Noether symmetries in extended gravity quantum cosmology

We summarize the use of Noether symmetries in Minisuperspace Quantum Cosmology. In particular, we consider minisuperspace models, showing that the existence of conserved quantities gives selection rules that allow to recover classical behaviors in cosmic evolution according to the so-called Hartle criterion. Such a criterion selects correlated regions in the configuration space of dynamical variables whose meaning is related to the emergence of classical observable universes. Some minisuperspace models are worked out starting from Extended Gravity, in particular coming from scalar-tensor, f(R) and f(T) theories. Exact cosmological solutions are derived.

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

The European Physical Journal C ◽

10.1140/epjc/s10052-020-7682-7 ◽

2020 ◽

Vol 80 (2) ◽

Cited By ~ 5

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.

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Isotropic exact solutions in $$F(R,Y,\phi )$$ gravity via Noether symmetries

The European Physical Journal C ◽

10.1140/epjc/s10052-021-08917-z ◽

2021 ◽

Vol 81 (2) ◽

Author(s):

Saira Waheed ◽

Iqra Nawazish ◽

M. Zubair

Keyword(s):

Scalar Field ◽

Exact Solutions ◽

Field Potential ◽

Noether Symmetries ◽

Dynamical Equations ◽

Curvature Invariant ◽

Gauge Symmetries ◽

Cosmic Evolution ◽

Fluid Matter ◽

Alpha Beta

AbstractThe present article investigates the existence of Noether and Noether gauge symmetries of flat Friedman–Robertson–Walker universe model with perfect fluid matter ingredients in a generalized scalar field formulation namely $$f(R,Y,\phi )$$ f ( R , Y , ϕ ) gravity, where R is the Ricci scalar and Y denotes the curvature invariant term defined by $$Y=R_{\alpha \beta }R^{\alpha \beta }$$ Y = R α β R α β , while $$\phi $$ ϕ represents scalar field. For this purpose, we assume different general cases of generic $$f(R,Y,\phi )$$ f ( R , Y , ϕ ) function and explore its possible forms along with field potential $$V(\phi )$$ V ( ϕ ) by taking constant and variable coupling function of scalar field $$\omega (\phi )$$ ω ( ϕ ) . In each case, we find non-trivial symmetry generator and its related first integrals of motion (conserved quantities). It is seen that due to complexity of the resulting system of Lagrange dynamical equations, it is difficult to find exact cosmological solutions except for few simple cases. It is found that in each case, the existence of Noether symmetries leads to power law form of scalar field potential and different new types of generic function. For the acquired exact solutions, we discuss the cosmology generated by these solutions graphically and discuss their physical significance which favors the accelerated expanding eras of cosmic evolution.

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

Applied Mathematics and Nonlinear Sciences ◽

10.2478/amns.2018.2.00040 ◽

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.

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Noether symmetries and conserved quantities for fractional Birkhoffian systems

Nonlinear Dynamics ◽

10.1007/s11071-015-2005-5 ◽

2015 ◽

Vol 81 (1-2) ◽

pp. 469-480 ◽

Cited By ~ 33

Author(s):

Yi Zhang ◽

Xiang-Hua Zhai

Keyword(s):

Conserved Quantities ◽

Noether Symmetries

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Noether symmetries and anisotropic universe models in f(R, T) gravity

Modern Physics Letters A ◽

10.1142/s021773231750136x ◽

2017 ◽

Vol 32 (26) ◽

pp. 1750136 ◽

Cited By ~ 4

Author(s):

M. Sharif ◽

Iqra Nawazish

Keyword(s):

Cosmological Parameters ◽

Graphical Analysis ◽

Momentum Tensor ◽

Anisotropic Universe ◽

Conserved Quantities ◽

Noether Symmetries ◽

Exponential Expansion ◽

Symmetry Generators ◽

Dust Distribution ◽

Universe Models

This paper investigates the existence of Noether symmetries of some anisotropic homogeneous universe models in non-minimally coupled f(R, T) gravity (R and T represent Ricci scalar and trace of the energy–momentum tensor). We evaluate symmetry generators and the corresponding conserved quantities for two models of this theory admitting direct and indirect non-minimal curvature–matter coupling. We also discuss exact solutions for dust as well as non-dust matter distribution and study the physical behavior of some cosmological parameters through these solutions. For dust distribution, the exact solution corresponds to power-law expansion and Einstein universe while exponential expansion appears for non-dust matter. The graphical analysis of these solutions and cosmological parameters provide consistent results with recent observations about accelerated cosmic expansion. We conclude that Noether symmetry generators and conserved quantities exist for both models.

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Conserved quantities for the geodesic motion on the configuration space of non-abelian Yang-Mills theory

Physics Letters B ◽

10.1016/0370-2693(81)90190-8 ◽

1981 ◽

Vol 103 (1) ◽

pp. 45-47 ◽

Cited By ~ 9

Author(s):

O. Babelon ◽

C.M. Viallet

Keyword(s):

Configuration Space ◽

Conserved Quantities ◽

Geodesic Motion ◽

Yang Mills

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

International Journal of Modern Physics D ◽

10.1142/s0218271817410061 ◽

2017 ◽

Vol 26 (05) ◽

pp. 1741006 ◽

Cited By ~ 3

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.

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Variational principles and symmetries on fibered multisymplectic manifolds

Communications in Mathematics ◽

10.1515/cm-2016-0010 ◽

2016 ◽

Vol 24 (2) ◽

pp. 137-152 ◽

Cited By ~ 3

Author(s):

Jordi Gaset ◽

Pedro D. Prieto-Martínez ◽

Narciso Román-Roy

Keyword(s):

Variational Principles ◽

Variational Calculus ◽

Field Theories ◽

Fiber Bundles ◽

Conserved Quantities ◽

Noether Symmetries ◽

Geometric Framework ◽

Gauge Symmetries ◽

Special Cases ◽

Standard Techniques

Abstract The standard techniques of variational calculus are geometrically stated in the ambient of fiber bundles endowed with a (pre)multi-symplectic structure. Then, for the corresponding variational equations, conserved quantities (or, what is equivalent, conservation laws), symmetries, Cartan (Noether) symmetries, gauge symmetries and different versions of Noether's theorem are studied in this ambient. In this way, this constitutes a general geometric framework for all these topics that includes, as special cases, first and higher order field theories and (non-autonomous) mechanics.

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