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

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.

Canonical Transformations and Poisson Theory for Dynamics with Non-Standard Lagrangians

“…Symmetry and conserved quantity have always been a hot spot in analytical mechanics [6][7][8] . Recently, Zhang and his collaborators have studied Noether theorems for systems with non-standard Lagrangians [9,10] , non-standard Hamiltonians [11] , non-standard Birkhoffians [12] , and Lie symmetry [13,14] , Mei symmetry [15] , first integral and method of reduction [16,17] . There have been some results about nonlinear dynamical equations and their symmetries with non-standard Lagrangians [18][19][20][21][22][23][24][25][26] , but canonical transformation and Poisson theory for non-standard Lagrangian dynamics have not been involved.…”

Asymptotic Stability of Nonlinear Discrete Fractional Pantograph Equations with Non-Local Initial Conditions

“…The fractional non-standard Lagrangians have been effective in various areas of physics like astrophysics, cosmology, quantum and classical dynamical systems. Recent works can be seen in [14][15][16][17][18]. Discrete fractional calculus is gaining its importance in recent years.…”

Mei Symmetry and Conservation Laws for Time-Scale Nonshifted Hamilton Equations

“…From Mei symmetry, a new kind of conservation law can be brought about, which is different from the Noether or Hojman one and called the Mei conserved quantity. The Mei symmetry theorem has been extended to fractional-order mechanical systems [18,19] and nonstan-dard Lagrangian dynamics [20]. Recently, we studied Mei symmetries on time scales [21,22], but the research is preliminary and limited to Lagrange equations and Birkhoff equations.…”