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.

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.

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.

Study of Theory about Large Unified Symmetries for Hamilton Systems

2011 ◽  
Vol 27 (2) ◽  
pp. 245-252
Author(s):  
Y.-P. Luo
Keyword(s):  
Lie Symmetry ◽  
Lie Symmetries ◽  
Noether Symmetry ◽  
Theoretical Physics ◽  
Conserved Quantities ◽  
Noether Symmetries ◽  
Mei Symmetry

ABSTRACTIn this paper, the new concept of theory about Large Unified Symmetries for Hamilton systems are presented. The Large Unified Symmetries and conserved quantities for Hamilton systems are studied by the relation between the three kinds of symmetries and the three kinds of conserved quantities. We worked on the Large Unified Symmetries and conserved quantities by Noether symmetry, Lie symmetry and Mei symmetry, including the definition and criterion of the Large Unified Symmetries and the conserved quantities deduced from them. The Large Unified Symmetries are a intersection set among the Noether symmetries, the Lie symmetries and the Mei symmetries. The theory about Large Unified Symmetries will play an important role in the fields of modern theoretical physics.

Stellar structures admitting Noether symmetries in f(ℛ,𝒯 ) gravity

2021 ◽  
Author(s):  
M. Sharif ◽  
M. Zeeshan Gul
Keyword(s):  
Specific Model ◽  
Compact Objects ◽  
Noether Symmetry ◽  
Model Parameters ◽  
Conserved Quantities ◽  
Noether Symmetries ◽  
Stellar Objects ◽  
Astrophysical Data ◽  
Symmetry Generators

This paper investigates the geometry of compact stellar objects via Noether symmetry strategy in the framework of curvature-matter coupled gravity. For this purpose, we assume the specific model of this theory to evaluate Noether equations, symmetry generators and corresponding conserved parameters. We use conserved parameters to examine some fascinating attributes of the compact objects for suitable values of the model parameters. It is analyzed that compact objects in this theory depend on the conserved quantities and model parameters. We find that the obtained solutions provide the viability of this process as they are compatible with the astrophysical data.

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 $$f(\mathcal {R},\varphi ,\chi )$$ cosmology with Noether symmetry

2020 ◽  
Vol 80 (2) ◽  
Author(s):  
M. Farasat Shamir
Keyword(s):  
Scalar Field ◽  
Modified Gravity ◽  
Vital Role ◽  
Gravity Models ◽  
Noether Symmetry ◽  
Accelerated Expansion ◽  
Conserved Quantities ◽  
Noether Symmetries ◽  
Cosmic Evolution ◽  
Friedmann Robertson Walker

Abstract This paper is devoted to explore modified $$f(\mathcal {R})$$f(R) theories of gravity using Noether symmetry approach. For this purpose, Friedmann–Robertson–Walker spacetime is chosen to investigate the cosmic evolution. The study is mainly divided into two parts: Firstly Noether symmetries of metric $$f(\mathcal {R})$$f(R) gravity are revisited and some new class of solutions with the help of conserved quantities are reported. It is shown that different scenarios of cosmic evolution can be discussed using Noether symmetries and one of the case indicates the chances for the existence of Big Rip singularity. Secondly, $$f(\mathcal {R})$$f(R) theory coupled with scalar field has been discussed in detail. The Noether equations of modified gravity are reported with three subcases for flat Friedmann–Robertson–Walker universe. It is concluded that conserved quantities are quite helpful to find some important exact solutions in the cosmological contexts. Moreover, the scalar field involved in the modified gravity plays a vital role in the cosmic evolution and an accelerated expansion phase can be observed for some suitable choices of $$f(\mathcal {R},\varphi ,\chi )$$f(R,φ,χ) gravity models.

scholarly journals Noether’s theorems for the relative motion systems on time scales

2018 ◽  
Vol 3 (2) ◽  
pp. 513-526
Author(s):  
Sheng-nan Gong ◽  
Jing-li Fu
Keyword(s):  
Time Scales ◽  
Relative Motion ◽  
Conserved Quantities ◽  
Noether Symmetries ◽  
Hamilton’S Principle ◽  
Lagrange Equations ◽  
Hamilton's Principle ◽  
Generalized Coordinates ◽  
Delta Derivatives ◽  
Infinitesimal Transformations

AbstractThis paper propose Noether symmetries and the conserved quantities of the relative motion systems on time scales. The Lagrange equations with delta derivatives on time scales are presented for the system. Based upon the invariance of Hamilton action on time scales, under the infinitesimal transformations with respect to the time and generalized coordinates, the Hamilton’s principle, the Noether theorems and conservation quantities are given for the systems on time scales. Lastly, an example is given to show the application the conclusion.

Noether symmetries of Bianchi type I spacetimes via Rif tree approach

2021 ◽  
Author(s):  
Ashfaque H. Bokhari ◽  
Muhammad Farhan ◽  
Tahir Hussain
Keyword(s):  
Bianchi Type ◽  
Noether Symmetry ◽  
Type I ◽  
Noether Symmetries ◽  
Bianchi Type I ◽  
Tree Approach ◽  
Symmetry Algebras

In this paper, we have studied Noether symmetries of the general Bianchi type I spacetimes. The Lagrangian associated with the most general Bianchi type I metric is used to find the set of Noether symmetry equations. These equations are analyzed using an algorithm, developed in Maple, to get all possible Bianchi type I metrics admitting different Noether symmetries. The set of Noether symmetry equations is then solved for each metric to obtain the Noether symmetry algebras of dimensions 4, 5, 6 and 9.

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