**1. Introduction**

The study of symmetry and invariants for non-conservative or nonlinear dynamics is of grea<sup>t</sup> significance. It is also a frontier research field of analytical mechanics. In a classical sense, the symmetries we refer to mainly include Noether symmetries [1] and Lie symmetries [2,3]. Noether symmetry and Lie symmetry are two different symmetries. After infinitesimal transformation, the former means the invariant property of the Hamilton action functional, and the latter means the invariant property of the differential equation. Unlike Noether symmetry or Lie symmetry, Mei proposed a new symmetry called form invariance in 2000 [4]. Form invariance, also known as Mei symmetry, refers to an invariant property, that is, the dynamical functions (such as Lagrangian, Hamiltonian, Birkhoffian, generalized force, etc.) that appear in the dynamical equations of the mechanical system still satisfy the original equations after the infinitesimal transformation. Under certain conditions, symmetry can lead to invariants, which are also called conserved quantities. Noether symmetry, Lie symmetry, and Mei symmetry of dynamical systems described by standard Lagrangian may lead to Noether conserved quantities or Mei conserved quantities [5], etc. Conserved quantities can also be independent of Lagrangian. For example, conserved quantities can be directly constructed from Lie symmetry neither utilizing Lagrangian nor Hamiltonian, or can be formulated for systems of differential equations by using symmetries and adjoint symmetries together regardless of the existence of a Lagrangian, see [5–8] and references therein. So far, much progress has been made in the study of the symmetries and corresponding invariants [9–22]. However, there are few reports on the symmetries and invariants of dynamical systems based on non-standard Lagrangians.

The concept of non-standard Lagrangian was first mentioned in Arnold's works in 1978 as non-natural Lagrangian [23], but it was ignored due to the lack of Hamilton form corresponding to it. Until 1984, when discussing the region adaptability of classical theories in Yang–Mills quantum field

theory [24], it was found that non-standard Lagrangians were directly related to the color constraint problem, which led to their renewed attention. The advantage of non-standard Lagrangians is that it can better describe nonlinear problems and plays an important role in non-conservative systems, dissipative systems, quantum field theory, etc. [25–34].

Fractional calculus can better describe natural phenomena and engineering problems [35–37]. Since Riewe [38,39] introduced fractional calculus into the modeling of non-conservative systems, fractional Lagrangian mechanics, fractional Hamiltonian mechanics and fractional Birkho ffian mechanics have been proposed and studied, and important progress has been made in fractional dynamics modeling, analysis, and calculation, see for example [40–47] and references therein. In 2005, El-Nabulsi proposed the fractional action-like variational approach to study non-conservative dynamical problem, in which the action is constructed by using the Riemann-Liouville definition of fractional integral [48,49], and extended it to the case of non-standard Lagrangians [28,50]. Considering the characteristic of fractional action-like variational approach, we call the non-conservative model obtained in this way as quasi-fractional order dynamical system. Here, we propose and study Mei symmetry and its invariants for the quasi-fractional order dynamical system with non-standard Lagrangians. New conserved quantities and new adiabatic invariants are derived from Mei symmetry of the quasi-fractional dynamical systems.

### **2. Mei Symmetry and Invariants of Quasi-Fractional Dynamical System Based on Exponential Lagrangians**

For the quasi-fractional dynamical system whose action functional depends on exponential Lagrangian, the Euler–Lagrange equations that are derived in Appendix A can be expressed as

$$(t - \tau)^{\alpha - 1} \exp L \Big( \frac{\partial L}{\partial q\_s} - \frac{\text{d}}{\text{d}\tau} \frac{\partial L}{\partial \dot{q}\_s} - \frac{\partial L}{\partial \dot{q}\_s} \frac{\text{d}L}{\text{d}\tau} + \frac{\alpha - 1}{t - \tau} \frac{\partial L}{\partial \dot{q}\_s} \Big) = 0,\\ (s = 1, 2, \cdots, n), \tag{1}$$

where *qs*(*s* = 1, 2, ··· , *n*) are the generalized coordinates, *L* = *L* τ, *qs*, . *qs* is the standard Lagrangian, 0 < α ≤ 1, τ is the intrinsic time, *t* is the observer time, and τ is not equal to *t*.

Let us introduce the infinitesimal transformations as

$$\tau^\* = \tau + \varepsilon \varepsilon\_0(\tau, q\_k, \dot{q}\_k), \\ q\_s^\*(\tau^\*) = q\_s(\tau) + \varepsilon \dot{\varepsilon}\_s(\tau, q\_k, \dot{q}\_k), \\ \text{ (s = 1, 2, \dots, n; k = 1, 2, \dots, n)}, \tag{2}$$

where ε is a small parameter, ς0 and ξ*s* are the infinitesimals. After the transformation of Equation (2), exp *L* is transformed into the following form

$$\exp L^\* = \exp L \big( \tau^\*, \dot{q}\_{s\prime}^\* \frac{\mathrm{d}q\_s^\*}{\mathrm{d}\tau^\*} \big) = \exp L \big( \tau, q\_{s\prime} \dot{q}\_s \big) + \varepsilon X^{(1)} (\exp L) + O \big( \varepsilon^2 \big), \tag{3}$$

where *X*(1) is the first extension of the infinitesimal generator *X*, that is [4]

$$X = \varepsilon\_0 \frac{\partial}{\partial \tau} + \underline{\boldsymbol{\zeta}}\_s \frac{\partial}{\partial q\_s}, \; X^{(1)} = \underline{\boldsymbol{\zeta}}\_0 \frac{\partial}{\partial \tau} + \underline{\boldsymbol{\zeta}}\_s \frac{\partial}{\partial q\_s} + \left(\dot{\underline{\boldsymbol{\zeta}}}\_s - \dot{q}\_s \dot{\underline{\boldsymbol{\zeta}}}\_0\right) \frac{\partial}{\partial \dot{q}\_s}.\tag{4}$$

