{\partial L_B}{\partial q_i}\right)\left({\xi }_i^s-{\dot{q}}_i{\xi }_0^s\right)\right]}=-{\epsilon }^{z+1}{W}_i\left({\xi }_i^z-{\dot{q}}_i{\xi }_0^z\right)\hfill \end{array}$$

Thus, this proof is completed. □

Remark 1. 

Conserved quantity, which is also called an exact invariant, is a special case for the adiabatic invariant. For example, let $z=0$ and $G^0=0$. Then, the adiabatic invariant in Theorem 3 becomes the conserved quantity in Theorem 1.

In the following text, two applications are given to illustrate the results obtained above.

3.4. Examples

Example 1. 

Consider the classical Hamilton action of

$$I\left[x\left(\cdot \right)\right]={\displaystyle {\int }_a^b{L}_x\left(t,x,\dot{x}\right)}\mathrm{d}t$$

(32)

with $x\left(a\right)=x_a$ and $x\left(b\right)$ unspecified.

For this problem, we have several known results.

Result 1. 

The Euler–Lagrange equation and the terminal condition are [29]

$$\frac{\partial L_x}{\partial x}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_x}{\partial \dot{x}}=0,$$

(33)

and

$${\left.\frac{\partial L_x}{\partial \dot{x}}\right|}_{t=b}=0.$$

(34)

Result 2. 

When $L_x$ is not an explicit function of $x$, i.e., $L_x\ne L_x\left(x\right)$, then $x$ is called cycle or ignorable; in this case, we determine from Equation (33) that $\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_x}{\partial \dot{x}}=0$, hence

$$\frac{\partial L_x}{\partial \dot{x}}=\mathrm{const}.$$

(35)

Equation (35) is the cyclic integral of Equation (33). The cyclic integral represents the conservation of the generalized momentum. It can be the conservation of momentum, or conservation of moments, or not.

When $L_x$ is not an explicit function of $t$, i.e., $L_x\ne L_x\left(t\right)$, then we have $\frac{\mathrm{d}}{\mathrm{d}t}\left(L_x-\frac{\partial L_x}{\partial \dot{x}}\dot{x}\right)=0$. Therefore,

$$L_x-\frac{\partial L_x}{\partial \dot{x}}\dot{x}=\mathrm{const}.$$

(36)

Equation (36) is called the generalized integral of energy and plays an important role in studying the problems of dynamics and physics. Under certain conditions, Equation (36) represents that the mechanical energy of the system is conserved.

Result 3 

([29]). The classical Noether identity and the classical conserved quantity have the following forms: for any interval $\left[a,b\right]$

$${\displaystyle {\int }_a^b\left[\frac{\partial L_x}{\partial t}\Delta t+\frac{\partial L_x}{\partial x}\Delta x+\frac{\partial L_x}{\partial \dot{x}}\cdot \frac{\mathrm{d}}{\mathrm{d}t}\Delta x+\left(L_x-\frac{\partial L_x}{\partial \dot{x}}\dot{x}\right)\frac{\mathrm{d}}{\mathrm{d}t}\Delta t\right]\mathrm{d}t}=0,$$

(37)

and

$${\displaystyle {\int }_a^b\frac{\mathrm{d}}{\mathrm{d}t}\left[\left(L_x-\frac{\partial L_x}{\partial \dot{x}}\dot{x}\right)\Delta t+\frac{\partial L_x}{\partial \dot{x}}\cdot \Delta x\right]\mathrm{d}t}=0.$$

(38)

The results listed above can be obtained using the methods presented here. We restate Equation (32) as

$$I_B\left[x\left(\cdot \right)\right]={\displaystyle {\int }_a^b{L}_B\left(t,\dot{q},z\right)\mathrm{d}t},$$

(39)

where $\dot{q}=\dot{x}$, and

$$x=z=x_a+{\displaystyle {\int }_a^t\dot{q}\left(\tau \right)\mathrm{d}\tau }=x_a+B_M^{\alpha }q\left(t\right).$$

(40)

In this case, $x\left(a\right)=x_a$, $h\left(t\right)=x_a$, $M=<a,t,b,1,0>$ and ${\kappa }_{1-\alpha }\left(t,\tau \right)={\kappa }_{1-\alpha }\left(\tau ,t\right)=1$. By making use of Equations (12) and (14), we obtain

$$\begin{array}{cc}0\hfill & =\frac{\partial L_B}{\partial q_i}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_B}{\partial {\dot{q}}_i}-A_{M^{*}}^{\alpha }\frac{\partial L_B}{\partial z_i}=-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_B}{\partial {\dot{q}}_i}-\frac{\mathrm{d}}{\mathrm{d}t}\left(K_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}\right)\hfill \\ & =-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_x}{\partial \dot{x}}-\frac{\mathrm{d}}{\mathrm{d}t}{\displaystyle {\int }_t^b\frac{\partial L_x}{\partial x}\left(\tau \right)\mathrm{d}\tau }=-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_x}{\partial \dot{x}}+\frac{\partial L_x}{\partial x}\hfill \end{array}$$

(41)

and

$$0={\left.\left\{\frac{\partial L_B}{\partial {\dot{q}}_i}+K_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}\right\}\right|}_{t=b}={\left.\left\{\frac{\partial L_x}{\partial \dot{x}}+{\displaystyle {\int }_t^b\frac{\partial L_x}{\partial x}\left(\tau \right)\mathrm{d}\tau }\right\}\right|}_{t=b}={\left.\frac{\partial L_x}{\partial \dot{x}}\right|}_{t=b}.$$

(42)

Equations (41) and (42) are consistent with Result 1 above.

