Noether Symmetry Research Articles

Noether symmetries of the minimal surface Lagrangian for Godel-type spacetimes

Noether symmetries of the minimal surface Lagrangian for Godel-type spacetimes

Kinetic Energy and Angular Momentum of Free Particles in a Class of Rotating Cylindrical Gravitational Waves using the Noether Symmetry Approach

Kinetic Energy and Angular Momentum of Free Particles in a Class of Rotating Cylindrical Gravitational Waves using the Noether Symmetry Approach

Quantum Cosmology in Bianchi-I model: A study of symmetry analysis

The paper deals with Noether symmetry analysis for anisotropic Bianchi-I space–time as well as conformal Symmetry of the physical space. Subsequently, quantum cosmology has been formulated for this model, using canonical quantization programme and also by causal interpretation. Finally, it has been examined whether quantum description may avoid the classical singularity or not.

The equivalence principle is NOT a Noether symmetry

The connection between the equivalence principle and Noether’s theorem was discussed in Capozziello and Ferrara (Int J Geom Methods Mod Phys 21:2440014, 2024). However, it is known that the Noether symmetry condition is independent of the equations of motions as follows from Hamilton’s principle. In this paper, we critically examine the analysis presented in the aforementioned work, highlighting its flaws, and provide a detailed demonstration that there is no connection between Noether’s theorem and the equivalence principle. Furthermore, we offer various insights on symmetry analysis to prevent the perpetuation of inaccuracies regarding Noether’s work.

Symmetry analysis in multi scalar-torsion cosmological model with quantum description

A detailed symmetry analysis has been done for the present multi scalar-torsion gravity theory in the background of flat Friedmann Lemai^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hat{\ ext{ i }}$$\\end{document}tre Robertson Walker (FLRW) space-time. Not only the Noether symmetry from the invariance of the Lagrangian has been determined and associated conserved quantities are evaluated, but also the conformal symmetry of the physical metric (i.e., metric of the kinetic part) has been investigated to evaluate homothetic and killing vector fields. Finally, a through study of quantum cosmology both by canonical approach and by considering quantum trajectories has been presented.

In search of hidden symmetries

Abstract This paper exemplifies the importance of finding hidden symmetries of differential equations that are models of physical phenomena. The hidden symmetries (Lie symmetries) may be determined by either linking together different equations for certain values of their parameters or transforming the original model into another equivalent system of equations that may have more symmetries. Therefore, hidden symmetries may help to solve the original model or yield its hidden properties, e.g. linearity and conservation laws. Moreover Noether symmetries are shown to be preserved by going from classical to quantum mechanics, namely from Lagrangian systems to the corresponding time-dependent Schrödinger equation.

Probing the effect of time conformal factor on the particle dynamics around AdS Reissner–Nordström black hole

This study entails the effect of time on particle dynamics (both charged and uncharged) in the surrounding of the Reissner–Nordström AdS black hole in light of a general time conformal factor by acquiring Noether symmetry relation using the Lagrangian of the considered black hole spacetime. Upon acquiring the required time conformal spacetime as well as the Noether and approximated Noether symmetries along with their respective laws of conservation, the study further aims at investigating different parameters, like effective potential, effective force, and trajectories of escape velocity for the particle dynamics of the specified black hole. Additionally, in order to investigate orbit stability, the study employs the Lyapunov exponent.

Noether symmetries, conservation laws and geometric properties of the Banados-Teitelboim-Zanelli black hole metric

This study provides a comprehensive analysis of the Banados-Teitelboim-Zanelli (BTZ) black hole, a significant solution in three-dimensional gravity. We begin with a detailed exploration of the BTZ black hole metric, highlighting its derivation and physical properties, such as its horizon structure and thermodynamic characteristics. The thermodynamics of BTZ black holes, including temperature, entropy, and free energy, are analyzed to understand their stability and behavior. A central focus of this work is the application of Noether symmetry principles to the BTZ black hole. We systematically compute the Noether symmetries of the BTZ metric and discuss their physical implications, including their role in conservation laws and stability analysis. This includes an examination of perturbations and their impact on the stability of the black hole, as well as a detailed geodesic analysis that leverages Noether symmetries to simplify the study of particle motion in this spacetime. Additionally, we investigate the first integrals associated with the BTZ black hole metric, elucidating their significance for the dynamics and conservation laws within this framework. The geometric analysis is further expanded through the Cartan structure equations, providing insight into the intrinsic geometric properties of the BTZ black hole and associated curvature tensors. Overall, this work integrates thermodynamic, symmetric, and geometric analyses to offer a unified understanding of the BTZ black hole’s structure and behavior, contributing to the broader field of black hole physics in lower-dimensional spacetimes.

