Adiabatic Invariants for Generalized Fractional Birkhoffian Mechanics and Their Applications

Song, Chunwei

doi:10.1155/2018/6414960

Cited by 5 publications

(3 citation statements)

References 36 publications

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“…Combined fractional derivatives, which contain the combined Riemann-Liouville fractional derivative and the combined Caputo fractional derivative, are listed below [5,7,19,25,59].…”

Section: Preliminaries On Fractional Derivativesmentioning

confidence: 99%

Conserved Quantities for Constrained Hamiltonian System within Combined Fractional Derivatives

Song

2022

Fractal Fract

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Singular systems, which can be applied to gauge field theory, condensed matter theory, quantum field theory of anyons, and so on, are important dynamic systems to study. The fractional order model can describe the mechanical and physical behavior of a complex system more accurately than the integer order model. Fractional singular systems within mixed integer and combined fractional derivatives are established in this paper. The fractional Lagrange equations, fractional primary constraints, fractional constrained Hamilton equations, and consistency conditions are analyzed. Then Noether and Lie symmetry methods are studied for finding the integrals of the fractional constrained Hamiltonian systems. Finally, an example is given to illustrate the methods and results.

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“…Combined fractional derivatives, which contain the combined Riemann-Liouville fractional derivative and the combined Caputo fractional derivative, are listed below [5,7,19,25,59].…”

Section: Preliminaries On Fractional Derivativesmentioning

confidence: 99%

Conserved Quantities for Constrained Hamiltonian System within Combined Fractional Derivatives

Song

2022

Fractal Fract

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“…Zhang and Zhou [52] proposed the quasi-fractional Pfaff-Birkhoff principle and derived corresponding quasi-fractional Birkhoff's equations which is based on the quasi-fractional model given by [38]. Up to now, some results have been obtained on Noether symmetry of fractional or quasi-fractional Birkhoffian systems, such as [52,[59][60][61][62][63][64][65][66]. However, the results of these quasi-fractional Birkhoffian systems are limited to the Pfaff action containing only integral-order derivative terms.…”

Section: Introductionmentioning

confidence: 99%

Fractional Birkhoffian Mechanics Based on Quasi-Fractional Dynamics Models and Its Noether Symmetry

Jia

Zhang

2021

Mathematical Problems in Engineering

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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.

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“…In 1996, Riewe started the theory of the fractional calculus of variations by considering that the first-order derivative terms should come from the half-order derivative terms [25,26]. From then on, Noether symmetry for the Hamiltonian system [53] as well as the Birkhoffian system [62]. Besides, the results on perturbation to Noether symmetry with RLVO for the fractional generalized Birkhoffian system [63] were also presented.…”

Section: Introductionmentioning

confidence: 99%

Conserved Quantity and Adiabatic Invariant for Hamiltonian System with Variable Order

Song

Yao

2019

Symmetry

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Hamiltonian mechanics plays an important role in the development of nonlinear science. This paper aims for a fractional Hamiltonian system of variable order. Several issues are discussed, including differential equation of motion, Noether symmetry, and perturbation to Noether symmetry. As a result, fractional Hamiltonian mechanics of variable order are established, and conserved quantity and adiabatic invariant are presented. Two applications, fractional isotropic harmonic oscillator model of variable order and fractional Lotka biochemical oscillator model of variable order are given to illustrate the Methods and Results.2 of 23 scholars began to use fractional derivatives to describe the dissipative forces such as frictional forces for the nonconservative systems. Then, fractional calculus of variations was studied by several scholars such as Klimek [27], Agrawal [28,29], Baleanu et al. [30,31], Torres [32], etc. Moreover, the development of the fractional calculus of variations with variable order also made great progress. In [33][34][35], several results for the fractional calculus of variations with variable order were obtained. Particularly, motivated by references [36][37][38], a linear combination of the fractional derivative of variable order was introduced. Based on a combined Caputo fractional derivative of variable order (CCVO), the necessary optimality conditions for variational problems were established [39,40], the fractional variational problem of Herglotz type with variable order was studied [41], and two fractional isoperimetric problems and a new variational problem subject to a holonomic constraint were presented [42].After differential equations of motion are established through the calculus of variations, the next task is to find the solutions to them in dynamics. In fact, the solution to the equations can be given if one can find all of the integrals of the equations. An integral is a conserved quantity; therefore, people try their best to find all of the conserved quantities of a mechanical system. By means of the analysis of forces, Newtonian mechanics gives three conservation laws, i.e., the conservation of momentum, the conservation of mechanical energy, and the conservation of moment of momentum. By means of the analysis of the form of Lagrangian, Lagrangian mechanics gives two conservation laws, i.e., the conservation of generalized energy and the conservation of generalized momentum. The conservation of generalized momentum may be a conservation of momentum, a conservation of moment of momentum, or neither. The physical meaning of the conservation law in Lagrangian mechanics is less clear than that in Newtonian mechanics, whose three conservation laws have very clear physical meaning. However, the conserved quantities deduced by Lagrangian mechanics are more than those deduced by Newtonian mechanics. Since German mathematician Emmy Noether published her famous paper [43], the Noether symmetry method has become a modern method for seeking the conservation law of mechanical systems (see...

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Adiabatic Invariants for Generalized Fractional Birkhoffian Mechanics and Their Applications

Cited by 5 publications

References 36 publications

Conserved Quantities for Constrained Hamiltonian System within Combined Fractional Derivatives

Conserved Quantities for Constrained Hamiltonian System within Combined Fractional Derivatives

Fractional Birkhoffian Mechanics Based on Quasi-Fractional Dynamics Models and Its Noether Symmetry

Conserved Quantity and Adiabatic Invariant for Hamiltonian System with Variable Order

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