Next, pay attention to Result 2. When $L_x$ is not an explicit function of $x$, it follows from Equation (40) that $L_B$ is not an explicit function of $z$, i.e., $L_B\ne L_B\left(z\right)$; then, from Equation (15), we obtain

$$\frac{\partial L_B}{\partial {\dot{q}}_i}+K_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}=\frac{\partial L_B}{\partial {\dot{q}}_i}=\frac{\partial L_x}{\partial \dot{x}}=\mathrm{const},$$

(43)

which is consistent with Equation (35). When $t$ is not explicitly contained in $L_x$, it means that $t$ is not explicitly contained in $L_B$, i.e., $L_B\ne L_B\left(t\right)$, then ${\dot{z}}_i\left(t\right)=\frac{\mathrm{d}}{\mathrm{d}t}{B}_M^{\alpha }{q}_i$. From Equation (18), for any interval $\left[a,b\right]$, we obtain

$$\begin{array}{c}0={\displaystyle {\int }_a^b\frac{\mathrm{d}}{\mathrm{d}t}\left[L_B-{\dot{q}}_i\left(\frac{\partial L_B}{\partial {\dot{q}}_i}+K_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}\right)\right]\mathrm{d}t}\hfill \\ ={\displaystyle {\int }_a^b\left(\frac{\partial L_B}{\partial q_i}{\dot{q}}_i+\frac{\partial L_B}{\partial {\dot{q}}_i}{\ddot{q}}_i+\frac{\partial L_B}{\partial z_i}\cdot \frac{\mathrm{d}}{\mathrm{d}t}{B}_M^{\alpha }{q}_i-{\dot{q}}_i\cdot \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_B}{\partial {\dot{q}}_i}-{\ddot{q}}_i\frac{\partial L_B}{\partial {\dot{q}}_i}-{\dot{q}}_i\cdot \frac{\mathrm{d}}{\mathrm{d}t}{K}_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}-\frac{\partial L_B}{\partial z_i}{B}_M^{\alpha }{\dot{q}}_i\right)\mathrm{d}t}\hfill \\ ={\displaystyle {\int }_a^b\left(\frac{\partial L_x}{\partial \dot{x}}\ddot{x}-\dot{x}\cdot \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_x}{\partial \dot{x}}-\ddot{x}\frac{\partial L_x}{\partial \dot{x}}+\dot{x}\frac{\partial L_x}{\partial x}\right)\mathrm{d}t}={\displaystyle {\int }_a^b\frac{\mathrm{d}}{\mathrm{d}t}\left(L_x-\dot{x}\frac{\partial L_x}{\partial \dot{x}}\right)\mathrm{d}t}.\hfill \end{array}$$

(44)

From Equation (44), Equation (36) can be obtained.

Finally, the Noether identity and the conserved quantity should be checked. For any interval $\left[a,b\right]$ from Equation (24), we have

$$\begin{array}{cc}0\hfill & ={\displaystyle {\int }_a^b\left[\frac{\partial L_B}{\partial t}\Delta t+\frac{\partial L_B}{\partial q_i}\Delta q_i+\frac{\partial L_B}{\partial {\dot{q}}_i}\cdot \Delta {\dot{q}}_i+\frac{\partial L_B}{\partial z_i}\cdot \left({\dot{h}}_i\left(t\right)+\frac{\mathrm{d}}{\mathrm{d}t}{B}_M^{\alpha }{q}_i\right)\Delta t+L_B\cdot \frac{\mathrm{d}}{\mathrm{d}t}\Delta t+K_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}\cdot \delta {\dot{q}}_i\right]\mathrm{d}t}\hfill \\ & ={\displaystyle {\int }_a^b\left[\frac{\partial L_x}{\partial t}\Delta t+\frac{\partial L_x}{\partial \dot{x}}\cdot \Delta \dot{x}+\frac{\partial L_x}{\partial x}\cdot \dot{x}\Delta t+L_x\cdot \frac{\mathrm{d}}{\mathrm{d}t}\Delta t-\frac{\partial L_x}{\partial \dot{x}}\cdot \delta \dot{x}\right]\mathrm{d}t}\hfill \\ & ={\displaystyle {\int }_a^b\left[\frac{\partial L_x}{\partial t}\Delta t+\frac{\partial L_x}{\partial x}\Delta x+\frac{\partial L_x}{\partial \dot{x}}\cdot \frac{\mathrm{d}}{\mathrm{d}t}\Delta x+\left(L_x-\frac{\partial L_x}{\partial \dot{x}}\dot{x}\right)\frac{\mathrm{d}}{\mathrm{d}t}\Delta t\right]\mathrm{d}t}.\hfill \end{array}$$

(45)

