Special Issue "Noether's Theorem and Symmetry"

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: 31 August 2019.

Special Issue Editors

Guest Editor
Prof. P.G.L. Leach

Department of Mathematics and Statistics, University of Cyprus, Lefkosia, Cyprus
E-Mail
Interests: Symmetries; Integrability; Differential Equations
Guest Editor
Dr. Andronikos Paliathanasis

Department of Mathematics and Natural Sciences, Core Curriculum Program, Prince Mohammad Bin Fahd University, Al Khobar 31952, Kingdom of Saudi Arabia
E-Mail
Interests: Symmetries; Integrability; Collineations; Gravitational Physics

Special Issue Information

Dear Colleagues,

In Noether's original presentation of her celebrated theorm of 1918 allowance was made for the dependence of the coefficient functions of the differential operator which generated the infinitesimal transformation of the Action Integral upon the derivatives of the depenent variable(s), the so-called generalised, or dynamical, symmetries. A similar allowance is to be found in the variables of the boundary function, often termed a gauge function by those who have not read the original paper. This generality was lost after texts such as those of Courant and Hilbert or Lovelock and Rund confined attention to point transformations only. In recent decades this dimunition of the power of Noether's Theorem has been partly countered, in particular in the review of Sarlet and Cantrijn.

In this special issue we emphasise the generality of Noether's Theorem in its original form and explore the applicability of even more general coefficient functions by alowing for nonlocal terms.   We also look for the application of these more general symmetries to problems in which parameters or parametric functions have a more general dependence upon the independent variables.

Prof. P.G.L. Leach
Dr. Andronikos Paliathanasis
Guest Editors

Manuscript Submission Information

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Keywords

  • Noether’s Theorem
  • Action Integral
  • Variational principle
  • Continuous transformations
  • Nonlocal transformations
  • Invariant
  • First integral
  • Generalized Symmetry
  • Boundary term
  • Integrability

Published Papers (9 papers)

