A new type of adiabatic invariants for disturbed Birkhoffian system of Herglotz type

Xu, Xin-Xin; Zhang, Yi

doi:10.1016/j.cjph.2019.12.012

Cited by 6 publications

(3 citation statements)

References 40 publications

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“…Theorem 3. For the Hamilton system (40) with delayed arguments, if the infinitesimal transformation (41) corresponds to its Noether symmetry, then r linearly independent conserved quantities of Herglotz type exist, such as…”

Section: Hamilton Generalization Of Herglotz Type Noether's Theoremmentioning

confidence: 99%

“…Third, HGVP unifies conservative and non-conservative processes into the same dynamics model, and thus can systematically deal with the actual dynamical problems. Noether's theorems [23,24] based on HGVP were extended to fractional order models [25][26][27][28][29], non-conservative Hamilton systems [30][31][32], non-holonomic systems [33], Birkhoff systems [34][35][36][37], non-conservative classical and quantum systems [38][39][40], and adiabatic invariants [41,42], etc. Although some advances have been made in the study of HGVP and Noether's theorems, but little work has been done on the HGVP with time delay and its symmetry and conservation laws.…”

Section: Introductionmentioning

confidence: 99%

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

Zhang

2020

Symmetry

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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.

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Section: Hamilton Generalization Of Herglotz Type Noether's Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Zhang

2020

Symmetry

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“…Furthermore, perturbation to symmetry and adiabatic invariants will also affect the integration of the differential equations of motion [55][56][57][58][59][60][61][62]. Therefore, perturbation to Noether symmetry is also a topic of the paper.…”

Section: Introductionmentioning

confidence: 99%

Further Research for Lagrangian Mechanics within Generalized Fractional Operators

Song

2023

Fractal Fract

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In this article, the problems of the fractional calculus of variations are discussed based on generalized fractional operators, and the corresponding Lagrange equations are established. Then, the Noether symmetry method and the perturbation to Noether symmetry are analyzed in order to find the integrals of the equations. As a result, the conserved quantities and the adiabatic invariants are obtained. Due to the universality of the generalized fractional operators, the results achieved here can be used to solve other specific problems. Several examples are given to illustrate the universality of the methods and results.

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Adiabatic invariants for disturbed fractional Hamiltonian system in terms of Herglotz differential variational principle

Zhang

2020

Acta Mech

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A new type of adiabatic invariants for disturbed Birkhoffian system of Herglotz type

Cited by 6 publications

References 40 publications

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

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

Further Research for Lagrangian Mechanics within Generalized Fractional Operators

Adiabatic invariants for disturbed fractional Hamiltonian system in terms of Herglotz differential variational principle

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