Additionally, from Equation (25), we have

$$\begin{array}{cc}0\hfill & ={\displaystyle {\int }_a^b\frac{\mathrm{d}}{\mathrm{d}t}\left[L_B\Delta t+\left(\frac{\partial L_B}{\partial {\dot{q}}_i}+K_{M^{*}}^{1-\alpha }\frac{\partial L_B}{\partial z_i}\right)\cdot \delta q_i\right]\mathrm{d}t}={\displaystyle {\int }_a^b\frac{\mathrm{d}}{\mathrm{d}t}\left(L_x\Delta t+\frac{\partial L}{\partial \dot{x}}\cdot \delta x\right)\mathrm{d}t}+{\displaystyle {\int }_t^b\frac{\partial L_x}{\partial x}\left(\tau \right)\mathrm{d}\tau }{\left.\cdot \delta x\right|}_{t=a}^{t=b}\hfill \\ & ={\displaystyle {\int }_a^b\frac{\mathrm{d}}{\mathrm{d}t}\left[\left(L_x-\frac{\partial L_x}{\partial \dot{x}}\dot{x}\right)\Delta t+\frac{\partial L_x}{\partial \dot{x}}\cdot \Delta x\right]\mathrm{d}t}.\hfill \end{array}$$

(46)

Equations (45) and (46) coincide with Result 3 above.

Therefore, both the current and the standard schemes give the same results, including the Euler–Lagrange equation, the Noether identity and the conserved quantity.

Example 2. 

Let $L_B=z^2+z\dot{q}+\frac{1}{2}{\dot{q}}^2$, $z=B_M^{\alpha }q$, try to find its conserved quantity.

Equation (12) thus gives

$$\frac{\mathrm{d}}{\mathrm{d}t}\left(z+\dot{q}\right)+A_{M^{*}}^{\alpha }\left(2z+\dot{q}\right)=0.$$

(47)

Then, Equation (15) gives

$$z+\dot{q}+K_{M^{*}}^{1-\alpha }\left(2z+\dot{q}\right)=\mathrm{const}.$$

(48)

Finally, under the condition  $\frac{\mathrm{d}}{\mathrm{d}\tau }{\kappa }_{\alpha }\left(t,\tau \right)=-\frac{\mathrm{d}}{\mathrm{d}t}{\kappa }_{\alpha }\left(t,\tau \right)$  and  ${\dot{q}}_i\left(a\right)={\dot{q}}_i\left(b\right)=0$, Equation (18) gives

$$z^2-\frac{1}{2}{\dot{q}}^2-\dot{q}{K}_{M^{*}}^{1-\alpha }\left(2z+\dot{q}\right)=\mathrm{const}.$$

(49)

4. Noether Theorem within the Generalized Fractional Operator $K_M^{\alpha }$

4.1. Euler–Lagrange Equation and Terminal Condition

Considering the following variational problem, find the extremum of the functional

$$I_K\left[{\mathit{q}}_{\mathit{K}}\left(\cdot \right)\right]={\displaystyle {\int }_a^b{L}_K\left(t,{\mathit{q}}_{\mathit{k}},{\dot{\mathit{q}}}_{\mathit{K}},{\mathit{z}}_{\mathit{K}}\right)\mathrm{d}t},$$

(50)

where Equation (50) is called Hamilton action with the generalized fractional operator $K_M^{\alpha }$, ${\mathit{z}}_{\mathit{K}}={\mathit{h}}_{\mathit{K}}\left(t\right)+{\mathit{K}}_{\mathit{M}}^{\mathbf{\alpha }}{\mathit{q}}_{\mathit{K}}\left(t\right)$, ${\mathit{q}}_{\mathit{K}}=\left(q_{K1},q_{K2},\cdots ,q_{Kn}\right)$, ${\dot{\mathit{q}}}_{\mathit{K}}=\left({\dot{q}}_{K1},{\dot{q}}_{K2},\cdots ,{\dot{q}}_{Kn}\right)$, ${\mathit{h}}_{\mathit{K}}=\left(h_{K1},h_{K2},\cdots ,h_{Kn}\right)$; where $h_{Ki}$ is an arbitrary function of time, ${\mathit{K}}_{\mathit{M}}^{\alpha }{\mathit{q}}_K=\left(K_M^{\alpha }{q}_{K1},K_M^{\alpha }{q}_{K2},\cdots ,K_M^{\alpha }{q}_{Kn}\right)$, ${\mathit{z}}_{\mathit{K}}=\left(z_{K1},z_{K2},\cdots ,z_{Kn}\right)$, and the Lagrangian $L_K:\left[t_1,t_2\right]\times {ℝ}^n\times {ℝ}^n\times {ℝ}^n\to ℝ$ and $q_{Ki}:\left[t_1,t_2\right]\to ℝ$ are assumed to be $C^2$ functions, $i=1,2,\cdots ,n$. Agrawal [28] obtained the Euler–Lagrange equation and the terminal conditions with the generalized fractional operator $K_M^{\alpha }$ as

$$\frac{\partial L_K}{\partial q_{Ki}}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_K}{\partial {\dot{q}}_{Ki}}+K_{M^{*}}^{\alpha }\frac{\partial L_K}{\partial z_{Ki}}=0,$$

(51)

$${\left.\frac{\partial L_K}{\partial {\dot{q}}_{Ki}}\right|}_{t=a}=0, \mathrm{if} {\mathit{q}}_{\mathit{K}}\left(a\right) \mathrm{is} \mathrm{not} \mathrm{specified}, \mathrm{and}$$

(52)

$${\left.\frac{\partial L_K}{\partial {\dot{q}}_{Ki}}\right|}_{t=b}=0, \mathrm{if} {\mathit{q}}_{\mathit{K}}\left(b\right) \mathrm{is} \mathrm{not} \mathrm{specified}, i=1,2,\cdots ,n.$$

(53)

4.2. Noether Symmetry and Conserved Quantity

For a dynamic system, Noether symmetry can lead to a conserved quantity.

