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

Zhang Zhang,Wang Wang

doi:10.3390/sym11081061

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.

Highlights

The study of symmetry and invariants for non-conservative or nonlinear dynamics is of great significance
Lie symmetry are two different symmetries. The former means the invariant property of the Hamilton action functional, and the latter means the invariant property of the differential equation
Known as Mei symmetry, refers to an invariant property, that is, the dynamical functions that appear in the dynamical equations of the mechanical system still satisfy the original equations after the infinitesimal transformation

Summary

Introduction

The study of symmetry and invariants for non-conservative or nonlinear dynamics is of great significance. Since Riewe [38,39] introduced fractional calculus into the modeling of non-conservative systems, fractional Lagrangian mechanics, fractional Hamiltonian mechanics and fractional Birkhoffian mechanics have been proposed and studied, and important progress has been made in fractional dynamics modeling, analysis, and calculation, see for example [40,41,42,43,44,45,46,47] and references therein. We propose and study Mei symmetry and its invariants for the quasi-fractional order dynamical system with non-standard. New conserved quantities and new adiabatic invariants are derived from Mei symmetry of the quasi-fractional dynamical systems. If the system is affected by small disturbance υQs , its Mei symmetry and the corresponding conserved quantity (8) will change correspondingly.

The differential equation of the disturbed motion is

Take then it is easy to verify

Conclusions

Findings

Due to s