Perturbation to Mei Symmetry and Adiabatic Invariants for Disturbed El-Nabulsi's Fractional Birkhoff System*

Abstract

For El-Nabulsi's fractional Birkhoff system, Mei symmetry perturbation, the corresponding Mei-type adiabatic invariants and Noether-type adiabatic invariants are investigated in this paper. Firstly, based on El-Nabulsi-Birkhoff fractional equations, Mei symmetry and the corresponding Mei conserved quantity, Noether conserved quantity deduced indirectly by Mei symmetry are studied. Secondly, Mei-type exact invariants and Noether-type exact invariants are given on the basis of the definition of adiabatic invatiant. Thirdly, Mei symmetry perturbation, Mei-type adiabatic invariants and Noether-type adiabatic invariants for the disturbed El-Nabulsi's fractional Birkhoff system are studied. Finally, two examples, Hojman-Urrutia problem for Mei-type adiabatic invariants and another for the Noether-type adiabatic invariants, are given to illustrate the application of the results.

Lie symmetries and conserved quantities of fractional nonconservative singular systems

“…Zhang and colleagues 11,12 studied the Noether symmetry theory of fractional order Hamiltonian systems using the El‐Nabulsi model by using the invariance of variational problem. Song and Zhang 13 also obtained the Lie symmetries and the perturbation for fractional order Birkhoff systems. For fractional order Hamiltonian systems with Caputo fractional derivative, Wang and Fu 14 revealed the relationship between Lie symmetry and the conserved quantity of the system.…”

A new type of adiabatic invariants for disturbed non-conservative nonholonomic system*

“…The research on the adiabatic invariants has aroused the interest of scholars and achieved some important results. [25][26][27][28][29][30][31][32][33][34][35][36] To the author's knowledge, the method of the differential variational principle of Herglotz type has not been applied to study the adiabatic invariant.…”