scholarly journals Variational principles and symmetries on fibered multisymplectic manifolds

2016 ◽  
Vol 24 (2) ◽  
pp. 137-152 ◽  
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.

GAUGE-NATURAL PARAMETRIZED VARIATIONAL PROBLEMS, VAKONOMIC FIELD THEORIES AND RELATIVISTIC HYDRODYNAMICS OF A CHARGED FLUID

2006 ◽  
Vol 03 (08) ◽  
pp. 1573-1608 ◽  
Author(s):  
E. BIBBONA ◽  
L. FATIBENE ◽  
M. FRANCAVIGLIA
Keyword(s):  
Variational Formulation ◽  
Variational Principles ◽  
Variational Problems ◽  
The Other ◽  
Field Theories ◽  
Relativistic Hydrodynamics ◽  
Charged Fluid ◽  
Gauge Symmetries ◽  
Relative Conservation ◽  
Conserved Currents

Variational principles for field theories where variations of fields are restricted along a parametrization are considered. In particular, gauge-natural parametrized variational problems are defined as those in which both the Lagrangian and the parametrization are gauge covariant and some further conditions are satisfied in order to formulate a Nöther theorem that links horizontal and gauge symmetries to the relative conservation laws (generalizing what Fernández, García and Rodrigo did in some recent papers). The case of vakonomic constraints in field theory is also studied within the framework of parametrized variational problems, defining and comparing two different concepts of criticality of a section, one arising directly from the vakonomic schema, the other making use of an adapted parametrization. The general theory is then applied to the case of hydrodynamics of a charged fluid coupled with its gravitational and electromagnetic field. A variational formulation including conserved currents and superpotentials is given that turns out to be computationally much easier than the standard one.

Symmetries, conservation laws and variational principles for vector field theories†

1996 ◽  
Vol 120 (2) ◽  
pp. 369-384 ◽  
Author(s):  
Ian M. Anderson ◽  
Juha Pohjanpelto
Keyword(s):  
Conservation Laws ◽  
Variational Principles ◽  
Field Theories ◽  
Conserved Quantities ◽  
Lagrange Equations ◽  
Differential Identities ◽  
Infinite Dimensional ◽  
Independent Variables ◽  
Generalized Symmetries ◽  
And Variational Principles

The interplay between symmetries, conservation laws, and variational principles is a rich and varied one and extends well beyond the classical Noether's theorem. Recall that Noether's first theorem asserts that to every r dimensional Lie algebra of (generalized) symmetries of a variational problem there are r conserved quantities for the corresponding Euler-Lagrange equations. Noether's second theorem asserts that infinite dimensional symmetry algebras (depending upon arbitrary functions of all the independent variables) lead to differential identities for the Euler-Lagrange equations.

scholarly journals Perturbation to Noether Symmetries and Adiabatic Invariants for Birkhoffian Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Yi Zhang
Keyword(s):  
Fractional Integral ◽  
Conservative System ◽  
Dynamical Model ◽  
Lagrangian System ◽  
Adiabatic Invariants ◽  
Conserved Quantities ◽  
Noether Symmetries ◽  
Small Disturbance ◽  
Birkhoffian System ◽  
Special Cases

Based on El-Nabulsi dynamical model for a non-conservative system, the problem of perturbation to Noether symmetries and adiabatic invariants of a Birkhoffian system under the action of a small disturbance is proposed and studied. Firstly, the El-Nabulsi-Pfaff variational problem from extended exponentially fractional integral is presented and the El-Nabulsi-Birkhoff equations are established. Secondly, the definitions and the criterions criteria of the Noether symmetric transformations and quasisymmetric transformations of the Birkhoffian system are given, and the Noether theorems of the system are established, which reveal the inner relationship between the Noether symmetries and the conserved quantities. Thirdly, the perturbation of Noether symmetries under a small disturbance is studied, and corresponding adiabatic invariants are obtained. As special cases, the deductions in nonconservative Hamiltonian system and nonconservative Lagrangian system and standard Birkhoffian system are given. At the end of the paper, the case known as Hojman-Urrutia problem is discussed to investigate the Noether symmetries and the adiabatic invariants, the perturbation to Noether symmetries and the adiabatic invariants under El-Nabulsi dynamical model.

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

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.

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

2017 ◽  
Vol 32 (26) ◽  
pp. 1750136 ◽  
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.

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 Gauge symmetries of the free supersymmetric string field theories

1985 ◽  
Vol 165 (1-3) ◽  
pp. 63-66 ◽  
Author(s):  
A. Neveu ◽  
P.C. West
Keyword(s):  
Field Theories ◽  
Gauge Symmetries

scholarly journals Noether Gauge Symmetries for Petrov Type D-Levi-Civita Space-Time in Spherical and Cylindrical Coordinates

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Adil Jhangeer ◽  
Tayyaba Naz
Keyword(s):  
Space Time ◽  
Petrov Type ◽  
Conserved Quantities ◽  
Restricted Range ◽  
Cylindrical Coordinates ◽  
Type D ◽  
Gauge Symmetries ◽  
Petrov Type D

Petrov Type D-Levi-Civita (DLC) space-time is considered in two different coordinates, that is, spherical and cylindrical. Noether gauge symmetries and their corresponding conserved quantities for respective metric with the restricted range of parameters and coordinates are discussed.

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