We express the infinitesimal transformations as

$$\overline{t}=t+\Delta t, {\overline{q}}_{Ki}\left(\overline{t}\right)=q_{Ki}\left(t\right)+\Delta q_{Ki} (\mathrm{or} {\overline{\mathit{q}}}_{\mathit{K}}\left(\overline{\mathit{t}}\right)={\mathit{q}}_{\mathit{K}}\left(\mathit{t}\right)+\Delta {\mathit{q}}_{\mathit{K}}),$$

(54)

whose expanded forms are

$$\overline{t}=t+{\theta }_K{\xi }_{K0}^0\left(t,{\mathit{q}}_{\mathit{K}},{\dot{\mathit{q}}}_{\mathit{K}},{\mathit{z}}_{\mathit{K}}\right)+o\left({\theta }_K\right),{\overline{q}}_{Ki}\left(\overline{t}\right)=q_{Ki}\left(t\right)+{\theta }_K{\xi }_{Ki}^0\left(t,{\mathit{q}}_{\mathit{K}},{\dot{\mathit{q}}}_{\mathit{K}},{\mathit{z}}_{\mathit{K}}\right)+o\left({\theta }_K\right),$$

(55)

where ${\theta }_K$ is an infinitesimal parameter, and ${\xi }_{K0}^0$ and ${\xi }_{Ki}^0$ are called infinitesimal generators of the infinitesimal transformations.

Let $\Delta I_K$ be the linear part of ${\overline{I}}_K-I_K$ and let it be neglected by the higher order of ${\theta }_K$, we then have

$$\begin{array}{c}\Delta I_K={\overline{I}}_K-I_K={\displaystyle {\int }_{\overline{a}}^{\overline{b}}{L}_K\left(\overline{t},{\overline{\mathit{q}}}_{\mathit{K}},{\dot{\overline{\mathit{q}}}}_{\mathit{K}},{\overline{\mathit{z}}}_{\mathit{K}}\right)}\mathrm{d}\overline{t}-{\displaystyle {\int }_a^b{L}_K\left(t,{\mathit{q}}_{\mathit{K}},{\dot{\mathit{q}}}_{\mathit{K}},{\mathit{z}}_{\mathit{K}}\right)}\mathrm{d}t\hfill \\ ={\displaystyle {\int }_a^b\left(\frac{\partial L_K}{\partial t}\Delta t+\frac{\partial L_K}{\partial q_{Ki}}\Delta q_{Ki}+\frac{\partial L_K}{\partial {\dot{q}}_{Ki}}\Delta {\dot{q}}_{Ki}+\frac{\partial L_K}{\partial z_{Ki}}\Delta z_{Ki}+L_K\cdot \frac{\mathrm{d}}{\mathrm{d}t}\Delta t\right)}\mathrm{d}t\hfill \\ ={\theta }_K{\displaystyle {\int }_a^b\left[\frac{\partial L_K}{\partial t}{\xi }_{K0}^0+\frac{\partial L_K}{\partial q_{Ki}}{\xi }_{Ki}^0+\frac{\partial L_K}{\partial {\dot{q}}_{Ki}}\left({\dot{\xi }}_{Ki}^0-{\dot{q}}_{Ki}{\dot{\xi }}_{K0}^0\right)+\frac{\partial L_K}{\partial z_{Ki}}\cdot \left({\dot{h}}_{Ki}\left(t\right)+\frac{\mathrm{d}}{\mathrm{d}t}{K}_M^{\alpha }{q}_{Ki}\right){\xi }_{K0}^0\right.}\hfill \\ \hspace{1em}\hspace{1em}\left.+L_K\cdot {\dot{\xi }}_{K0}^0+\left({\xi }_{Ki}^0-{\dot{q}}_{Ki}{\xi }_{K0}^0\right)\cdot K_{M^{*}}^{\alpha }\frac{\partial L_K}{\partial z_{Ki}}\right]\mathrm{d}t.\hfill \end{array}$$

(56)

Noether symmetry requires $\Delta I_K=0$. Hence, we obtain

Theorem 4. 

For the Lagrangian system (Equation (51)), if ${\xi }_{K0}^0$ and ${\xi }_{Ki}^0$ satisfy the following Noether identity

$$\begin{gathered} \frac{\partial L_K}{\partial t}{\xi }_{K0}^0+\frac{\partial L_K}{\partial q_{Ki}}{\xi }_{Ki}^0+\frac{\partial L_K}{\partial {\dot{q}}_{Ki}}\left({\dot{\xi }}_{Ki}^0-{\dot{q}}_{Ki}{\dot{\xi }}_{K0}^0\right)+\frac{\partial L_K}{\partial z_{Ki}}\cdot \left({\dot{h}}_{Ki}\left(t\right)+\frac{\mathrm{d}}{\mathrm{d}t}{K}_M^{\alpha }{q}_{Ki}\right){\xi }_{K0}^0 \\ +L_K\cdot {\dot{\xi }}_{K0}^0+\left({\xi }_{Ki}^0-{\dot{q}}_{Ki}{\xi }_{K0}^0\right)\cdot K_{M^{*}}^{\alpha }\frac{\partial L_K}{\partial z_{Ki}}=0, \end{gathered}$$

(57)

then there exists a conserved quantity

$$C_{K0}=L_K{\xi }_{K0}^0+\frac{\partial L_K}{\partial {\dot{q}}_{Ki}}\cdot \left({\xi }_{Ki}^0-{\dot{q}}_{Ki}{\xi }_{K0}^0\right).$$

(58)

Proof. 