If *L* is replaced with *L*<sup>∗</sup>, Equation (1) still holds, namely

$$\mathbb{L}\left(t-\tau\right)^{\alpha-1}\exp L^\*\left(\frac{\partial L^\*}{\partial q\_\delta} - \frac{\mathrm{d}}{\mathrm{d}\tau}\frac{\partial L^\*}{\partial \dot{q}\_\delta} - \frac{\partial L^\*}{\partial \dot{q}\_\delta}\frac{\mathrm{d}L^\*}{\mathrm{d}\tau} + \frac{\alpha-1}{t-\tau}\frac{\partial L^\*}{\partial \dot{q}\_\delta}\right) = 0,\\ (\mathrm{s} = 1, 2, \cdots, n), \tag{5}$$

then this invariance is called Mei symmetry of quasi-fractional dynamical system (1). Substituting the formula (3) into Equation (5), and considering Equation (1), we have

$$\begin{array}{ll} \left(t-\tau\right)^{\alpha-1} \exp L \Big( \frac{\partial X^{(1)}(L)}{\partial q\_s} - \frac{\partial}{\partial \tau} \frac{\partial X^{(1)}(L)}{\partial \dot{q}\_s} - \frac{\partial X^{(1)}(L)}{\partial \dot{q}\_s} \Big) \overset{\text{Id}}{\text{d}\tau} \\\ -\frac{\partial L}{\partial \dot{q}\_s} \frac{\text{d}X^{(1)}(L)}{\text{d}\tau} + \frac{\alpha-1}{t-\tau} \frac{\partial X^{(1)}(L)}{\partial \dot{q}\_s} \Big) = 0, \ (s=1,2,\cdots,n). \end{array} \tag{6}$$

Equation (6) is the criterion for Mei symmetry of system (1).

**Theorem 1.** *For the quasi-fractional dynamical system (1), if there is a gauge function G* = *<sup>G</sup>*<sup>τ</sup>, *qk*, .*qksuch that the structural equation*

$$\left(\frac{1-\alpha}{t-\tau}\varepsilon\_0 + \dot{\varepsilon}\_0\right) \mathbf{X}^{(1)} (\exp L) (t-\tau)^{\alpha-1} + \mathbf{X}^{(1)} \left[\mathbf{X}^{(1)} (\exp L)\right] (t-\tau)^{\alpha-1} + \dot{G} = 0 \tag{7}$$

*holds, the Mei symmetry directly leads to the new conserved quantity*

$$I\_0 = \left(t - \tau\right)^{a-1} X^{(1)} \left(\exp L\right) \varepsilon\_0 + \left(t - \tau\right)^{a-1} \frac{\partial X^{(1)} \left(\exp L\right)}{\partial \dot{q}\_s} \left(\xi\_s - \dot{q}\_s \varepsilon\_0\right) + G = \text{const.}\tag{8}$$

**Proof.**

d*I*0 dτ = 1−<sup>α</sup> *t*−τ (*t* − τ)<sup>α</sup>−1*X*(1)(exp *<sup>L</sup>*)<sup>ς</sup>0 + (*t* − τ)<sup>α</sup>−<sup>1</sup> <sup>d</sup>*X*(1)(exp *L*) dτ ς0 + (*t* − τ)<sup>α</sup>−1*X*(1)(exp *<sup>L</sup>*).ς0 +1−<sup>α</sup> *t*−τ (*t* − τ)<sup>α</sup>−<sup>1</sup> <sup>∂</sup>*X*(1)(exp *L*) <sup>∂</sup>.*qs* ξ*s* − .*qs*ς0 + (*t* − τ)<sup>α</sup>−<sup>1</sup> ddτ <sup>∂</sup>*X*(1)(exp *L*) <sup>∂</sup>.*qs* ξ*s* − .*qs*ς0 +(*t* − τ)<sup>α</sup>−<sup>1</sup> <sup>∂</sup>*X*(1)(exp *L*) <sup>∂</sup>.*qs* .ξ*s* − .*qs* .ς0 − ..*qs*ς0 − (*t* − τ)<sup>α</sup>−1*X*(1)(exp *L*) 1−<sup>α</sup> *t*−τ ς0 + .ς0 −(*<sup>t</sup>* − τ)<sup>α</sup>−1*X*(1)*X*(1)(exp *L*) = −∂*X*(1)(*L*) ∂*qs* + ∂*X*(1)(*L*) <sup>∂</sup>.*qs* d*L*dτ + ∂*L*<sup>∂</sup>.*qs* d*X*(1)(*L*) dτ + ddτ ∂*X*(1)(*L*) <sup>∂</sup>.*qs* + 1−<sup>α</sup> *t*−τ ∂*X*(1)(*L*) <sup>∂</sup>.*qs* × <sup>×</sup><sup>ξ</sup>*s* − .*qs*ς0(*<sup>t</sup>* − τ)<sup>α</sup>−<sup>1</sup> exp *L* + 1−<sup>α</sup> *t*−τ ς0 + .<sup>ς</sup>0*X*(1)(exp *L*)(*t* − τ)<sup>α</sup>−<sup>1</sup> <sup>+</sup>*X*(1)*X*(1)(exp *L*) (*t* − τ)<sup>α</sup>−<sup>1</sup> + − ∂*L*∂*qs* + ddτ ∂*L*<sup>∂</sup>.*qs* + ∂*L*<sup>∂</sup>.*qs* d*L*dτ + 1−<sup>α</sup> *t*−τ ∂*L*<sup>∂</sup>.*qs* × <sup>×</sup><sup>ξ</sup>*s* − .*qs*ς0(*<sup>t</sup>* − τ)<sup>α</sup>−1*X*(1)(exp *L*) + .*G*. (9)

Substituting Equations (1) and (6) into the formula (9), and using Equation (7), we obtain

$$\frac{\mathrm{d}I\_{0}}{\mathrm{d}\tau} = \left\{ \left( \frac{1-a}{t-\tau} \boldsymbol{\varepsilon}\_{0} + \dot{\boldsymbol{\varepsilon}}\_{0} \right) \mathrm{X}^{(1)}(\exp L) + \mathrm{X}^{(1)} \left[ \mathrm{X}^{(1)}(\exp L) \right] \right\} (t-\tau)^{a-1} + \dot{\mathcal{G}} = 0. \tag{10}$$

