Abstract
This paper focuses on the exploration of fractional Birkhoffian mechanics and its fractional Noether theorems under quasi-fractional dynamics models. The quasi-fractional dynamics models under study are nonconservative dynamics models proposed by El-Nabulsi, including three cases: extended by Riemann–Liouville fractional integral (abbreviated as ERLFI), extended by exponential fractional integral (abbreviated as EEFI), and extended by periodic fractional integral (abbreviated as EPFI). First, the fractional Pfaff–Birkhoff principles based on quasi-fractional dynamics models are proposed, in which the Pfaff action contains the fractional-order derivative terms, and the corresponding fractional Birkhoff’s equations are obtained. Second, the Noether symmetries and conservation laws of the systems are studied. Finally, three concrete examples are given to demonstrate the validity of the results.
Highlights
Symmetry theory plays an important role in mathematics, physics, and mechanics, and the study of symmetry properties of dynamic systems has become a very effective method to solve some practical problems. e most important and common symmetries are mainly of two kinds, namely, Noether symmetry and Lie symmetry
We prove Noether’s theorems for fractional Birkhoffian systems under three quasi-fractional dynamics models
By introducing fractional calculus into the dynamic modeling of nonconservative systems, the dynamic behavior and physical process of complex systems can be described more accurately, which provides the possibility for the quantization of nonconservative problems
Summary
Symmetry theory plays an important role in mathematics, physics, and mechanics, and the study of symmetry properties of dynamic systems has become a very effective method to solve some practical problems. e most important and common symmetries are mainly of two kinds, namely, Noether symmetry and Lie symmetry. Nonconservative dynamical systems based on quasi-fractional dynamical models have been studied deeply, and the corresponding dynamical equations and Noether conservation laws have been obtained [52,53,54,55]. The results of these quasi-fractional Birkhoffian systems are limited to the Pfaff action containing only integral-order derivative terms. We will further extend fractional Birkhoffian mechanics on the basis of three quasi-fractional dynamical models given in [38,39,40], where for the Pfaff actions, we consider contain fractional-order derivative terms. − Rμ τ, a]aDβτ aμ − B τ, a](cosht − coshτ)α− 1dτ
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