From Equations (51) and (57), we have

$$\begin{gathered} \frac{\mathrm{d}{C}_{K0}}{\mathrm{d}t}=L_K{\dot{\xi }}_{K0}^0+{\xi }_{K0}^0\left(\frac{\partial L_K}{\partial t}+\frac{\partial L_K}{\partial q_{Ki}}{\dot{q}}_{Ki}+\frac{\partial L_K}{\partial {\dot{q}}_{Ki}}{\ddot{q}}_{Ki}+\frac{\partial L_K}{\partial z_{Ki}}{\dot{z}}_{Ki}\right)+\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_K}{\partial {\dot{q}}_{Ki}}\cdot \left({\xi }_{Ki}^0-{\dot{q}}_{Ki}{\xi }_{K0}^0\right) \\ +\frac{\partial L_K}{\partial {\dot{q}}_{Ki}}\cdot \left({\dot{\xi }}_{Ki}^0-{\dot{q}}_{Ki}{\dot{\xi }}_{K0}^0-{\ddot{q}}_{Ki}{\xi }_{K0}^0\right)=\left(\frac{\partial L_K}{\partial q_{Ki}}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_K}{\partial {\dot{q}}_{Ki}}+K_{M^{*}}^{\alpha }\frac{\partial L_K}{\partial z_{Ki}}\right)\cdot \left({\xi }_{Ki}^0-{\dot{q}}_{Ki}{\xi }_{K0}^0\right)=0 \end{gathered}$$

Therefore, this proof is completed. □

Let $\Delta I_K=-{\displaystyle {\int }_a^b\frac{\mathrm{d}}{\mathrm{d}t}\Delta G_K^0\mathrm{d}t}$, $\Delta G_K^0={\theta }_K{G}_K^0\left(t,{\mathit{q}}_{\mathit{K}},{\dot{\mathit{q}}}_{\mathit{K}},{\mathit{z}}_{\mathit{K}}\right)+o\left({\theta }_K\right)$, then we obtain

Theorem 5. 

For the Lagrangian system (Equation (51)), if ${\xi }_{K0}^0$, ${\xi }_{Ki}^0$ and a function $G_K^0$ satisfy the following Noether quasi-identity

$$\begin{gathered} \frac{\partial L_K}{\partial t}{\xi }_{K0}^0+\frac{\partial L_K}{\partial q_{Ki}}{\xi }_{Ki}^0+\frac{\partial L_K}{\partial {\dot{q}}_{Ki}}\left({\dot{\xi }}_{Ki}^0-{\dot{q}}_{Ki}{\dot{\xi }}_{K0}^0\right)+\frac{\partial L_K}{\partial z_{Ki}}\cdot \left({\dot{h}}_{Ki}\left(t\right)+\frac{\mathrm{d}}{\mathrm{d}t}{K}_M^{\alpha }{q}_{Ki}\right){\xi }_{K0}^0 \\ +L_K\cdot {\dot{\xi }}_{K0}^0+\left({\xi }_{Ki}^0-{\dot{q}}_{Ki}{\xi }_{K0}^0\right)\cdot K_{M^{*}}^{\alpha }\frac{\partial L_K}{\partial z_{Ki}}+{\dot{G}}_K^0=0, \end{gathered}$$

(59)

then a conserved quantity

$$C_{KG0}=L_K{\xi }_{K0}^0+\frac{\partial L_K}{\partial {\dot{q}}_{Ki}}\cdot \left({\xi }_{Ki}^0-{\dot{q}}_{Ki}{\xi }_{K0}^0\right)+G_K^0$$

(60)

exists.

Proof. 

From Equations (51) and (59), we can obtain $\mathrm{d}{C}_{KG0}/\mathrm{d}t=0$. □

Remark 2. 

It is obvious that when the function $G_K^0=0$, Theorem 5 reduces to Theorem 4.

4.3. Perturbation to Noether Symmetry and Adiabatic Invariant

When the dynamic system in terms of the generalized fractional operator (Equation (51)) is disturbed, the original conserved quantity may also change. In this section, we study the perturbation to Noether symmetry and the corresponding adiabatic invariant.

Under the small disturbance forces ${\epsilon }_K{W}_{Ki}\left(t,{\mathit{q}}_{\mathit{K}},{\dot{\mathit{q}}}_{\mathit{K}},{\mathit{z}}_{\mathit{K}}\right)$, $i=1,2,\cdots ,n$, the Lagrangian system, the infinitesimal generators ${\xi }_{K0}\left(t,{\mathit{q}}_{\mathit{K}},{\dot{\mathit{q}}}_{\mathit{K}},{\mathit{z}}_{\mathit{K}}\right)$, ${\xi }_K\left(t,{\mathit{q}}_{\mathit{K}},{\dot{\mathit{q}}}_{\mathit{K}},{\mathit{z}}_{\mathit{K}}\right)$ and the function $G_K\left(t,{\mathit{q}}_{\mathit{K}},{\dot{\mathit{q}}}_{\mathit{K}},{\mathit{z}}_{\mathit{K}}\right)$ are assumed to have the following forms

$$\frac{\partial L_K}{\partial q_{Ki}}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_K}{\partial {\dot{q}}_{Ki}}+K_{M^{*}}^{\alpha }\frac{\partial L_K}{\partial z_{Ki}}-{\epsilon }_K{W}_{Ki}=0,$$