Thus, we ge<sup>t</sup> the desired result. -

The new conserved quantity (8) is called Mei conserved quantity. Since the system is not disturbed, it is an exact invariant. However, in nature and engineering, it is often affected by disturbing forces. If the system is affected by small disturbance <sup>υ</sup>*Qs*, its Mei symmetry and the corresponding conserved quantity (8) will change correspondingly. The infinitesimals of transformations (2) without disturbance is denoted as ς00, ξ<sup>0</sup>*s*, while the infinitesimals are changed into ς0, ξ*s* when disturbed, and we have

$$\boldsymbol{\varepsilon}\_{0} = \boldsymbol{\varepsilon}\_{0}^{0} + \nu \boldsymbol{\varepsilon}\_{0}^{1} + \nu^{2} \boldsymbol{\varepsilon}\_{0}^{2} + \cdots, \ \boldsymbol{\varepsilon}\_{s} = \boldsymbol{\xi}\_{s}^{0} + \nu \boldsymbol{\xi}\_{s}^{1} + \nu^{2} \boldsymbol{\xi}\_{s}^{2} + \cdots, \ (\mathbf{s} = 1, 2, \cdots, n). \tag{11}$$

Meanwhile, we let *G*<sup>0</sup> represent the gauge function without disturbance, and *G* represent the gauge function of the disturbed system, which is the small perturbation on the basis of *G*0, i.e.,

$$G = G^0 + \nu G^1 + \nu^2 G^2 + \dotsb \,\tag{12}$$

If a small disturbance υ*Qs* is applied, Equation (1) is changed to

$$\mathbb{E}\left((t-\tau)^{\alpha-1}\exp L\Big|\frac{\partial L}{\partial \dot{q}\_s} - \frac{\mathbf{d}}{\mathbf{d}\tau}\frac{\partial L}{\partial \dot{q}\_s} - \frac{\partial L}{\partial \dot{q}\_s}\frac{\mathbf{d}L}{\mathbf{d}\tau} + \frac{\alpha-1}{t-\tau}\frac{\partial L}{\partial \dot{q}\_s}\right) = \nu Q\_s. \tag{13}$$

Accordingly, Equation (6) is changed to

$$\begin{split} (t-\tau)^{a-1} \exp L \Big[ \frac{\partial X^{(1)}(L)}{\partial q\_{\mu}} - \frac{d}{d\tau} \frac{\partial X^{(1)}(L)}{\partial \dot{q}\_{s}} - \frac{\partial X^{(1)}(L)}{\partial \dot{q}\_{s}} \Big] \overset{\text{Id}}{\text{d}\tau} \\ -\frac{\partial L}{\partial \dot{q}\_{s}} \frac{d X^{(1)}(L)}{\partial \tau} + \frac{a-1}{t-\tau} \frac{\partial \dot{X}^{(1)}(L)}{\partial \dot{q}\_{s}} \Big] = \nu X^{(1)}(Q\_{s}), \ (s = 1, 2, \cdots, n). \end{split} \tag{14}$$

Substituting the formulae (11) into Equation (14), we ge<sup>t</sup>

$$\begin{split} (t-\tau)^{a-1} \upsilon^{m} \exp L \Big[ \frac{\partial X\_{\text{m}}^{(1)}(L)}{\partial \dot{q}\_{s}} - \frac{d}{d\tau} \frac{\partial X\_{\text{m}}^{(1)}(L)}{\partial \dot{q}\_{s}} - \frac{\partial X\_{\text{m}}^{(1)}(L)}{\partial \dot{q}\_{s}} \frac{dL}{d\tau} \\ -\frac{\partial L}{\partial \dot{q}\_{s}} \frac{d X\_{\text{m}}^{(1)}(L)}{\partial \tau} + \frac{a-1}{t-\tau} \frac{\partial X\_{\text{m}}^{(1)}(L)}{\partial \dot{q}\_{s}} \end{split} \tag{15}$$

where

$$X^{(1)} = \upsilon^m X^{(1)}\_m \,, \ X^{(1)}\_m = \varepsilon^m\_0 \frac{\partial}{\partial \tau} + \xi^m\_s \frac{\partial}{\partial q\_s} + \left(\dot{\xi}^m\_s - \dot{q}\_s \dot{\xi}^m\_0\right) \frac{\partial}{\partial \dot{q}\_s} \,. \tag{16}$$

As a result, we have

**Theorem 2.** *If the quasi-fractional dynamical system (1) is disturbed by a small disturbance* υ*Qs, and there is a gauge function G* = *<sup>G</sup>*<sup>τ</sup>, *qk*, .*qk such that the structural equation*

$$\begin{aligned} & \left( \frac{1-\underline{\boldsymbol{\alpha}}}{t-\tau} \boldsymbol{\varepsilon}\_{0}^{\boldsymbol{m}} + \boldsymbol{\varepsilon}\_{0}^{\boldsymbol{m}} \right) \mathbf{X}\_{m}^{(1)} (\exp L) (t-\tau)^{\boldsymbol{a}-1} + \mathbf{X}\_{m}^{(1)} \Big[ \mathbf{X}\_{m}^{(1)} (\exp L) \Big] (t-\tau)^{\boldsymbol{a}-1} + \dot{\mathcal{G}}^{\boldsymbol{m}} \\ & - \left[ \mathbf{X}\_{m-1}^{(1)} (\boldsymbol{Q}\_{s}) + \mathbf{Q}\_{s} \mathbf{X}\_{m-1}^{(1)} (L) \right] \Big( \boldsymbol{\xi}\_{s}^{\boldsymbol{m}-1} - \dot{\boldsymbol{q}}\_{s} \boldsymbol{\varepsilon}\_{0}^{\boldsymbol{m}-1} \Big) = \mathbf{0}, \ (\boldsymbol{s} = 1, 2, \cdots, \boldsymbol{n}; \boldsymbol{m} = 0, 1, 2, \cdots \cdot), \end{aligned} \tag{17}$$

*holds, where G* = *z <sup>m</sup>*=0 υ*mGm and* ς<sup>−</sup><sup>1</sup> 0 = ξ−<sup>1</sup> *s* = 0*, the Mei symmetry directly leads to the new adiabatic invariant*

$$I\_z = \sum\_{m=0}^{z} v^m \left[ (t - \tau)^{a-1} X\_m^{(1)} (\exp L) \varepsilon\_0^m + (t - \tau)^{a-1} \frac{\partial X\_m^{(1)} (\exp L)}{\partial \dot{q}\_s} \left( \xi\_s^m - \dot{q}\_s \varepsilon\_0^m \right) + G^m \right]. \tag{18}$$

