**4. Conclusions**

Symmetry is closely related to invariants, and it is of grea<sup>t</sup> significance to find the invariants of complex system dynamics. First, even if the equations of motion are difficult to solve, the existence of some conserved quantity makes it possible to understand the local physical state or dynamical behavior of the system. Secondly, we can reduce the differential equations of motion by using conserved quantities. Thirdly, we can study the motion stability of complex dynamical systems by using conserved quantities. Based on the quasi-fractional dynamical model proposed by El-Nabulsi according to the Riemann–Liouville definition of fractional integral, we studied Mei symmetry and its corresponding invariants of quasi-fractional dynamics system whose action functional is composed of non-standard Lagrangians. The main results of this paper are its four theorems. In this paper, we provided a method

to study nonlinear or non-conservative dynamics and obtained new conserved quantities and new adiabatic invariants, and the results are expected to be generalized or applied to the dynamics of constrained systems, such as those of nonholonomic systems.

**Author Contributions:** The authors equally contributed to this research work.

**Funding:** This work was supported by the National Natural Science Foundation of China (grant nos. 11,572,212 and 11,272,227).

**Conflicts of Interest:** The authors declare no conflict of interest.

### **Appendix A. Derivation of the Euler–Lagrange Equations for Quasi-Fractional Dynamical System with Exponential Lagrangians**

Consider a nonlinear dynamical system whose configuration is determined by *n* generalized coordinates *qs*(*s* = 1, 2, ··· , *<sup>n</sup>*), its action functional based on exponential Lagrangian is

$$S = \frac{1}{\Gamma(\alpha)} \int\_{t\_1}^{t\_2} \exp[L(\tau, q\_s, \dot{q}\_s)] (t - \tau)^{\alpha - 1} d\tau. \tag{A1}$$

where *L* = *<sup>L</sup>*<sup>τ</sup>, *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*.

The isochronous variational principle

$$
\delta S = 0,\tag{A2}
$$

which satisfies the following commutation relation

$$\mathsf{cd}\delta q\_{\mathbb{S}} = \delta \mathsf{d}q\_{\mathbb{S}\prime} \,(\mathsf{s} = \mathsf{1}, \mathsf{2}, \cdots, \mathsf{n}),\tag{\mathsf{A3}}$$

and given boundary condition

$$\left. \delta q\_s \right|\_{t=t\_1} = \left. \delta q\_s \right|\_{t=t\_2} = 0, \left( \mathbf{s} = 1, 2, \cdots, n \right) \tag{A4}$$

can be called the Hamilton principle of the quasi-fractional dynamical system with exponential Lagrangians.

Expanding the Hamilton principle (A2), we have

$$\begin{split} 0 &= \delta S = \frac{1}{\Gamma(a)} \int\_{t\_1}^{t\_2} \delta \Big[ \exp L(t - \tau)^{a - 1} \Big] d\tau \\ &= \frac{1}{\Gamma(a)} \int\_{t\_1}^{t\_2} \left( t - \tau \right)^{a - 1} \exp L \Big( \frac{\partial L}{\partial q\_s} \delta q\_s + \frac{\partial L}{\partial \dot{q}\_s} \delta \dot{q}\_s \Big) d\tau \end{split} \tag{A5}$$

Due to

$$\begin{split} & \left. \int\_{t\_1}^{t\_2} (t - \tau)^{\alpha - 1} \exp L \frac{\partial L}{\partial \dot{q}\_s} \delta \dot{q}\_s d\tau = \left[ (t - \tau)^{\alpha - 1} \exp L \frac{\partial L}{\partial \dot{q}\_s} \delta q\_s \right]\_{t\_1}^{t\_2} \\ & - \int\_{t\_1}^{t\_2} (t - \tau)^{\alpha - 1} \exp L \left( \frac{-\alpha - 1}{t - \tau} \frac{\partial L}{\partial \dot{q}\_s} + \frac{\dddot{q}L}{\text{d\tau}} \frac{\partial L}{\partial \dot{q}\_s} + \frac{\dddot{q}}{\text{d\tau}} \frac{\partial L}{\partial \dot{q}\_s} \right) \delta q\_s d\tau \\ & = - \int\_{t\_1}^{t\_2} (t - \tau)^{\alpha - 1} \exp L \Big( -\frac{\alpha - 1}{t - \tau} \frac{\partial L}{\partial \dot{q}\_s} + \frac{\dddot{q}L}{\text{d\tau}} \frac{\partial L}{\partial \dot{q}\_s} + \frac{\dddot{q}}{\text{d\tau}} \frac{\partial L}{\partial \dot{q}\_s} \Big) \delta q\_s d\tau. \end{split} \tag{A6}$$