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Research

Open AccessArticle
Hamiltonian Structure, Symmetries and Conservation Laws for a Generalized (2 + 1)-Dimensional Double Dispersion Equation
Symmetry 2019, 11(8), 1031; https://doi.org/10.3390/sym11081031
Received: 15 July 2019 / Revised: 5 August 2019 / Accepted: 6 August 2019 / Published: 9 August 2019
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Abstract
This paper considers a generalized double dispersion equation depending on a nonlinear function f(u) and four arbitrary parameters. This equation describes nonlinear dispersive waves in 2 + 1 dimensions and admits a Lagrangian formulation when it is expressed in terms [...] Read more.
This paper considers a generalized double dispersion equation depending on a nonlinear function f ( u ) and four arbitrary parameters. This equation describes nonlinear dispersive waves in 2 + 1 dimensions and admits a Lagrangian formulation when it is expressed in terms of a potential variable. In this case, the associated Hamiltonian structure is obtained. We classify all of the Lie symmetries (point and contact) and present the corresponding symmetry transformation groups. Finally, we derive the conservation laws from those symmetries that are variational, and we discuss the physical meaning of the corresponding conserved quantities. Full article
(This article belongs to the Special Issue Noether's Theorem and Symmetry)
Open AccessArticle
Quasi-Noether Systems and Quasi-Lagrangians
Symmetry 2019, 11(8), 1008; https://doi.org/10.3390/sym11081008
Received: 5 July 2019 / Revised: 1 August 2019 / Accepted: 2 August 2019 / Published: 5 August 2019
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Abstract
We study differential systems for which it is possible to establish a correspondence between symmetries and conservation laws based on Noether identity: quasi-Noether systems. We analyze Noether identity and show that it leads to the same conservation laws as Lagrange (Green–Lagrange) identity. We [...] Read more.
We study differential systems for which it is possible to establish a correspondence between symmetries and conservation laws based on Noether identity: quasi-Noether systems. We analyze Noether identity and show that it leads to the same conservation laws as Lagrange (Green–Lagrange) identity. We discuss quasi-Noether systems, and some of their properties, and generate classes of quasi-Noether differential equations of the second order. We next introduce a more general version of quasi-Lagrangians which allows us to extend Noether theorem. Here, variational symmetries are only sub-symmetries, not true symmetries. We finally introduce the critical point condition for evolution equations with a conserved integral, demonstrate examples of its compatibility, and compare the invariant submanifolds of quasi-Lagrangian systems with those of Hamiltonian systems. Full article
(This article belongs to the Special Issue Noether's Theorem and Symmetry)
Open AccessArticle
Symmetries and Reductions of Integrable Nonlocal Partial Differential Equations
Symmetry 2019, 11(7), 884; https://doi.org/10.3390/sym11070884
Received: 8 June 2019 / Revised: 24 June 2019 / Accepted: 2 July 2019 / Published: 5 July 2019
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Abstract
In this paper, symmetry analysis is extended to study nonlocal differential equations. In particular, two integrable nonlocal equations are investigated, the nonlocal nonlinear Schrödinger equation and the nonlocal modified Korteweg–de Vries equation. Based on general theory, Lie point symmetries are obtained and used [...] Read more.
In this paper, symmetry analysis is extended to study nonlocal differential equations. In particular, two integrable nonlocal equations are investigated, the nonlocal nonlinear Schrödinger equation and the nonlocal modified Korteweg–de Vries equation. Based on general theory, Lie point symmetries are obtained and used to reduce these equations to nonlocal and local ordinary differential equations, separately; namely, one symmetry may allow reductions to both nonlocal and local equations, depending on how the invariant variables are chosen. For the nonlocal modified Korteweg–de Vries equation, analogously to the local situation, all reduced local equations are integrable. We also define complex transformations to connect nonlocal differential equations and differential-difference equations. Full article
(This article belongs to the Special Issue Noether's Theorem and Symmetry)
Open AccessArticle
Symmetry Analysis and Conservation Laws of a Generalization of the Kelvin-Voigt Viscoelasticity Equation
Symmetry 2019, 11(7), 840; https://doi.org/10.3390/sym11070840
Received: 30 May 2019 / Revised: 22 June 2019 / Accepted: 24 June 2019 / Published: 28 June 2019
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Abstract
In this paper, we study a generalization of the well-known Kelvin-Voigt viscoelasticity equation describing the mechanical behaviour of viscoelasticity. We perform a Lie symmetry analysis. Hence, we obtain the Lie point symmetries of the equation, allowing us to transform the partial differential equation [...] Read more.
In this paper, we study a generalization of the well-known Kelvin-Voigt viscoelasticity equation describing the mechanical behaviour of viscoelasticity. We perform a Lie symmetry analysis. Hence, we obtain the Lie point symmetries of the equation, allowing us to transform the partial differential equation into an ordinary differential equation by using the symmetry reductions. Furthermore, we determine the conservation laws of this equation by applying the multiplier method. Full article
(This article belongs to the Special Issue Noether's Theorem and Symmetry)
Open AccessArticle
Conservation Laws and Stability of Field Theories of Derived Type
Symmetry 2019, 11(5), 642; https://doi.org/10.3390/sym11050642
Received: 4 April 2019 / Revised: 29 April 2019 / Accepted: 5 May 2019 / Published: 7 May 2019
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Abstract
We consider the issue of correspondence between symmetries and conserved quantities in the class of linear relativistic higher-derivative theories of derived type. In this class of models the wave operator is a polynomial in another formally self-adjoint operator, while each isometry of space-time [...] Read more.
We consider the issue of correspondence between symmetries and conserved quantities in the class of linear relativistic higher-derivative theories of derived type. In this class of models the wave operator is a polynomial in another formally self-adjoint operator, while each isometry of space-time gives rise to the series of symmetries of action functional. If the wave operator is given by n-th-order polynomial then this series includes n independent entries, which can be explicitly constructed. The Noether theorem is then used to construct an n-parameter set of second-rank conserved tensors. The canonical energy-momentum tensor is included in the series, while the other entries define independent integrals of motion. The Lagrange anchor concept is applied to connect the general conserved tensor in the series with the original space-time translation symmetry. This result is interpreted as existence of multiple energy-momentum tensors in the class of derived systems. To study stability we seek for bounded-conserved quantities that are connected with the time translations. We observe that the derived theory is stable if its wave operator is defined by a polynomial with simple and real roots. The general constructions are illustrated by the examples of the Pais–Uhlenbeck oscillator, higher-derivative scalar field, and extended Chern–Simons theory. Full article