**Proof.** By using Equations (13), (15), and (17), we have

d*Iz* dτ = *z <sup>m</sup>*=0 υ*m* 1−<sup>α</sup> *t*−τ (*t* − τ)<sup>α</sup>−1*X*(1) *m* (exp *<sup>L</sup>*)ς*<sup>m</sup>*0 + (*t* − τ)<sup>α</sup>−<sup>1</sup> d*X*(1) *m* (exp *L*) dτ ς*m*0 +(*t* − τ)<sup>α</sup>−1*X*(1) *m* (exp *<sup>L</sup>*).ς*<sup>m</sup>*0 + 1−<sup>α</sup> *t*−τ (*t* − τ)<sup>α</sup>−<sup>1</sup> ∂*X*(1) *m* (exp *L*) <sup>∂</sup>.*qs* ξ*ms* − .*qs*ς*<sup>m</sup>*0 +(*t* − τ)<sup>α</sup>−<sup>1</sup> ddτ ∂*X*(1) *m* (exp *L*) <sup>∂</sup>.*qs* ξ*ms* − .*qs*ς*<sup>m</sup>*0 + (*t* − τ)<sup>α</sup>−<sup>1</sup> ∂*X*(1) *m* (exp *L*) <sup>∂</sup>.*qs* .ξ*ms* − .*qs* .ς*m*0 − ..*qs*ς*<sup>m</sup>*0 + .*Gm* = *z <sup>m</sup>*=0 <sup>υ</sup>*<sup>m</sup>*−∂*X*(1) *m* (*L*) ∂*qs* + ∂*X*(1) *m* (*L*) <sup>∂</sup>.*qs* d*L*dτ + ∂*L*<sup>∂</sup>.*qs* d*X*(1) *m* (*L*) dτ + ddτ ∂*X*(1) *m* (*L*) <sup>∂</sup>.*qs* + 1−<sup>α</sup> *t*−τ ∂*X*(1) *m* (*L*) <sup>∂</sup>.*qs* × ×ξ*ms* − .*qs*ς*<sup>m</sup>*0 (*t* − τ)<sup>α</sup>−<sup>1</sup> exp *L* + 1−<sup>α</sup> *t*−τ ς*m*0 + .ς*m*0 *X*(1) *m* (exp *L*)(*t* − τ)<sup>α</sup>−<sup>1</sup> +*X*(1) *m X*(1) *m* (exp *L*)(*t* − τ)<sup>α</sup>−<sup>1</sup> − ∂*L*∂*qs* − ddτ ∂*L*<sup>∂</sup>.*qs* − ∂*L*<sup>∂</sup>.*qs* d*L*dτ − 1−<sup>α</sup> *t*−τ ∂*L*<sup>∂</sup>.*qs* × ×ξ*ms* − .*qs*ς*<sup>m</sup>*0 (*t* − τ)<sup>α</sup>−1*X*(1) *m* (*L*) exp *L* + .*Gm* = *z <sup>m</sup>*=0 <sup>υ</sup>*<sup>m</sup>*−<sup>υ</sup>*X*(1) *m* (*Qs*)ξ*ms* − .*qs*ς*<sup>m</sup>*0 + *X*(1) *<sup>m</sup>*−<sup>1</sup>(*Qs*)ξ*<sup>m</sup>*−<sup>1</sup> *s* − .*qs*ς*<sup>m</sup>*−<sup>1</sup> 0 +*QsX*(1) *<sup>m</sup>*−<sup>1</sup>(*L*)ξ*<sup>m</sup>*−<sup>1</sup> *s* − .*qs*ς*<sup>m</sup>*−<sup>1</sup> 0 − υ*QsX*(1) *m* (*L*)ξ*ms* − .*qs*ς*<sup>m</sup>*0 = −υ*z*+<sup>1</sup> *X*(1) *z* (*Qs*) + *QsX*(1) *z* (*L*)ξ*zs* − .*qs*ς*<sup>z</sup>*0. (19)

According to the definition of adiabatic invariant [51], *Iz* is an adiabatic invariant of order *z*. This completes the proof. -

**Example 1.** *Considering the nonlinear dynamical system, its action functional based on exponential Lagrangian is*

$$S = \frac{1}{\Gamma(a)} \int\_{t\_1}^{t\_2} \exp[L(\tau, q\_{s\prime} \dot q\_s)](t - \tau)^{a-1} d\tau,\tag{20}$$

*where L* = τ*q* .*q.*

> Equation (1) gives

$$\left[\tau \left(t - \tau\right)^{\alpha - 1} \exp\left(\tau q \dot{q}\right)\right] \tau q \left(\frac{\alpha - 1}{t - \tau} - q \dot{q} - \tau \dot{q}^2 - \tau q \ddot{q}\right) - q\right] = 0. \tag{21}$$

By calculation, we have

$$X\_0^{(1)}(L) = \eta \dot{q} \varepsilon\_0^0 + \pi \dot{q} \xi^0 + \pi q \left(\dot{\xi}^0 - \dot{q} \dot{\varepsilon}\_0^0\right) \tag{22}$$

$$X\_0^{(1)}(\exp L) = \exp(\pi q \dot{q}) \Big[q \dot{q} \varepsilon\_0^0 + \pi \dot{q} \dot{\xi}^0 + \pi q \dot{\left(\dot{\xi}^0 - \dot{q} \dot{\varepsilon}\_0^0\right)}\Big].\tag{23}$$

If we take

$$
\omega\_0^0 = \tau, \ \xi^0 = \frac{1}{q'} \tag{24}
$$

then we have

$$X\_0^{(1)}(L) = 0, \ X\_0^{(1)}(\exp L) = 0. \tag{25}$$

According to the criterion (6), the infinitesimals (24) correspond to Mei symmetry. Substituting (24) into Equation (7), we ge<sup>t</sup>

$$G^0 = \pi \exp(\pi q \dot{q}) (t - \pi)^{a-1}. \tag{26}$$

From Theorem 1, we have

$$I\_0 = \tau \exp(\tau q \dot{q}) (t - \tau)^{a-1} = \text{const.} \tag{27}$$

*Symmetry* **2019**, *11*, 1061

> Let the small disturbance be

$$\upsilon Q = \upsilon q \dot{q} \exp\big(q^2/2\big). \tag{28}$$

The di fferential equation of the disturbed motion is

$$\left[ (t - \tau)^{\alpha - 1} \exp(\tau q \dot{q}) \right] \left[ \tau q \left( \frac{\alpha - 1}{t - \tau} - q \dot{q} - \tau \dot{q}^2 - \tau q \ddot{q} \right) - q \right] = \nu q \dot{q} \exp(q^2 / 2). \tag{29}$$

