Noether theorem for Birkhoffian systems on time scales

Song, Chunwei; Zhang, Yi

doi:10.1063/1.4932607

Cited by 43 publications

(19 citation statements)

References 19 publications

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“…Some results for the Birkhoffian system on time scales with delta derivatives are reviewed, see [66]. We consider the following problem…”

Section: Results With Delta Derivativesmentioning

confidence: 99%

Noether theory for Birkhoffian systems with nabla derivatives

Song¹,

Yi²

2017

J. Nonlinear Sci. Appl.

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There are discrete phenomena which happen only on discrete time or hold discrete space structures such as economy series, population dynamics et al.. Then there is a tool needed for these discrete issues or applications. Time scale is one of the useful tools to solve some discrete problems. In this paper, time scale is used to establish discrete Pfaff-Birkhoff principle and achieve discrete Birkhoff equations, discrete Noether identity and discrete conserved quantity for the discrete Birkhoffian system. Firstly, Birkhoff equations, Noether identity and Noether theorem with nabla derivatives on time scales are investigated by using the isochronous variational principle. Secondly, some special cases, especially the discrete Birkhoffian system are discussed. Thirdly, another method, i.e., the duality principle is introduced for the Birkhoffian system on time scales. And finally, an example is given to illustrate the results and methods.

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“…Some results for the Birkhoffian system on time scales with delta derivatives are reviewed, see [66]. We consider the following problem…”

Section: Results With Delta Derivativesmentioning

confidence: 99%

Noether theory for Birkhoffian systems with nabla derivatives

Song¹,

Yi²

2017

J. Nonlinear Sci. Appl.

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“…e Pfaff-Birkhoff principle and Birkhoff's equations were studied in a time-scale analogue, and their corresponding Noether symmetry and Lie symmetry as well as the conserved quantities have been studied [39,40] recently.…”

Section: Introductionmentioning

confidence: 99%

Mei Symmetry and New Conserved Quantities of Time-Scale Birkhoff’s Equations

Zhai

Zhang

2020

Complexity

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The time-scale dynamic equations play an important role in modeling complex dynamical processes. In this paper, the Mei symmetry and new conserved quantities of time-scale Birkhoff’s equations are studied. The definition and criterion of the Mei symmetry of the Birkhoffian system on time scales are given. The conditions and forms of new conserved quantities which are found from the Mei symmetry of the system are derived. As a special case, the Mei symmetry of time-scale Hamilton canonical equations is discussed and new conserved quantities for the Hamiltonian system on time scales are derived. Two examples are given to illustrate the application of results.

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“…Cai and Fu established the Noether symmetries of the non-conservative and non-holonomic systems on time scales, and obtained the symmetry theorem for constrained mechanical systems on time scales [4,5]. More recently, Noether theory for Bikhoffian systems on time scales was established by Song and Zhang [6]. Zhai and Zhang obtain the Noether theorem for non-conservative systems with time delay on time scales [7].The time scales has a tremendous potential for applications and has recently received much attention in other areas such as engineering, biology, economics, and physics [9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Noether’s theorems of variable mass systems on time scales

2018

Applied Mathematics and Nonlinear Sciences

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This paper deals with the Noether’s theory for variable mass system on time scales. The calculus on time scales unifies and extends variable mass system continuous model and discrete model into a single theory. Firstly, Hamilton’s principle of the variable mass system on time scales is given. Secondly, based on the quasi-invariance of the Hamilton’s action under a group of infinitesimal transformations, Noether’s theorem and its inverse theorem of the variable mass system on time scales are presented. Finally, two examples are given to illustrate the applications of the results.

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Noether theorem for Birkhoffian systems on time scales

Cited by 43 publications

References 19 publications

Noether theory for Birkhoffian systems with nabla derivatives

Noether theory for Birkhoffian systems with nabla derivatives

Mei Symmetry and New Conserved Quantities of Time-Scale Birkhoff’s Equations

Noether’s theorems of variable mass systems on time scales

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