*Article* **Mei Symmetry and Invariants of Quasi-Fractional Dynamical Systems with Non-Standard Lagrangians**

**Yi Zhang 1,\* and Xue-Ping Wang 2**


Received: 15 July 2019; Accepted: 15 August 2019; Published: 18 August 2019

**Abstract:** Non-standard Lagrangians play an important role in the systems of non-conservative dynamics or nonlinear differential equations, quantum field theories, etc. This paper deals with quasi-fractional dynamical systems from exponential non-standard Lagrangians and power-law non-standard Lagrangians. Firstly, the definition, criterion, and corresponding new conserved quantity of Mei symmetry in this system are presented and studied. Secondly, considering that a small disturbance is applied on the system, the differential equations of the disturbed motion are established, the definition of Mei symmetry and corresponding criterion are given, and the new adiabatic invariants led by Mei symmetry are proposed and proved. Examples also show the validity of the results.

**Keywords:** Mei symmetry; conserved quantity; adiabatic invariant; quasi-fractional dynamical system; non-standard Lagrangians