Take

$$
\zeta\_0^1 = \tau,\ \xi^1 = \frac{1}{q},\tag{30}
$$

then we have

$$X\_1^{(1)}(L) = 0, \; X\_1^{(1)}(\exp L) = 0, \; X\_0^{(1)}(Q) = X\_1^{(1)}(Q) = 0. \tag{31}$$

According to the criterion (15), the infinitesimals (30) correspond to Mei symmetry. Substituting (30) into Equation (17), we have

$$\mathcal{G}^1 = \pi \exp(\pi q \dot{q}) (t - \tau)^{\alpha - 1} + \nu \int \exp(q^2 / 2) d\eta. \tag{32}$$

By Theorem 2, we obtain

$$I\_1 = \tau \exp(\tau q \dot{q}) (t - \tau)^{a-1} + \nu \left[ \tau \exp(\tau q \dot{q}) (t - \tau)^{a-1} + \nu \int \exp(q^2 / 2) d\eta \right]. \tag{33}$$

The formula (33) is an adiabatic invariant led by Mei symmetry.

### **3. Mei Symmetry and Invariants of Quasi-Fractional Dynamical System Based on Power-Law Lagrangians**

For the quasi-fractional dynamical system whose action functional depends on power-law Lagrangian, the Euler–Lagrange equations that are derived in Appendix B can be expressed as

$$(1+\gamma)(t-\tau)^{a-1}L^{\gamma}\Big(\frac{\partial L}{\partial q\_s} - \frac{\mathbf{d}}{\mathbf{d}\tau}\frac{\partial L}{\partial \dot{q}\_s} - \frac{\gamma}{L}\frac{\partial L}{\partial \dot{q}\_s}\frac{\mathbf{d}L}{\mathbf{d}\tau} + \frac{a-1}{t-\tau}\frac{\partial L}{\partial \dot{q}\_s}\Big) = 0,\ (s=1,2,\cdots,n),\tag{34}$$

where γ is not equal to −1.

> After the transformation of (2), *L*1+γ is transformed into the following form

$$L^{\*1+\gamma} = L^{1+\gamma} \left( \pi^\*, q\_{s\prime}^\* \frac{\mathrm{d}q\_s^\*}{\mathrm{d}\pi^\*} \right) = L^{1+\gamma} \left( \pi, q\_{\prime\prime} \dot{q}\_{\prime} \right) + \varepsilon X^{(1)} \left( L^{1+\gamma} \right) + O \left( \varepsilon^2 \right). \tag{35}$$

If *L* is replaced with *L*<sup>∗</sup>, Equation (34) still holds, namely

$$(1+\gamma)(t-\tau)^{a-1}L^{\ast\gamma}\Big(\frac{\partial L^{\ast}}{\partial q\_{\boldsymbol{s}}} - \frac{\mathbf{d}}{\mathbf{d}\tau}\frac{\partial L^{\ast}}{\partial \dot{q}\_{\boldsymbol{s}}} - \frac{\gamma}{L^{\ast}}\frac{\partial L^{\ast}}{\partial \dot{q}\_{\boldsymbol{s}}}\frac{\mathbf{d}L^{\ast}}{\mathbf{d}\tau} + \frac{\alpha-1}{t-\tau}\frac{\partial L^{\ast}}{\partial \dot{q}\_{\boldsymbol{s}}}\Big) = 0,\ (\mathbf{s} = 1, 2, \cdots, n). \tag{36}$$

then this invariance is called Mei symmetry of quasi-fractional dynamical system (34). Substituting the formula (35) into Equation (36), and considering Equation (34), we have

$$\begin{split} \mathbb{E}\left(\mathbbm{1} + \gamma\right)(t-\tau)^{a-1}L^{\gamma}\Big[\frac{\partial X^{(1)}(L)}{\partial q\_{s}} - \frac{q\_{1}}{\mathrm{d}\tau}\frac{\partial X^{(1)}(L)}{\partial \dot{q}\_{s}} - \frac{\gamma}{L}\frac{\mathrm{d}L}{\mathrm{d}\tau}\frac{\partial X^{(1)}(L)}{\partial \dot{q}\_{s}} - \frac{\gamma}{L}\frac{\mathrm{d}L}{\partial \dot{q}\_{s}}\frac{\mathrm{d}X^{(1)}(L)}{\mathrm{d}\tau} \\ + \frac{\gamma}{L^{2}}\frac{\mathrm{d}L}{\partial \dot{q}\_{s}}\frac{\mathrm{d}L}{\mathrm{d}\tau}X^{(1)}(L) + \frac{q-1}{t-\tau}\frac{\mathrm{d}X^{(1)}(L)}{\partial \dot{q}\_{s}} \bigg] = 0, \ (s=1,2,\cdots,n). \end{split} \tag{37}$$

Equation (37) is the criterion for Mei symmetry of system (34). Hence, we have *Symmetry* **2019**, *11*, 1061

**Theorem 3.** *For the quasi-fractional dynamical system (34), if there is a gauge function G* = *<sup>G</sup>*<sup>τ</sup>, *qk*, .*qk such that the structural equation*

$$\left\{ \frac{1-\alpha}{t-\tau}\varepsilon\_0 + \dot{\varepsilon}\_0 \right\} \mathbf{X}^{(1)} \left( L^{1+\gamma} \right) + \mathbf{X}^{(1)} \left\{ \mathbf{X}^{(1)} \left( L^{1+\gamma} \right) \right\} + \left( t - \tau \right)^{1-\alpha} \dot{\mathbf{G}} = \mathbf{0} \tag{38}$$

*holds, the Mei symmetry directly leads to the new conserved quantity*

$$I\_0 = (t - \tau)^{a-1} X^{(1)} \Big( L^{1+\gamma} \Big) \varsigma\_0 + (t - \tau)^{a-1} \frac{\partial X^{(1)} \Big( L^{1+\gamma} \Big)}{\partial \dot{q}\_s} \Big( \underline{\varsigma}\_s - \dot{q}\_s \underline{\varsigma}\_0 \Big) + G = \text{const.} \tag{39}$$

