Time-Scale Version of Generalized Birkhoffian Mechanics and Its Symmetries and Conserved Quantities of Noether Type

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Jin-Yue Chen ◽  
Yi Zhang
Keyword(s):  
Conservation Laws ◽  
Time Scales ◽  
Time Scale ◽  
Symmetry And Conservation Laws ◽  
Conserved Quantity ◽  
Noether Symmetry ◽  
Conserved Quantities ◽  
Birkhoffian System

The time-scale version of Noether symmetry and conservation laws for three Birkhoffian mechanics, namely, nonshifted Birkhoffian systems, nonshifted generalized Birkhoffian systems, and nonshitfed constrained Birkhoffian systems, are studied. Firstly, on the basis of the nonshifted Pfaff-Birkhoff principle on time scales, Birkhoff’s equations for nonshifted variables are deduced; then, Noether’s quasi-symmetry for the nonshifted Birkhoffian system is proved and time-scale conserved quantity is presented. Secondly, the nonshifted generalized Pfaff-Birkhoff principle on time scales is proposed, the generalized Birkhoff’s equations for nonshifted variables are derived, and Noether’s symmetry for the nonshifted generalized Birkhoffian system is established. Finally, for the nonshifted constrained Birkhoffian system, Noether’s symmetry and time-scale conserved quantity are proposed and proved. The validity of the result is proved by examples.

scholarly journals Symmetries for Nonconservative Field Theories on Time Scale

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 552
Author(s):  
Octavian Postavaru ◽  
Antonela Toma
Keyword(s):  
Time Scales ◽  
Dynamic Systems ◽  
Time Scale ◽  
Selection Criteria ◽  
Noether Symmetry ◽  
Field Theories ◽  
Conserved Quantities ◽  
Hamilton’S Principle ◽  
Lagrange Equations ◽  
Infinitesimal Transformations

Symmetries and their associated conserved quantities are of great importance in the study of dynamic systems. In this paper, we describe nonconservative field theories on time scales—a model that brings together, in a single theory, discrete and continuous cases. After defining Hamilton’s principle for nonconservative field theories on time scales, we obtain the associated Lagrange equations. Next, based on the Hamilton’s action invariance for nonconservative field theories on time scales under the action of some infinitesimal transformations, we establish symmetric and quasi-symmetric Noether transformations, as well as generalized quasi-symmetric Noether transformations. Once the Noether symmetry selection criteria are defined, the conserved quantities for the nonconservative field theories on time scales are identified. We conclude with two examples to illustrate the applicability of the theory.

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

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Xiang-Hua Zhai ◽  
Yi Zhang
Keyword(s):  
Hamiltonian System ◽  
Time Scales ◽  
Time Scale ◽  
Dynamic Equations ◽  
Conserved Quantities ◽  
Dynamical Processes ◽  
Birkhoffian System ◽  
Special Case ◽  
Mei Symmetry

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.

Conservation laws corresponding to the Noether symmetries of the geodetic Lagrangian in spherically symmetric spacetimes

2017 ◽  
Vol 26 (05) ◽  
pp. 1741006 ◽  
Author(s):  
Bismah Jamil ◽  
Tooba Feroze
Keyword(s):  
Conservation Laws ◽  
Noether Symmetry ◽  
Complete List ◽  
Spherically Symmetric ◽  
Conserved Quantities ◽  
Noether Symmetries ◽  
Symmetric Spacetimes ◽  
Spherically Symmetric Spacetimes

In this paper, we present a complete list of spherically symmetric nonstatic spacetimes along with the generators of all Noether symmetries of the geodetic Lagrangian for such metrics. Moreover, physical and geometrical interpretations of the conserved quantities (conservation laws) corresponding to each Noether symmetry are also given.

scholarly journals Adiabatic Invariants for Generalized Fractional Birkhoffian Mechanics and Their Applications

2018 ◽  
Vol 2018 ◽  
pp. 1-18 ◽  
Author(s):  
C. J. Song
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Equations Of Motion ◽  
Conserved Quantity ◽  
Noether Symmetry ◽  
Adiabatic Invariants ◽  
Birkhoffian System ◽  
Liouville Fractional Derivative

Perturbation to Noether symmetry and adiabatic invariants are investigated for the generalized fractional Birkhoffian system with the combined Riemann-Liouville fractional derivative and the combined Caputo fractional derivative, respectively. Firstly, differential equations of motion for the generalized fractional Birkhoffian system are established. Secondly, Noether symmetry and conserved quantity are studied. Thirdly, perturbation to Noether symmetry and adiabatic invariants are presented for the generalized fractional Birkhoffian mechanics. And finally, several applications are discussed to illustrate the results and methods.

scholarly journals Approximate Mei Symmetries and Invariants of the Hamiltonian

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2910
Author(s):  
Umara Kausar ◽  
Tooba Feroze
Keyword(s):  
Conserved Quantity ◽  
Noether Symmetry ◽  
Conserved Quantities ◽  
Noether Symmetries ◽  
Fields Of Study

It is known that corresponding to each Noether symmetry there is a conserved quantity. Another class of symmetries that corresponds to conserved quantities is the class of Mei symmetries. However, the two sets of symmetries may give different conserved quantities. In this paper, a procedure of finding approximate Mei symmetries and invariants of the perturbed/approximate Hamiltonian is presented that can be used in different fields of study where approximate Hamiltonians are under consideration. The results are presented in the form of theorems along with their proofs. A simple example of mechanics is considered to elaborate the method of finding these symmetries and the related Mei invariants. At the end, a comparison of approximate Mei symmetries and approximate Noether symmetries is also given. The comparison shows that there is only one common symmetry in both sets of symmetries. Hence, rest of the symmetries in the two sets correspond to two different sets of conserved quantities.

scholarly journals Mei Symmetry and Conservation Laws for Time-Scale Nonshifted Hamilton Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Yi Zhang
Keyword(s):  
Phase Space ◽  
Conservation Laws ◽  
Time Scale ◽  
Symmetry And Conservation Laws ◽  
Dynamic Equations ◽  
Constrained Mechanical Systems ◽  
Hamilton Principle ◽  
Hamilton Equations ◽  
Mei Symmetry ◽  
Case Based

The Mei symmetry and conservation laws for time-scale nonshifted Hamilton equations are explored, and the Mei symmetry theorem is presented and proved. Firstly, the time-scale Hamilton principle is established and extended to the nonconservative case. Based on the Hamilton principles, the dynamic equations of time-scale nonshifted constrained mechanical systems are derived. Secondly, for the time-scale nonshifted Hamilton equations, the definitions of Mei symmetry and their criterion equations are given. Thirdly, Mei symmetry theorems are proved, and the Mei-type conservation laws in time-scale phase space are driven. Two examples show the validity of the results.

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