Substituting the formula (A6) into Equation (A5), we have

$$\frac{1}{\Gamma(\alpha)} \int\_{t\_1}^{t\_2} (t - \tau)^{\alpha - 1} \exp L \Big( \frac{\partial L}{\partial q\_s} - \frac{\mathbf{d}}{\mathbf{d}\tau} \frac{\partial L}{\partial \dot{q}\_s} - \frac{\partial L}{\partial \dot{q}\_s} \frac{\mathbf{d}L}{\mathbf{d}\tau} + \frac{\alpha - 1}{t - \tau} \frac{\partial L}{\partial \dot{q}\_s} \Big) \delta q\_s d\tau = 0. \tag{A7}$$

Because of the arbitrariness of the interval [*<sup>t</sup>*1, *t*2] and the independence of δ*qs* (*s* = 1, 2, ··· , *<sup>n</sup>*), using the fundamental lemma [23] of the calculus of variations, we ge<sup>t</sup>

$$\mathbb{E}\left((t-\tau)^{\alpha-1}\exp\mathcal{L}\Big(\frac{\partial\mathcal{L}}{\partial q\_{\mathcal{S}}}-\frac{\mathrm{d}}{\mathrm{d}\tau}\frac{\partial\mathcal{L}}{\partial\dot{q}\_{\mathcal{S}}}-\frac{\partial\mathcal{L}}{\partial\dot{q}\_{\mathcal{S}}}\frac{\mathrm{d}\mathcal{L}}{\mathrm{d}\tau}+\frac{\alpha-1}{t-\tau}\frac{\partial\mathcal{L}}{\partial\dot{q}\_{\mathcal{S}}}\Big)=0,\ (s=1,2,\cdots,n).\tag{A8}$$

Equation (A8) can be called the Euler–Lagrange equations for quasi-fractional dynamical system with exponential Lagrangians.

### **Appendix B. Derivation of the Euler–Lagrange Equations for Quasi-Fractional Dynamical System with Power-Law Lagrangians**

Consider a nonlinear dynamical system whose configuration is determined by *n* generalized coordinates *qs*(*s* = 1, 2, ··· , *<sup>n</sup>*), its action functional based on power-law Lagrangian is

$$A = \frac{1}{\Gamma(\alpha)} \int\_{t\_1}^{t\_2} \left[ L^{1+\gamma} \left( \tau, q\_s, \dot{q}\_s \right) \right] (t-\tau)^{\alpha-1} d\tau \tag{A9}$$

where *L* = *<sup>L</sup>*<sup>τ</sup>, *qs*, .*qs*is the standard Lagrangian, γ is not equal to −1, 0 < α ≤ 1, τ is the intrinsic time, *t* is the observer time, and τ is not equal to *t*.

The isochronous variational principle

$$
\delta A = 0,\tag{A10}
$$

which satisfies the following commutation relation

$$\mathsf{cd}\delta q\_{\mathsf{s}} = \delta \mathsf{d}q\_{\mathsf{s}\prime} \,(\mathsf{s} = 1, 2, \cdots, n),\tag{A11}$$

and given boundary condition

$$
\left.\delta q\_s\right|\_{t=t\_1} = \left.\delta q\_s\right|\_{t=t\_2} = 0,\\
\left(s = 1, 2, \cdots, n\right) \tag{A12}
$$

can be called the Hamilton principle of the quasi-fractional dynamical system with power-law Lagrangians.

