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.

scholarly journals Herglotz’s Variational Problem for Non-Conservative System with Delayed Arguments under Lagrangian Framework and Its Noether’s Theorem

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 845
Author(s):  
Yi Zhang
Keyword(s):  
Variational Problem ◽  
Conservative System ◽  
Noether Symmetry ◽  
Dynamic Processes ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Lagrangian Framework ◽  
Delayed Arguments ◽  
Damped Oscillators ◽  
Action Case

Because Herglotz’s variational problem achieves the variational representation of non-conservative dynamic processes, its research has attracted wide attention. The aim of this paper is to explore Herglotz’s variational problem for a non-conservative system with delayed arguments under Lagrangian framework and its Noether’s theorem. Firstly, we derive the non-isochronous variation formulas of Hamilton–Herglotz action containing delayed arguments. Secondly, for the Hamilton–Herglotz action case, we define the Noether symmetry and give the criterion of symmetry. Thirdly, we prove Herglotz type Noether’s theorem for non-conservative system with delayed arguments. As a generalization, Birkhoff’s version and Hamilton’s version for Herglotz type Noether’s theorems are presented. To illustrate the application of our Noether’s theorems, we give two examples of damped oscillators.

scholarly journals Noether’s theorem for non-smooth extremals of variational problems with time delay

2013 ◽  
Vol 93 (1) ◽  
pp. 153-170 ◽  
Author(s):  
Gastão S.F. Frederico ◽  
Tatiana Odzijewicz ◽  
Delfim F.M. Torres
Keyword(s):  
Time Delay ◽  
Variational Problems ◽  
Noether’S Theorem ◽  
Noether's Theorem

Second Noether’s theorem with time delay

2016 ◽  
Vol 96 (8) ◽  
pp. 1358-1378 ◽  
Author(s):  
Agnieszka B. Malinowska ◽  
Tatiana Odzijewicz
Keyword(s):  
Time Delay ◽  
Noether’S Theorem ◽  
Noether's Theorem

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 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.

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