Noether symmetries and conservation laws in some analytic spherically symmetric spacetimes of f(R, L m ) gravity

In general relativity, dark energy is usually illustrated by a cosmological constant(Λ), but f(R, L m ) gravity provides a different approach to cosmic acceleration by modifying the gravitational theory. In the present paper, the non-static spherically symmetric spacetimes have been derived by taking into account f(R, L m ) = f 1(R) + H(L m )f 2(R). Lie symmetry approach is operated to reduce the order of the partial differential equations corresponding to the field equations, which are further solved. Killing and Noether symmetries provides insights into the conservation laws. So, the obtained spacetimes have been investigated to obtain the Killing and Noether symmetries. The Lagrangian approach have been used to obtain the Noether symmetries. This study is well-structured, as it provides a justification for the well-established result that Noether symmetries encompass Killing symmetries [1]. Also the conserved quantities and commutators of Noether symmetries have been calculated.

Geometric Linearization for Constraint Hamiltonian Systems

This study investigates the geometric linearization of constraint Hamiltonian systems using the Jacobi metric and the Eisenhart lift. We establish a connection between linearization and maximally symmetric spacetimes, focusing on the Noether symmetries admitted by the constraint Hamiltonian systems. Specifically, for systems derived from the singular Lagrangian LN,qk,q˙k=12Ngijq˙iq˙j−NV(qk), where N and qi are dependent variables and dimgij=n, the existence of nn+12 Noether symmetries is shown to be equivalent to the linearization of the equations of motion. The application of these results is demonstrated through various examples of special interest. This approach opens new directions in the study of differential equation linearization.

General analysis of Noether symmetries in Horndeski gravity

We explore Noether symmetries of Horndeski gravity, extending the classification of general scalar–tensor theories. Starting from the minimally coupled scalar field and the first-generation scalar–tensor gravity, the discussion is generalised to kinetic gravity braiding and Horndeski gravity. We highlight the main findings by focusing on the non-minimally coupled Gauss–Bonnet term and the extended cuscuton model. Finally, we discuss how the presence of matter can influence Noether symmetries. It turns out that the selected Horndeski functions are unchanged with respect to the vacuum case.

Classical and Quantum Cosmology in Weyl Integrable Gravity: A Noether Symmetry Approach

Classical and Quantum Cosmology in Weyl Integrable Gravity: A Noether Symmetry Approach

Canonical lifts in multisymplectic De Donder–Weyl Hamiltonian field theories

We define canonical lifts of vector fields to the multisymplectic multimomentum bundles of De Donder–Weyl Hamiltonian (first-order) field theories and to the appropriate premultisymplectic embedded constraint submanifolds on which singular field theories are studied. These new canonical lifts are used to study the so-called natural Noether symmetries present in both regular and singular Hamiltonian field theories along with their associated conserved quantities obtained from Noether’s theorem. The Klein–Gordon field, the Polyakov bosonic string, and Einstein–Cartan gravity in 3+1 dimensions are analyzed in depth as applications of these concepts; as a peripheral result obtained in the analysis of the bosonic string, we provide a new geometrical interpretation of the well-known Virasoro constraint.

Noether Symmetries of the Triple Degenerate DNLS Equations

In this paper, Lie symmetries and Noether symmetries along with the corresponding conservation laws are derived for weakly nonlinear dispersive magnetohydrodynamic wave equations, also known as the triple degenerate derivative nonlinear Schrödinger equations. The main goal of this study is to obtain Noether symmetries of the second-order Lagrangian density for these equations using the Noether symmetry approach with a gauge term. For this Lagrangian density, we compute the conserved densities and fluxes corresponding to the Noether symmetries with a gauge term, which differ from the conserved densities obtained using Lie symmetries in Webb et al. (J. Plasma Phys. 1995, 54, 201–244; J. Phys. A Math. Gen. 1996, 29, 5209–5240). Furthermore, we find some new Lie symmetries of the dispersive triple degenerate derivative nonlinear Schrödinger equations for non-vanishing integration functions Ki(t) (i=1,2,3).