(This article belongs to the Special Issue Noether's Theorem and Symmetry)
Open AccessArticle
Noether’s Theorem and Symmetry
Symmetry 2018, 10(12), 744; https://doi.org/10.3390/sym10120744
Received: 20 November 2018 / Revised: 7 December 2018 / Accepted: 9 December 2018 / Published: 12 December 2018
PDF Full-text (791 KB) | HTML Full-text | XML Full-text
Abstract
In Noether’s original presentation of her celebrated theorem of 1918, allowance was made for the dependence of the coefficient functions of the differential operator, which generated the infinitesimal transformation of the action integral upon the derivatives of the dependent variable(s), the so-called generalized, [...] Read more.
In Noether’s original presentation of her celebrated theorem of 1918, allowance was made for the dependence of the coefficient functions of the differential operator, which generated the infinitesimal transformation of the action integral upon the derivatives of the dependent variable(s), the so-called generalized, or dynamical, symmetries. A similar allowance is to be found in the variables of the boundary function, often termed a gauge function by those who have not read the original paper. This generality was lost after texts such as those of Courant and Hilbert or Lovelock and Rund confined attention to point transformations only. In recent decades, this diminution of the power of Noether’s theorem has been partly countered, in particular in the review of Sarlet and Cantrijn. In this Special Issue, we emphasize the generality of Noether’s theorem in its original form and explore the applicability of even more general coefficient functions by allowing for nonlocal terms. We also look for the application of these more general symmetries to problems in which parameters or parametric functions have a more general dependence on the independent variables. Full article
(This article belongs to the Special Issue Noether's Theorem and Symmetry)
Open AccessArticle
F(R,G) Cosmology through Noether Symmetry Approach
Symmetry 2018, 10(12), 719; https://doi.org/10.3390/sym10120719
Received: 2 November 2018 / Revised: 28 November 2018 / Accepted: 3 December 2018 / Published: 5 December 2018
Cited by 2 | PDF Full-text (321 KB) | HTML Full-text | XML Full-text
Abstract
The F(R,G) theory of gravity, where R is the Ricci scalar and G is the Gauss-Bonnet invariant, is studied in the context of existence the Noether symmetries. The Noether symmetries of the point-like Lagrangian of F(R [...] Read more.
The F ( R , G ) theory of gravity, where R is the Ricci scalar and G is the Gauss-Bonnet invariant, is studied in the context of existence the Noether symmetries. The Noether symmetries of the point-like Lagrangian of F ( R , G ) gravity for the spatially flat Friedmann-Lemaitre-Robertson-Walker cosmological model is investigated. With the help of several explicit forms of the F ( R , G ) function it is shown how the construction of a cosmological solution is carried out via the classical Noether symmetry approach that includes a functional boundary term. After choosing the form of the F ( R , G ) function such as the case ( i ) : F ( R , G ) = f 0 R n + g 0 G m and the case ( i i ) : F ( R , G ) = f 0 R n G m , where n and m are real numbers, we explicitly compute the Noether symmetries in the vacuum and the non-vacuum cases if symmetries exist. The first integrals for the obtained Noether symmetries allow to find out exact solutions for the cosmological scale factor in the cases (i) and (ii). We find several new specific cosmological scale factors in the presence of the first integrals. It is shown that the existence of the Noether symmetries with a functional boundary term is a criterion to select some suitable forms of F ( R , G ) . In the non-vacuum case, we also obtain some extra Noether symmetries admitting the equation of state parameters w p / ρ such as w = 1 , 2 / 3 , 0 , 1 etc. Full article
(This article belongs to the Special Issue Noether's Theorem and Symmetry)
Open AccessArticle
Positive Energy Condition and Conservation Laws in Kantowski-Sachs Spacetime via Noether Symmetries
Symmetry 2018, 10(12), 712; https://doi.org/10.3390/sym10120712
Received: 16 November 2018 / Revised: 29 November 2018 / Accepted: 30 November 2018 / Published: 4 December 2018
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Abstract
In this paper, we have investigated Noether symmetries of the Lagrangian of Kantowski–Sachs spacetime. The associated Lagrangian of the Kantowski–Sachs metric is used to derive the set of determining equations. Solving the determining equations for several values of the metric functions, it is [...] Read more.
In this paper, we have investigated Noether symmetries of the Lagrangian of Kantowski–Sachs spacetime. The associated Lagrangian of the Kantowski–Sachs metric is used to derive the set of determining equations. Solving the determining equations for several values of the metric functions, it is observed that the Kantowski–Sachs spacetime admits the Noether algebra of dimensions 5, 6, 7, 8, 9, and 11. A comparison of the obtained Noether symmetries with Killing and homothetic vectors is also presented. With the help of Noether’s theorem, we have presented the expressions for conservation laws corresponding to all Noether symmetries. It is observed that the positive energy condition is satisfied for most of the obtained metrics. Full article
(This article belongs to the Special Issue Noether's Theorem and Symmetry)
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Open AccessArticle
Invariant Solutions of the Wave Equation on Static Spherically Symmetric Spacetimes Admitting G7 Isometry Algebra
Symmetry 2018, 10(12), 665; https://doi.org/10.3390/sym10120665
Received: 30 October 2018 / Revised: 17 November 2018 / Accepted: 19 November 2018 / Published: 23 November 2018
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Abstract
Algorithms to construct the optimal systems of dimension of at most three of Lie algebras are given. These algorithms are applied to determine the Lie algebra structure and optimal systems of the symmetries of the wave equation on static spherically symmetric spacetimes admitting [...] Read more.
Algorithms to construct the optimal systems of dimension of at most three of Lie algebras are given. These algorithms are applied to determine the Lie algebra structure and optimal systems of the symmetries of the wave equation on static spherically symmetric spacetimes admitting G7 as an isometry algebra. Joint invariants and invariant solutions corresponding to three-dimensional optimal systems are also determined. Full article
(This article belongs to the Special Issue Noether's Theorem and Symmetry)
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