scholarly journals A new type of adiabatic invariants for Birkhoffian system

2006 ◽  
Vol 55 (8) ◽  
pp. 3833
Author(s):  
Zhang Yi
Keyword(s):  
Adiabatic Invariants ◽  
Birkhoffian System ◽  
New Type

scholarly journals Adiabatic Invariants for Generalized Fractional Birkhoffian Mechanics and Their Applications

2018 ◽  
Vol 2018 ◽  
pp. 1-18 ◽  
Author(s):  
C. J. Song
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Equations Of Motion ◽  
Conserved Quantity ◽  
Noether Symmetry ◽  
Adiabatic Invariants ◽  
Birkhoffian System ◽  
Liouville Fractional Derivative

Perturbation to Noether symmetry and adiabatic invariants are investigated for the generalized fractional Birkhoffian system with the combined Riemann-Liouville fractional derivative and the combined Caputo fractional derivative, respectively. Firstly, differential equations of motion for the generalized fractional Birkhoffian system are established. Secondly, Noether symmetry and conserved quantity are studied. Thirdly, perturbation to Noether symmetry and adiabatic invariants are presented for the generalized fractional Birkhoffian mechanics. And finally, several applications are discussed to illustrate the results and methods.

scholarly journals Perturbation to Noether Symmetries and Adiabatic Invariants for Birkhoffian Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Yi Zhang
Keyword(s):  
Fractional Integral ◽  
Conservative System ◽  
Dynamical Model ◽  
Lagrangian System ◽  
Adiabatic Invariants ◽  
Conserved Quantities ◽  
Noether Symmetries ◽  
Small Disturbance ◽  
Birkhoffian System ◽  
Special Cases

Based on El-Nabulsi dynamical model for a non-conservative system, the problem of perturbation to Noether symmetries and adiabatic invariants of a Birkhoffian system under the action of a small disturbance is proposed and studied. Firstly, the El-Nabulsi-Pfaff variational problem from extended exponentially fractional integral is presented and the El-Nabulsi-Birkhoff equations are established. Secondly, the definitions and the criterions criteria of the Noether symmetric transformations and quasisymmetric transformations of the Birkhoffian system are given, and the Noether theorems of the system are established, which reveal the inner relationship between the Noether symmetries and the conserved quantities. Thirdly, the perturbation of Noether symmetries under a small disturbance is studied, and corresponding adiabatic invariants are obtained. As special cases, the deductions in nonconservative Hamiltonian system and nonconservative Lagrangian system and standard Birkhoffian system are given. At the end of the paper, the case known as Hojman-Urrutia problem is discussed to investigate the Noether symmetries and the adiabatic invariants, the perturbation to Noether symmetries and the adiabatic invariants under El-Nabulsi dynamical model.

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