**Proof.**

d*I*0 d*t* = 1−<sup>α</sup> *t*−τ (*t* − <sup>τ</sup>)<sup>α</sup>−1*X*(1)*L*1+γ<sup>ς</sup>0 + (*t* − τ)<sup>α</sup>−<sup>1</sup> d*X*(1)(*L*1+γ) dτ ς0 + (*t* − <sup>τ</sup>)<sup>α</sup>−1*X*(1)*L*1+γ.ς0 +1−<sup>α</sup> *t*−τ (*t* − τ)<sup>α</sup>−<sup>1</sup> ∂*X*(1)(*L*1+γ) <sup>∂</sup>.*qs* ξ*s* − .*qs*ς0 + (*t* − τ)<sup>α</sup>−<sup>1</sup> ddτ ∂*X*(1)(*L*1+γ) <sup>∂</sup>.*qs* ξ*s* − .*qs*ς0 +(*t* − τ)<sup>α</sup>−<sup>1</sup> ∂*X*(1)(*L*1+γ) <sup>∂</sup>.*qs* .ξ*s* − .*qs* .ς0 − ..*qs*ς0 + .*G* = −∂*X*(1)(*L*) ∂*qs* + ddτ ∂*X*(1)(*L*) <sup>∂</sup>.*qs* + γ*L* d*L*dτ ∂*X*(1)(*L*) <sup>∂</sup>.*qs* + γ*L* ∂*L*<sup>∂</sup>.*qs* d*X*(1)(*L*) dτ − γ*L*<sup>2</sup> ∂*L*<sup>∂</sup>.*qs* d*L*<sup>d</sup>τ*X*(1)(*L*) +1−<sup>α</sup> *t*−τ ∂*X*(1)(*L*) <sup>∂</sup>.*qs* (1 + γ)(*<sup>t</sup>* − <sup>τ</sup>)<sup>α</sup>−1*L*<sup>γ</sup><sup>ξ</sup>*s* − .*qs*ς0 + 1−<sup>α</sup> *t*−τ ς0 + .<sup>ς</sup>0*X*(1)*L*1+γ(*<sup>t</sup>* − τ)<sup>α</sup>−<sup>1</sup> +− ∂*L*∂*qs* + ddτ ∂*L*<sup>∂</sup>.*qs* + γ*L* ∂*L*<sup>∂</sup>.*qs* d*L*dτ + 1−<sup>α</sup> *t*−τ ∂*L*<sup>∂</sup>.*qs* (1 + γ)(*<sup>t</sup>* − <sup>τ</sup>)<sup>α</sup>−1*X*(1)(*L*<sup>γ</sup>)<sup>ξ</sup>*s* − .*qs*ς0 +*X*(1)(*L*)*X*(1)(*L*)(<sup>1</sup> + γ)γ(*<sup>t</sup>* − τ)<sup>α</sup>−1*L*γ−<sup>1</sup> + *X*(1)*X*(1)(*L*) (1 + γ)(*<sup>t</sup>* − τ)<sup>α</sup>−1*L*<sup>γ</sup> + .*G*. (40)

Substituting Equations (34) and (37) into the formula (40), and using Equation (38), we obtain

$$\frac{d\mathrm{d}l\_{0}}{dt} = \left(\frac{1-\alpha}{t-\tau}\varepsilon\_{0} + \dot{\varepsilon}\_{0}\right) \mathrm{X}^{(1)}\left(\mathrm{L}^{1+\gamma}\right) \mathrm{(t-\tau)}^{a-1} + \mathrm{X}^{(1)}\left[\mathrm{X}^{(1)}\mathrm{(L}^{1+\gamma})\right] \mathrm{(t-\tau)}^{a-1} + \dot{\mathcal{G}} = 0.\tag{41}$$

Thus, we ge<sup>t</sup> the desired result. -

> Mei conserved quantity (39) is an exact invariant for the quasi-fractional dynamical systems (34). If a small disturbance υ*Qs* is applied, Equation (34) is changed to

$$\nu \left( (1+\gamma)(t-\tau)^{\alpha-1} L^{\gamma} \middle| \frac{\partial L}{\partial \dot{q}\_{\rm s}} - \frac{\text{d}}{\text{d}\tau} \frac{\partial L}{\partial \dot{q}\_{\rm s}} - \frac{\text{\underline{\gamma}}}{L} \frac{\partial L}{\partial \dot{q}\_{\rm s}} \frac{\text{d}L}{\text{d}\tau} + \frac{\alpha-1}{t-\tau} \frac{\partial L}{\partial \dot{q}\_{\rm s}} \right) = \nu Q\_{\rm s}, \text{ (s = 1,2,\cdots,n)}.\tag{42}$$

Accordingly, Equation (37) is changed to

$$\begin{split} & \left( (1+\gamma) \right) (t-\tau)^{a-1} L^{\gamma} \Big[ \frac{\partial X^{(1)}(L)}{\partial q\_s} - \frac{\mathbf{d}}{\mathbf{d}\tau} \frac{\partial X^{(1)}(L)}{\partial \dot{q}\_s} - \frac{\gamma}{L} \frac{\text{dI}}{\text{d}\tau} \frac{\partial X^{(1)}(L)}{\partial \dot{q}\_s} - \frac{\gamma}{L} \frac{\partial \mathcal{L}}{\partial \dot{q}\_s} \frac{\text{d}X^{(1)}(L)}{\text{d}\tau} \\ & + \frac{\gamma}{L^2} \frac{\partial \mathcal{L}}{\partial \dot{q}\_s} \frac{\text{dI}}{\text{d}\tau} X^{(1)}(L) + \frac{a-1}{t-\tau} \frac{\partial X^{(1)}(L)}{\partial \dot{q}\_s} \Big] = \nu X^{(1)}(Q\_s), \text{ (s = 1,2, \cdots, n)}. \end{split} \tag{43}$$

