Adiabatic Invariants for Generalized Fractional Birkhoffian Mechanics and Their Applications

C J Song

doi:10.1155/2018/6414960

Abstract

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.

Highlights

Compared with the integer order model, the fractional order model can describe the mechanical and physical behavior of the complex system more accurately
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
Birkhoffian mechanics is a generalization of Hamiltonian mechanics, which is described in detail in [14]

Summary

Introduction

Compared with the integer order model, the fractional order model can describe the mechanical and physical behavior of the complex system more accurately. Thereafter, perturbation to symmetry and adiabatic invariants were studied in many constrained mechanical systems such as the Lagrangian system [34], the Hamiltonian system [35], the nonholonomic system [36], and the Birkhoffian system [37], as well as the fractional constrained mechanical system [38, 39]. Because the generalized fractional Birkhoffian system and the combined fractional derivatives are general, generalized fractional Birkhoff equation, Noether symmetry and conserved quantity, and perturbation to Noether symmetry and adiabatic invariant with the combined Riemann-Liouville fractional derivative and the combined Caputo fractional derivative will be studied here.

Some Preliminaries on Fractional Derivatives

Generalized Fractional Birkhoffian System

Noether Symmetry and Conserved Quantity

Perturbation to Noether Symmetry and Adiabatic Invariants

Applications

Conclusions