(61)

$$\begin{gathered} {\xi }_{K0}={\xi }_{K0}^0+{\epsilon }_K{\xi }_{K0}^1+{\epsilon }_K^2{\xi }_{K0}^2+\cdots ={\epsilon }_K^s{\xi }_{K0}^s, {\xi }_{Ki}={\xi }_{Ki}^0+{\epsilon }_K{\xi }_{Ki}^1+{\epsilon }_K^2{\xi }_{Ki}^2+\cdots ={\epsilon }_K^s{\xi }_{Ki}^s, \\ {G}_K=G_K^0+{\epsilon }_K{G}_K^1+{\epsilon }_K^2{G}_K^2+\cdots ={\epsilon }_K^s{G}_K^s, s=0,1,2\cdots , i=1,2,\cdots ,n, \end{gathered}$$

(62)

where ${\epsilon }_K$ is a small parameter. Then, we obtain

Theorem 6. 

For the perturbed Lagrangian system (Equation (61)), if ${\xi }_{K0}^s$ and ${\xi }_{Ki}^s$ satisfy

$$\begin{gathered} \frac{\partial L_K}{\partial t}{\xi }_{K0}^s+\frac{\partial L_K}{\partial q_{Ki}}{\xi }_{Ki}^s+\frac{\partial L_K}{\partial {\dot{q}}_{Ki}}\left({\dot{\xi }}_{Ki}^s-{\dot{q}}_{Ki}{\dot{\xi }}_{K0}^s\right)+\frac{\partial L_K}{\partial z_{Ki}}\cdot \left({\dot{h}}_{Ki}\left(t\right)+\frac{\mathrm{d}}{\mathrm{d}t}{K}_M^{\alpha }{q}_{Ki}\right){\xi }_{K0}^s \\ +L_K\cdot {\dot{\xi }}_{K0}^s+\left({\xi }_{Ki}^s-{\dot{q}}_{Ki}{\xi }_{K0}^s\right)\cdot K_{M^{*}}^{\alpha }\frac{\partial L_K}{\partial z_{Ki}}-W_{Ki}\left({\xi }_{Ki}^{s-1}-{\dot{q}}_{Ki}{\xi }_{K0}^{s-1}\right)=0, \end{gathered}$$

(63)

where  ${\xi }_{Ki}^{s-1}={\xi }_{K0}^{s-1}=0$  when  $s=0$, then there exists an adiabatic invariant

$$C_{Kz}={\displaystyle \sum _{s=0}^z{\epsilon }_K^s\left[L_K{\xi }_{K0}^s+\frac{\partial L_K}{\partial {\dot{q}}_{Ki}}\cdot \left({\xi }_{Ki}^s-{\dot{q}}_{Ki}{\xi }_{K0}^s\right)\right]}.$$

(64)

Proof. 

From Equations (61) and (63), we have

$$\begin{array}{cc}\frac{\mathrm{d}{C}_{Kz}}{\mathrm{d}t}\hfill & ={\displaystyle \sum _{s=0}^z{\epsilon }_K^s\left[L_K{\dot{\xi }}_{K0}^s+{\xi }_{K0}^s\left(\frac{\partial L_K}{\partial t}+\frac{\partial L_K}{\partial q_{Ki}}{\dot{q}}_{Ki}+\frac{\partial L_K}{\partial {\dot{q}}_{Ki}}{\ddot{q}}_{Ki}+\frac{\partial L_K}{\partial z_{Ki}}{\dot{z}}_{Ki}\right)\right.}\hfill \\ & \hspace{1em}\left.+\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_K}{\partial {\dot{q}}_{Ki}}\cdot \left({\xi }_{Ki}^s-{\dot{q}}_{Ki}{\xi }_{K0}^s\right)+\frac{\partial L_K}{\partial {\dot{q}}_{Ki}}\cdot \left({\dot{\xi }}_{Ki}^s-{\dot{q}}_{Ki}{\dot{\xi }}_{K0}^s-{\ddot{q}}_{Ki}{\xi }_{K0}^s\right)\right]\hfill \\ & ={\displaystyle \sum _{s=0}^z{\epsilon }_K^s\left[\left({\xi }_{Ki}^s-{\dot{q}}_{Ki}{\xi }_{K0}^s\right)\left(-\frac{\partial L_K}{\partial q_{Ki}}+\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_K}{\partial {\dot{q}}_{Ki}}-K_{M^{*}}^{\alpha }\frac{\partial L_K}{\partial z_{Ki}}\right)+W_{Ki}\left({\xi }_{Ki}^{s-1}-{\dot{q}}_{Ki}{\xi }_{K0}^{s-1}\right)\right]}\hfill \\ & =-{\epsilon }_K^{z+1}{W}_{Ki}\left({\xi }_{Ki}^z-{\dot{q}}_{Ki}{\xi }_{K0}^z\right).\hfill \end{array}$$

Thus, this proof is completed. □

Theorem 7. 