Substituting the formulae (11) into Equation (43), we ge<sup>t</sup>

$$\begin{split} & \left( (1+\gamma) \left( t-\tau \right)^{\alpha-1} \nu^{m} L^{\prime} \left[ \frac{\partial X\_{m}^{(1)}(L)}{\partial \dot{q}\_{s}} - \frac{\mathbf{d}}{\mathrm{d}\tau} \frac{\partial X\_{m}^{(1)}(L)}{\partial \dot{q}\_{s}} - \frac{\mathbf{y}}{L} \frac{\mathrm{d}L}{\mathrm{d}\tau} \frac{\partial X\_{m}^{(1)}(L)}{\partial \dot{q}\_{s}} - \frac{\mathbf{y}}{L} \frac{\partial \mathbf{L}}{\partial \dot{q}\_{s}} \frac{\mathrm{d}X\_{m}^{(1)}(L)}{\mathrm{d}\tau} \right. \\ & \left. + \frac{\gamma}{L^{2}} \frac{\partial L}{\partial \dot{q}\_{s}} \frac{\mathrm{d}L}{\partial \tau} X\_{m}^{(1)}(L) + \frac{\alpha-1}{l-\tau} \frac{\partial X\_{m}^{(1)}(L)}{\partial \dot{q}\_{s}} \right] = \nu^{m+1} X\_{m}^{(1)}(Q\_{s}), \ (s=1,2,\cdots,n). \end{split} \tag{44}$$

Therefore, we have **Theorem 4.** *If the quasi-fractional dynamical system (34) is disturbed by small disturbance* υ*Qs, and there is a gauge function G* = *<sup>G</sup>*<sup>τ</sup>, *qk*, .*qk such that the structural equation*

$$\begin{cases} \frac{1-\underline{\boldsymbol{m}}}{t-\tau}\boldsymbol{\zeta}\_{0}^{\boldsymbol{m}} + \boldsymbol{\zeta}\_{0}^{\boldsymbol{m}} \Big] \boldsymbol{X}\_{m}^{(1)} \Big( \boldsymbol{L}^{1+\gamma} \Big) (t-\tau)^{\boldsymbol{a}-1} + \boldsymbol{X}\_{m}^{(1)} \Big[ \boldsymbol{X}\_{m}^{(1)} \Big( \boldsymbol{L}^{1+\gamma} \Big) \Big] (t-\tau)^{\boldsymbol{a}-1} - \boldsymbol{X}\_{m-1}^{(1)} \big( \boldsymbol{Q}\_{s} \big) \Big( \boldsymbol{\zeta}\_{s}^{\boldsymbol{m}-1} - \dot{q}\_{s} \boldsymbol{\zeta}\_{0}^{\boldsymbol{m}-1} \Big) \\\ - \boldsymbol{\check{\tau}}\_{0}^{\boldsymbol{\prime}} \boldsymbol{Q}\_{s} \boldsymbol{X}\_{m-1}^{(1)} \big( \boldsymbol{L} \big) \Big( \boldsymbol{\zeta}\_{s}^{\boldsymbol{m}-1} - \dot{q}\_{s} \boldsymbol{\zeta}\_{0}^{\boldsymbol{m}-1} \Big) + \boldsymbol{\check{G}} \ = \boldsymbol{0}, \ (\boldsymbol{s} = 1, 2, \cdots, \boldsymbol{n}; m = 0, 1, 2, \cdots \big), \end{cases} \tag{45}$$

*holds, where G* = *z <sup>m</sup>*=0 υ*mGm and* ς<sup>−</sup><sup>1</sup> 0 = ξ−<sup>1</sup> *s* = 0*, the Mei symmetry directly leads to the new adiabatic invariant*

$$I\_z = \sum\_{m=0}^{z} \upsilon^m \left[ (t - \tau)^{\alpha - 1} X\_m^{(1)} \big( L^{1 + \gamma} \big) \varepsilon\_0^m + (t - \tau)^{\alpha - 1} \frac{\partial}{\partial \dot{q}\_s} X\_m^{(1)} \big( L^{1 + \gamma} \big) \big( \xi\_s^m - \dot{q}\_s \varepsilon\_0^m \big) + G^m \right]. \tag{46}$$

**Proof.** By using Equations (42), (44), and (45), we have

d*Iz* dτ = *z <sup>m</sup>*=0 υ*<sup>m</sup>*<sup>1</sup>−<sup>α</sup> *t*−τ (*t* − τ)<sup>α</sup>−1*X*(1) *m L*1+γς*<sup>m</sup>*0 + (*t* − τ)<sup>α</sup>−<sup>1</sup> d*X*(1) *m* (*L*1+γ) dτ ς*m*0 +(*t* − τ)<sup>α</sup>−1*X*(1) *m L*1+γ.ς*<sup>m</sup>*0 + 1−<sup>α</sup> *t*−τ (*t* − τ)<sup>α</sup>−<sup>1</sup> ∂*X*(1) *m* (*L*1+γ) <sup>∂</sup>.*qs* ξ*ms* − .*qs*ς*<sup>m</sup>*0 +(*t* − τ)<sup>α</sup>−<sup>1</sup> ddτ ∂*X*(1) *m* (*L*1+γ) <sup>∂</sup>.*qs* ξ*ms* − .*qs*ς*<sup>m</sup>*0 + (*t* − τ)<sup>α</sup>−<sup>1</sup> ∂*X*(1) *m* (*L*1+γ) <sup>∂</sup>.*qs* .ξ*ms* − .*qs* .ς*m*0 − ..*qs*ς*<sup>m</sup>*0 + .*Gm* = *z <sup>m</sup>*=0 <sup>υ</sup>*<sup>m</sup>*−∂*X*(1) *m* (*L*) ∂*qs* + ddτ ∂*X*(1) *m* (*L*) <sup>∂</sup>.*qs* + γ*L* d*L*dτ ∂*X*(1) *m* (*L*) <sup>∂</sup>.*qs* + γ*L* ∂*L*<sup>∂</sup>.*qs* d*X*(1) *m* (*L*) dτ − γ *L*<sup>2</sup> ∂*L* <sup>∂</sup>.*qs* d*L* <sup>d</sup>τ*X*(1) *m* (*L*) + 1−<sup>α</sup> *t*−τ ∂*X*(1) *m* (*L*) <sup>∂</sup>.*qs* (1 + γ)(*<sup>t</sup>* − τ)<sup>α</sup>−1*L*<sup>γ</sup>ξ*ms* − .*qs*ς*<sup>m</sup>*0 +− ∂*L*∂*qs* + ddτ ∂*L*<sup>∂</sup>.*qs* + γ*L* ∂*L*<sup>∂</sup>.*qs* d*L*dτ + 1−<sup>α</sup> *t*−τ ∂*L*<sup>∂</sup>.*qs* (1 + γ)(*<sup>t</sup>* − τ)<sup>α</sup>−1*X*(1) *m* (*L*γ)ξ*ms* − .*qs*ς*<sup>m</sup>*0 + 1−<sup>α</sup> *t*−τ ς*m*0 + .ς*m*0 *X*(1) *m L*1+γ(*t* − τ)<sup>α</sup>−<sup>1</sup> + *X*(1) *m* (*L*)*X*(1) *m* (*L*)(1 + γ)γ(*<sup>t</sup>* − τ)<sup>α</sup>−1*L*γ−<sup>1</sup> +*X*(1) *m X*(1) *m* (*L*)(1 + γ)(*<sup>t</sup>* − τ)<sup>α</sup>−1*L*<sup>γ</sup> + .*Gm* = *z <sup>m</sup>*=0 <sup>υ</sup>*<sup>m</sup>*−<sup>υ</sup>*X*(1) *m* (*Qs*)ξ*ms* − .*qs*ς*<sup>m</sup>*0 − υ <sup>γ</sup>*LQsX*(1) *m* (*L*)ξ*ms* − .*qs*ς*<sup>m</sup>*0 +*X*(1) *<sup>m</sup>*−<sup>1</sup>(*Qs*)ξ*<sup>m</sup>*−<sup>1</sup> *s* − .*qs*ς*<sup>m</sup>*−<sup>1</sup> 0 + <sup>γ</sup>*LQsX*(1) *<sup>m</sup>*−<sup>1</sup>(*L*)ξ*<sup>m</sup>*−<sup>1</sup> *s* − .*qs*ς*<sup>m</sup>*−<sup>1</sup> 0 = −υ*z*+<sup>1</sup> *X*(1) *z* (*Qs*) + <sup>γ</sup>*LQsX*(1) *z* (*L*)ξ*zs* − .*qs*ς*<sup>z</sup>*0. (47)

