scholarly journals Noether Symmetry Method for Hamiltonian Mechanics Involving Generalized Operators

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Chuan-Jing Song ◽  
Yao Cheng
Keyword(s):  
Adiabatic Invariant ◽  
Conserved Quantity ◽  
Noether Symmetry ◽  
Hamiltonian Mechanics ◽  
Hamilton Equation ◽  
Hamilton Equations ◽  
Symmetry Method

Based on the generalized operators, Hamilton equation, Noether symmetry, and perturbation to Noether symmetry are studied. The main contents are divided into four parts, and every part includes two generalized operators. Firstly, Hamilton equations within generalized operators are established. Secondly, the Noether symmetry method and conserved quantity are studied. Thirdly, perturbation to the Noether symmetry and adiabatic invariant are presented. And finally, two applications are presented to illustrate the methods and results.

scholarly journals Conserved Quantity and Adiabatic Invariant for Hamiltonian System with Variable Order

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1270
Author(s):  
Song ◽  
Cheng
Keyword(s):  
Hamiltonian System ◽  
Equation Of Motion ◽  
Adiabatic Invariant ◽  
Conserved Quantity ◽  
Noether Symmetry ◽  
Oscillator Model ◽  
Variable Order ◽  
Hamiltonian Mechanics ◽  
Isotropic Harmonic Oscillator ◽  
Biochemical Oscillator

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.

Hojman conserved quantity deduced by weak Noether symmetry for Lagrange systems

2008 ◽  
Vol 17 (2) ◽  
pp. 390-393 ◽  
Author(s):  
Xie Jia-Fang ◽  
Gang Tie-Qiang ◽  
Mei Feng-Xiang
Keyword(s):  
Conserved Quantity ◽  
Noether Symmetry ◽  
Hojman Conserved Quantity

scholarly journals Noether's symmetry and conserved quantity for a time-delayed Hamiltonian system of Herglotz type

2018 ◽  
Vol 5 (10) ◽  
pp. 180208 ◽  
Author(s):  
Yi Zhang
Keyword(s):  
Time Delay ◽  
Phase Space ◽  
Hamiltonian System ◽  
Variational Problem ◽  
Conserved Quantity ◽  
Noether Symmetry ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Special Cases ◽  
The Relationship

The variational problem of Herglotz type and Noether's theorem for a time-delayed Hamiltonian system are studied. Firstly, the variational problem of Herglotz type with time delay in phase space is proposed, and the Hamilton canonical equations with time delay based on the Herglotz variational problem are derived. Secondly, by using the relationship between the non-isochronal variation and the isochronal variation, two basic formulae of variation of the Hamilton–Herglotz action with time delay in phase space are derived. Thirdly, the definition and criterion of the Noether symmetry for the time-delayed Hamiltonian system are established and the corresponding Noether's theorem is presented and proved. The theorem we obtained contains Noether's theorem of a time-delayed Hamiltonian system based on the classical variational problem and Noether's theorem of a Hamiltonian system based on the variational problem of Herglotz type as its special cases. At the end of the paper, an example is given to illustrate the application of the results.

Noether symmetry and conserved quantity for fractional Birkhoffian mechanics and its applications

2018 ◽  
Vol 21 (2) ◽  
pp. 509-526 ◽  
Author(s):  
Chuan-Jing Song ◽  
Yi Zhang
Keyword(s):  
Caputo Derivative ◽  
Conserved Quantity ◽  
Noether Symmetry ◽  
Conserved Quantities ◽  
Noether Theorem ◽  
Birkhoffian System ◽  
Special Cases ◽  
Liouville Derivative ◽  
Fractional Models ◽  
Biochemical Oscillator

AbstractNoether theorem is an important aspect to study in dynamical systems. Noether symmetry and conserved quantity for the fractional Birkhoffian system are investigated. Firstly, fractional Pfaff actions and fractional Birkhoff equations in terms of combined Riemann-Liouville derivative, Riesz-Riemann-Liouville derivative, combined Caputo derivative and Riesz-Caputo derivative are reviewed. Secondly, the criteria of Noether symmetry within combined Riemann-Liouville derivative, Riesz-Riemann-Liouville derivative, combined Caputo derivative and Riesz-Caputo derivative are presented for the fractional Birkhoffian system, respectively. Thirdly, four corresponding conserved quantities are obtained. The classical Noether identity and conserved quantity are special cases of this paper. Finally, four fractional models, such as the fractional Whittaker model, the fractional Lotka biochemical oscillator model, the fractional Hénon-Heiles model and the fractional Hojman-Urrutia model are discussed as examples to illustrate the results.

scholarly journals New static cylindrical symmetric solutions in GR for unfamiliar EoS parameter values via Noether symmetry

2021 ◽  
Vol 81 (10) ◽  
Author(s):  
Işıl Başaran Öz ◽  
Kazuharu Bamba
Keyword(s):  
Equations Of State ◽  
State Parameter ◽  
Space Time ◽  
Noether Symmetry ◽  
Field Equations ◽  
Eos Parameter ◽  
New Exact Solutions ◽  
Cylindrically Symmetric ◽  
Parameter Values ◽  
Symmetry Method

AbstractThe solutions for the field equations of f(R) gravity are investigated in static cylindrically symmetric space-time. Conserved quantities of the system, as well as unknown functions, can be determined with the help of the Noether symmetry method. In this article, some unknown values of the equations of state parameter (EoS) have emerged as a result of the constraints obtained by analyzing the Noether symmetry equations for the $$f(R)=f_0 R$$ f ( R ) = f 0 R case. Consequently, several new exact solutions have been found for cases of General Relativity in static cylindrically symmetrical space-time for the non-dust matter.

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