For the perturbed Lagrangian system (Equation (61)), if ${\xi }_{K0}^s$, ${\xi }_{Ki}^s$ and a function $G_K^s$ satisfy

$$\begin{gathered} \frac{\partial L_K}{\partial t}{\xi }_{K0}^s+\frac{\partial L_K}{\partial q_{Ki}}{\xi }_{Ki}^s+\frac{\partial L_K}{\partial {\dot{q}}_{Ki}}\left({\dot{\xi }}_{Ki}^s-{\dot{q}}_{Ki}{\dot{\xi }}_{K0}^s\right)+\frac{\partial L_K}{\partial z_{Ki}}\cdot \left({\dot{h}}_{Ki}\left(t\right)+\frac{\mathrm{d}}{\mathrm{d}t}{K}_M^{\alpha }{q}_{Ki}\right){\xi }_{K0}^s \\ +L_K\cdot {\dot{\xi }}_{K0}^s+\left({\xi }_{Ki}^s-{\dot{q}}_{Ki}{\xi }_{K0}^s\right)\cdot K_{M^{*}}^{\alpha }\frac{\partial L_K}{\partial z_{Ki}}+{\dot{G}}_K^s-W_{Ki}\left({\xi }_{Ki}^{s-1}-{\dot{q}}_{Ki}{\xi }_{K0}^{s-1}\right)=0, \end{gathered}$$

(65)

where  ${\xi }_{Ki}^{s-1}={\xi }_{K0}^{s-1}=0$  when  $s=0$, then there exists an adiabatic invariant

$$C_{KGz}={\displaystyle \sum _{s=0}^z{\epsilon }_K^s\left[L_K{\xi }_{K0}^s+\frac{\partial L_K}{\partial {\dot{q}}_{Ki}}\cdot \left({\xi }_{Ki}^s-{\dot{q}}_{Ki}{\xi }_{K0}^s\right)+G_K^s\right]}.$$

(66)

Proof. 

From Equations (61) and (65), we have $\mathrm{d}{C}_{KGz}/\mathrm{d}t=-{\epsilon }_K^{z+1}{W}_{Ki}\left({\xi }_{Ki}^z-{\dot{q}}_{Ki}{\xi }_{K0}^z\right)$. □

Remark 3. 

Theorem 7 reduces to Theorem 6 when $G_K^s=0$, $s=0,1,2\cdots$. In addition, when $z=0$, the adiabatic invariants in Theorem 6 and Theorem 7 reduce to the conserved quantities in Theorem 4 and Theorem 5, respectively.

4.4. An Example

Example 3. 

Given a functional

$$I\left[x\right]={\displaystyle {\int }_a^b{L}_x\left(t,x,\dot{x}\right)\mathrm{d}t},$$

(67)

which is subjected to $x\left(a\right)=x_a$ and $x\left(b\right)$ being unspecified, then there are several known results [29] for it.

Firstly, when we try to find its extremum, the necessary and the terminal condition for this problem can be found as

$$\frac{\partial L_x}{\partial x}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_x}{\partial \dot{x}}=0,$$

(68)

and

$${\left.\frac{\partial L_x}{\partial \dot{x}}\right|}_{t=b}=0.$$

(69)

Secondly, for the Lagrangian mechanics (Equation (68)), if ${\eta }_0^0$ and ${\eta }^0$ satisfy

$$\frac{\partial L_x}{\partial t}{\eta }_0^0+\frac{\partial L_x}{\partial x}{\eta }^0+\frac{\partial L_x}{\partial \dot{x}}\left({\dot{\eta }}^0-\dot{x}{\dot{\eta }}_0^0\right)+L_x\cdot {\dot{\eta }}_0^0=0,$$

(70)

then there exists a conserved quantity

$$C_0=L_x{\eta }_0^0+\frac{\partial L_x}{\partial \dot{x}}\cdot \left({\eta }^0-\dot{x}{\eta }_0^0\right),$$

(71)

where Equation (70) is called the standard Noether identity, and ${\eta }_0^0$ and ${\eta }^0$ are the infinitesimal generators of the following infinitesimal transformations

$$\overline{t}=t+\Delta t, \overline{x}\left(\overline{t}\right)=x\left(t\right)+\Delta x,$$

(72)

whose expanded forms are

$$\overline{t}=t+{\theta }_x{\eta }_0^0\left(t,x,\dot{x}\right)+o\left({\theta }_x\right), \overline{x}\left(\overline{t}\right)=x\left(t\right)+{\theta }_x{\eta }^0\left(t,x,\dot{x}\right)+o\left({\theta }_x\right),$$

(73)

where ${\theta }_x$ is an infinitesimal parameter.

Those results (Equations (68)–(71)) listed above can be obtained using the methods presented in this paper. We can also restate this problem: find the extremum of the functional

$$I_K\left[x\right]={\displaystyle {\int }_a^b{L}_K\left(t,q_K,z_K\right)\mathrm{d}t},$$

(74)

where $q_K=\dot{x}$, $x=z_K=x_a+{\displaystyle {\int }_a^{t}q\left(\tau \right)\mathrm{d}\tau }=x_a+K_M^{\alpha }{q}_K$, $x\left(a\right)=x_a$, $h_K\left(t\right)=x_a$, $M=<a,t,b,1,0>$. Then, it follows from Equation (51) that

$$0=\frac{\partial L_K}{\partial q_K}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L_K}{\partial {\dot{q}}_K}+K_{M^{*}}^{\alpha }\frac{\partial L_K}{\partial z_K}=\frac{\partial L_x}{\partial \dot{x}}+{\displaystyle {\int }_t^b\frac{\partial L_x}{\partial x}\left(\tau \right)\mathrm{d}\tau }.$$