One class of solutions to the equations of ideal plasticity

Much attention is given to the study and solution of nonlinear differential equations in the modern mathematical literature. Despite this, there are not many methods for researching and solving such equations. These are point and contact transformations of equations, various methods of separating variables, the method of differential connections, the search for various symmetries and their use to construct solutions, as well as conservation laws. The paper considers a nonlinear differential equation describing the plastic flow of a prismatic rod. A group of point symmetries is found for this equation. The optimal system of one-dimensional subalgebras is calculated. Conservation laws corresponding to Noetherian symmetries are given, and it is also shown that there are infinitely many non-Noetherian conservation laws. Several new invariant solutions of rank one, i. e. depending on one independent variable, are constructed. It is shown how classes of new solutions can be constructed from two exact solutions, passing to a linear equation. Thus, in this short article, almost all methods of modern research of nonlinear differential equations are involved.

Generalized Brans-Dicke cosmology, noether symmetries and quantization

A generalized Brans-Dicke theory of gravitation is applied to investigate a cosmological model for a primordial universe that includes a self-interacting bosonic field as source. This model permits, initially, to consider an arbitrary ω function, not usual for BD formulations. The imposition of Noether symmetries conditions identifies a reduced space for the ω function. This model is shown to present a variety of behaviors consistent with a primordial era, depending on the parameters chosen. We find that the bosonic field is capable of promoting a transition from an initially accelerated to a decelerated regime. The self-interaction potential is also restricted by the Noether conditions, and can be interpreted as a vacuum energy contribution. The model also permits a quantization process, going through a hamiltonian formulation and recognizing the presence of cyclic variables. The Bohm approach shows as result a consistent model with a smooth classical limit as the Universe leaves the Planck era.

Noether symmetry analysis in f(T,TG) gravity theory: A study of classical cosmology

This work deals with [Formula: see text] gravity in the background of homogeneous and isotropic flat Friedmann–Lemaître–Robertson–Walker (FLRW) space-time model. The main aim of this work is to examine whether the model supports the observational data or not. By using the Noether symmetry analysis, not only the symmetry vector is obtained, but also the coupling function [Formula: see text] in the Lagrangian has been determined. Also, the symmetry analysis identifies a transformation from [Formula: see text] to [Formula: see text] in such a way that one of the new variables become cyclic along the direction symmetry vector and as a result the field equations become solvable. Then the solution is analyzed from the observational point of view. Finally, quantum cosmology has been formulated for the present model and the wave function of the universe has been evaluated by solving the Wheeler–DeWitt equation with some assumptions.

Herglotz-type vakonomic dynamics and its Noether symmetry for nonholonomic constrained systems

In this paper, Herglotz-type vakonomic dynamics and Noether theory of nonholonomic systems are studied. Firstly, Herglotz-type vakonomic dynamical equations for nonholonomic systems are derived on the premise of Herglotz variational principle. Secondly, in terms of the Herglotz-type vakonomic dynamical equations, the Noether symmetry of Herglotz-type vakonomic dynamics is explored, and the Herglotz-type vakonomic dynamical Noether theorems and their inverse theorems are deduced. Finally, the conservation laws of Appell–Hamel case with non-conservative forces are analyzed to show the validity of our results.

Noether symmetry in new extended modified f(R,G,T) theory of gravity

This study is dedicated to investigate the Noether symmetry approach in a newly introduced gravitational theory known as [Formula: see text] gravity, where [Formula: see text] is the function of Ricci scalar [Formula: see text], trace of energy–momentum tensor [Formula: see text], and the Gauss–Bonnet invariant term [Formula: see text]. The Noether symmetry approach plays a supportive role in generating models and then determining the exact solutions by using the conserved quantities. For this purpose, the flat Friedmann–Robertson–Walker space–time is selected to observe cosmic evolution. The set of partial differential equations is calculated in the background [Formula: see text] modified theory gravity. We further examine the four different [Formula: see text] gravity models and calculate the conserved quantities to investigate the exact solutions. The three models show the increasing behavior of the cosmic scale factor that supports the expansion of universe. In fourth model, we are able to get the corresponding conserved quantity only and it is anticipated that this model will also support the expansion of universe.

Exact solutions of an anisotropic universe in a modified teleparallel gravity model via the Noether and B.N.S. approaches

In this paper, we present the Noether symmetries of locally rotationally symmetric Bianchi-type I (LRS BI), an anisotropic model, in the context of the teleparallel gravity. We study a certain modified teleparallel theory based on the action that, in particular, contains a coupling between the scalar field and field strength (magnetism part). We derive the symmetry generators and show that, by means of cyclic variables approach, one can not obtain a suitable solution for field equations. Hence by the use of beyond Noether symmetry (B.N.S.) approach, we solve the equations which carry Noether currents as well. By data analysis of the obtained results, compatible results with observational data at the last half of the age of universe which is accelerating are demonstrated.