Expanding the Hamilton principle (A10), we have

$$\begin{split} 0 &= \delta A = \frac{1}{\Gamma(a)} \int\_{t\_1}^{t\_2} \delta \Big[ L^{1+\gamma} (t-\tau)^{a-1} \Big] d\tau\\ &= \frac{1}{\Gamma(a)} \int\_{t\_1}^{t\_2} (1+\gamma)(t-\tau)^{a-1} L^{\gamma} \Big( \frac{\partial L}{\partial q\_s} \delta q\_s + \frac{\partial L}{\partial \dot{q}\_s} \delta \dot{q}\_s \Big) d\tau \end{split} \tag{A13}$$

Due to

$$\begin{split} & \int\_{t\_1}^{t\_2} (t-\tau)^{a-1} L^{\gamma} \frac{\partial L}{\partial \dot{q}\_s} \delta \dot{q}\_s d\tau = \left. \left[ (t-\tau)^{a-1} L^{\gamma} \frac{\partial L}{\partial \dot{q}\_s} \delta q\_s \right] \right|\_{t\_1}^{t\_2} \\ & - \int\_{t\_1}^{t\_2} (t-\tau)^{a-1} L^{\gamma} \left( -\frac{a-1}{t-\tau} \frac{\partial L}{\partial \dot{q}\_s} + \frac{\mathscr{V}}{L} \frac{\partial L}{\partial \tau} \frac{\partial L}{\partial \dot{q}\_s} + \frac{\mathbf{d}}{\mathbf{d}\tau} \frac{\partial L}{\partial \dot{q}\_s} \right) \delta q\_s d\tau \\ & = - \int\_{t\_1}^{t\_2} (t-\tau)^{a-1} L^{\gamma} \left( -\frac{a-1}{t-\tau} \frac{\partial L}{\partial \dot{q}\_s} + \frac{\mathscr{V}}{L} \frac{\partial L}{\partial \tau} \frac{\partial L}{\partial \dot{q}\_s} + \frac{\mathbf{d}}{\mathbf{d}\tau} \frac{\partial L}{\partial \dot{q}\_s} \right) \delta q\_s d\tau. \end{split} \tag{A14}$$

Substituting the formula (A14) into Equation (A13), we have

$$\frac{1}{\Gamma(\alpha)} \int\_{t\_1}^{t\_2} (1+\gamma)(t-\tau)^{\alpha-1} L^{\gamma} \left( \frac{\partial L}{\partial q\_s} - \frac{\mathbf{d}}{\mathbf{d}\tau} \frac{\partial L}{\partial \dot{q}\_s} - \frac{\gamma}{L} \frac{\mathbf{d}L}{\mathbf{d}\tau} \frac{\partial L}{\partial \dot{q}\_s} + \frac{\alpha-1}{t-\tau} \frac{\partial L}{\partial \dot{q}\_s} \right) \delta q\_s d\tau = 0. \tag{A15}$$

Because of the arbitrariness of the interval [*<sup>t</sup>*1, *t*2] and the independence of δ*qs* (*s* = 1, 2, ··· , *<sup>n</sup>*), using the fundamental lemma [23] of the calculus of variations, we ge<sup>t</sup>

$$\frac{1}{2}(1+\gamma)(t-\tau)^{a-1}L^{\flat}\Big(\frac{\partial L}{\partial q\_{\ $}}-\frac{\mathrm{d}}{\mathrm{d}\tau}\frac{\partial L}{\partial \dot{q}\_{\$ }}-\frac{\gamma}{L}\frac{\partial L}{\partial \dot{q}\_{\ $}}\frac{\mathrm{d}L}{\mathrm{d}\tau}+\frac{a-1}{t-\tau}\frac{\partial L}{\partial \dot{q}\_{\$ }}\Big)=0,\ (s=1,2,\cdots,n).\tag{A16}$$

Equation (A16) can be called the Euler–Lagrange equations for quasi-fractional dynamical system with power-law Lagrangians. Equation (A16) is consistent with the results given in [50].