(75)

When differentiating Equation (75) with respect to time, Equation (68) is obtained. Additionally, Equation (69) is obtained by substituting $t=b$ in Equation (75). Equations (68) and (69) were checked by Agrawal [28]. Let us verify Equations (70) and (71). In fact,

$$\begin{array}{c}{\theta }_K{\displaystyle {\int }_a^b\left[\frac{\partial L_K}{\partial t}{\xi }_{K0}^0+\frac{\partial L_K}{\partial q_K}{\xi }_K^0+\frac{\partial L_K}{\partial {\dot{q}}_K}\left({\dot{\xi }}_K^0-{\dot{q}}_K{\dot{\xi }}_{K0}^0\right)+\frac{\partial L_K}{\partial z_K}\cdot \left({\dot{h}}_K\left(t\right)+\frac{\mathrm{d}}{\mathrm{d}t}{K}_M^{\alpha }{q}_K\right){\xi }_{K0}^0\right.}\hfill \\ \hspace{1em}\hspace{1em} \left.+L_K\cdot {\dot{\xi }}_{K0}^0+\left({\xi }_K^0-{\dot{q}}_K{\xi }_{K0}^0\right)\cdot K_{M^{*}}^{\alpha }\frac{\partial L_K}{\partial z_K}\right]\mathrm{d}t\hfill \\ ={\displaystyle {\int }_a^b\left[\frac{\partial L_K}{\partial t}\Delta t+\frac{\partial L_K}{\partial q_K}\Delta q_K+\frac{\partial L_K}{\partial z_K}\cdot {\dot{z}}_K\Delta t+L_K\cdot \frac{\mathrm{d}}{\mathrm{d}t}\Delta t-\frac{\partial L_K}{\partial q_K}\cdot \delta q_K\right]\mathrm{d}t}\hfill \\ ={\displaystyle {\int }_a^b\left[\frac{\partial L_x}{\partial t}\Delta t+\frac{\partial L_x}{\partial \dot{x}}\Delta \dot{x}+\frac{\partial L_x}{\partial x}\cdot \dot{x}\Delta t+L_x\cdot \frac{\mathrm{d}}{\mathrm{d}t}\Delta t-\frac{\partial L_x}{\partial \dot{x}}\cdot \delta \dot{x}\right]\mathrm{d}t}\hfill \\ ={\theta }_x{\displaystyle {\int }_a^b\left[\frac{\partial L_x}{\partial t}{\eta }_0^0+\frac{\partial L_x}{\partial x}{\eta }^0+\frac{\partial L_x}{\partial \dot{x}}\left({\dot{\eta }}^0-\dot{x}{\dot{\eta }}_0^0\right)+L_x\cdot {\dot{\eta }}_0^0\right]\mathrm{d}t},\hfill \end{array}$$

(76)

and

$$\begin{array}{c}{\displaystyle {\int }_a^b\frac{\mathrm{d}}{\mathrm{d}t}[L_K\cdot \Delta t+\frac{\partial L_K}{\partial {\dot{q}}_K}·(\Delta q_K-{\dot{q}}_K\Delta t)]\mathrm{d}t}\hfill \\ ={\displaystyle {\int }_a^b\left(L_K\cdot \frac{\mathrm{d}}{\mathrm{d}t}\Delta t+\frac{\partial L_K}{\partial t}\cdot \Delta t-\frac{\partial L_K}{\partial q_K}\cdot \delta q_K+\frac{\partial L_K}{\partial q_K}\cdot \Delta q_K+\frac{\partial L_K}{\partial z_K}{\dot{z}}_K\cdot \Delta t\right)\mathrm{d}t}\hfill \\ ={\displaystyle {\int }_a^b\left(L_x\cdot \frac{\mathrm{d}}{\mathrm{d}t}\Delta t+\frac{\partial L_x}{\partial t}\cdot \Delta t-\frac{\partial L_x}{\partial \dot{x}}\cdot \delta \dot{x}+\frac{\partial L_x}{\partial \dot{x}}\cdot \Delta \dot{x}+\frac{\partial L_x}{\partial x}\dot{x}\cdot \Delta t\right)\mathrm{d}t}\hfill \\ {\displaystyle ={\int }_a^b\frac{\mathrm{d}}{\mathrm{d}t}[L_x\cdot \Delta t+\frac{\partial L_x}{\partial \dot{x}}·(\Delta x-\dot{x}\Delta t)]\mathrm{d}t.}\hfill \end{array}$$

(77)

Therefore, Equations (76) and (77) illustrate that both the current and the standard schemes deliver the same Noether theorem.

5. Conclusions

Conserved quantities and the adiabatic invariants in terms of the generalized fractional operators $K_M^{\alpha }$ and $B_M^{\alpha }$. Several simple examples were discussed to explain the universality of the generalized fractional operators in terms of the general kernels. Theorem 1–Theorem 7 are each new work. In fact, the conserved quantities for the optimal control problem, the fractional optimal control problem and some other more complex problems (these problems can be seen in ref. [28]) can also be solved using the methods and results obtained here.

Symmetry analysis and conservation laws can be adopted for application to many specific equations [63,64,65,66]. Particularly, for the constrained mechanics systems, the Lie symmetry method and the Mei symmetry method within generalized fractional operators (in terms of the general kernels) are also to be investigated in the near future.