According to the definition of adiabatic invariant [51], *Iz* is an adiabatic invariant of order *z*. So that ends the proof. -

**Example 2.** *Considering the nonconservative dynamical system, its action functional based on power-law Lagrangian is [50]*

$$A = \frac{1}{\Gamma(\alpha)} \int\_{t\_1}^{t\_2} \left[ L^{1+\gamma} \left( \tau, q\_s, \dot{q}\_s \right) \right] (t-\tau)^{\alpha-1} d\tau,\tag{48}$$

*where L* = .*q* − *q*(τ − *t*), γ = 1*.*

> Equation (34) gives

$$2\left(t-\tau\right)^{\alpha-1}\left[-\ddot{q}+\frac{\alpha-1}{t-\tau}\dot{q}+\left(\left(\tau-t\right)^{2}+\alpha\right)q\right]=0.\tag{49}$$

By calculation, we have

$$X\_0^{(1)}(L) = -q\varepsilon\_0^0 - (\tau - t)\dot{\xi}\_0^0 + \dot{\xi}\_0^0 - \dot{q}\dot{\varepsilon}\_0^0 \tag{50}$$

$$X\_0^{(1)}\{L^2\} = 2L\Big[ -q\varepsilon\_0^0 - (\tau - t)\dot{\varepsilon}\_0^0 + \dot{\xi}\_0^0 - \dot{q}\dot{\varepsilon}\_0^0 \Big].\tag{51}$$

Let

$$
\xi\_0^0 = 0, \ \xi\_0^0 = \exp\left[\left(t + \tau\right)^2 / 2\right]. \tag{52}
$$

then

$$X\_0^{(1)}(L) = 0, \; X\_0^{(1)}(L^2) = 0. \tag{53}$$

According to the criterion (37), the infinitesimals (52) correspond to Mei symmetry. Substituting (52) into Equation (38), we ge<sup>t</sup>

$$G^0 = 2\left[\dot{q} - q\left(\tau - t\right)\right] \left(t - \tau\right)^{\alpha - 1} \exp\left[\left(\tau - t\right)^2 / 2\right]. \tag{54}$$

From Theorem 3, we have

$$I\_0 = 2\left[\dot{q} - q(\tau - t)\right](t - \tau)^{\alpha - 1} \exp\left[(\tau - t)^2 / 2\right] = \text{const.}\tag{55}$$

Let the small disturbance be

$$
\upsilon Q = \upsilon \sin \tau \exp\left[-\left(\tau - t\right)^2/2\right]. \tag{56}
$$

The differential equation of the disturbed motion is

$$2(t - \tau)^{\alpha - 1} \left[ -\ddot{q} + \frac{\alpha - 1}{t - \tau} \dot{q} + \left( (\tau - t)^2 + a \right) q \right] = \nu \sin \tau \exp \left[ -(\tau - t)^2 / 2 \right]. \tag{57}$$

Take

$$\varepsilon\_0^1 = 0,\ \xi\_0^1 = \exp\left[\left(t + \tau\right)^2 / 2\right].\tag{58}$$

then it is easy to verify

$$X\_1^{(1)}(L) = 0, \; X\_1^{(1)}(L^2) = 0, \; X\_0^{(1)}(Q) = X\_1^{(1)}(Q) = 0. \tag{59}$$

According to the criterion (44), the infinitesimals (58) correspond to Mei symmetry. Substituting (58) into Equation (45), we have

$$G^1 = 2\left[\dot{q} - q(\tau - t)\right](t - \tau)^{\alpha - 1} \exp\left[\left(\tau - t\right)^2 / 2\right] - \nu \cos \tau. \tag{60}$$

By Theorem 4, we obtain

$$\begin{aligned} l\_1 &= 2 \left[ \dot{q} - q(\tau - t) \right] (t - \tau)^{a-1} \exp\left[ \left( \tau - t \right)^2 / 2 \right] \\ &+ \nu \left[ 2 \left[ \dot{q} - q(\tau - t) \right] (t - \tau)^{a-1} \exp\left[ \left( \tau - t \right)^2 / 2 \right] - \nu \cos \tau \right]. \end{aligned} \tag{61}$$

Formula (61) is an adiabatic invariant led by Mei symmetry.
