Noether's Theorem Research Articles

Application of symmetry analysis and conservation laws to a fractional-order nonlinear conduction-diffusion model

<abstract> <p>In this paper, the Lie symmetry analysis was executed for the nonlinear fractional-order conduction-diffusion Buckmaster model (BM), which involves the Riemann-Liouville (R-L) derivative of fractional-order 'β'. In the study of groundwater flow and oil reservoir engineering where fluid flow through porous materials is crucial, BM played an important role. The Lie point infinitesimal generators and Lie algebra were constructed for the equation. The Lie symmetries were acquired for the ordinary fractional-order BM. The power series solution and its convergence were also analyzed with the application of the implicit theorem. Noether's theorem was employed to ensure the consistency of a system by deriving the conservation laws of its physical model.</p> </abstract>

The Lie derivative and Noether's theorem on the aromatic bicomplex for the study of volume-preserving numerical integrators

The Lie derivative and Noether's theorem on the aromatic bicomplex for the study of volume-preserving numerical integrators

Exterior Boundary-Value Poincaré Problem for Elliptic Systems of the Second Order with Two Independent Variables

This paper offers a number of examples showing that in the case of two independent variables the uniform ellipticity of a linear system of differential equations with partial derivatives of the second order, which fulfills condition (3), do not always cause the normal solvability of formulated exterior elliptic problems in the sense of Noether.Nevertheless, from the system of differential equations with partial derivatives of elliptic type it is possible to choose, under certain additional conditions, classes which are normally solvable in the sense of Noether.This paper also shows that for the so-called decomposed system of differential equations, with partial derivatives of an elliptic type in the case of exterior regions, the Noether theorems are valid.

Noether theorem and its inverse for nonstandard generalized Chaplygin systems

Noether theorem and its inverse for nonstandard generalized Chaplygin systems

Emmy Noether and Her Theorems

AbstractToday Noether's principal theorem occupies a prominent place in theoretical physics, though for a long time its significance was largely overlooked. Even now, relatively few physicists realize that Emmy Noether's original paper from 1918 contains two fundamental theorems. Moreover, both theorems are essential for understanding her original motivation, namely to distinguish between proper and improper conservation laws in physics.

Conservation laws and variational structure of damped nonlinear wave equations

All low‐order conservation laws are found for a general class of nonlinear wave equations in one dimension with linear damping which is allowed to be time‐dependent. Such equations arise in numerous physical applications and have attracted much attention in analysis. The conservation laws describe generalized momentum and boost momentum, conformal momentum, generalized energy, dilational energy, and light cone energies. Both the conformal momentum and dilational energy have no counterparts for nonlinear undamped wave equations in one dimension. All of the conservation laws are obtainable through Noether's theorem, which is applicable because the damping term can be transformed into a time‐dependent self‐interaction term by a change of dependent variable. For several of the conservation laws, the corresponding variational symmetries have a novel form which is different than any of the well‐known variational symmetries admitted by nonlinear undamped wave equations in one dimension.

Loss of pseudo-momentum, energy-release rate and the effective mass of a moving dislocation.

In a seminal paper in the Philosophical Transactions of the Royal Society (A244, 87-112). Eshelby (Eshelby 1951 Phil. Trans. R. Soc. Lond. A 244, 87-112. (doi:10.1098/rsta.1951.0016)) introduced the concept of 'the force on an elastic singularity' and suggested that the extensions to dynamics include the application of the momentum flux. In this direction, it is shown that, in the non-uniformly motion of a dislocation there is a loss in quasi-momentum (or pseudo-momentum) across the scales ε to ε2, which induces an effective mass of the dislocation, and a loss in kinetic energy across the scales. It is shown through Noether's theorem that the rate of change of quasi-momentum in the volume enclosing the dislocation is equal to the flux of it through the surface minus a quasi-force, which is the dynamic J integral. The connection between the variation in the Hamiltonian and in the Lagrangian relates the quasi-force to the energy-release rate, yielding the same effective mass, whileproviding physical meaning both through momentum and energy. The effective mass of a dislocation is important in relating the energetics of defects from the micro to the macro scales, and the loss of quasi-momentum can have wider applications in continua. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'.

A certain counterpart in dissipative setting of the Noether theorem with no dissipation pseudo-potentials.

We look at a mechanical dissipation inequality differing from the standard one by what we call a relative power, a notion that is appropriate in the presence of material mutations. We prove that a requirement of structural invariance for such an inequality under the action of diffeomorphism-based changes of observers (covariance) implies (i) the representation of contact actions in terms of the first Piola-Kirchhoff stress, (ii) local balances of standard and configurational actions, (iii) a priori constitutive restrictions in terms of free energy, and(iv) a representation of viscous-type stress components. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'.

General fractional classical mechanics: Action principle, Euler–Lagrange equations and Noether theorem

Standard action principle and first Noether theorem are generalized to take into account general form of nonlocality in time (memory) and to describe dissipative and non-Lagrangian nonlinear systems. General fractional calculus (GFC) of the Luchko form is used to take into account a wide class of nonlocalities time compared to the usual fractional calculus. Nonlocality is described by a pair of operator kernels belonging to the Luchko set. Non-holonomic variation equations of Sedov type is used to take into account equations of motion for wide class of dissipative and non-Lagrangian systems. In addition, within the framework of the proposed approach, equations of motion are considered not only with general fractional derivatives, but also for general fractional integrals. Example of application is considered.

General Fractional Noether Theorem and Non-Holonomic Action Principle

Using general fractional calculus (GFC) of the Luchko form and non-holonomic variational equations of Sedov type, generalizations of the standard action principle and first Noether theorem are proposed and proved for non-local (general fractional) non-Lagrangian field theory. The use of the GFC allows us to take into account a wide class of nonlocalities in space and time compared to the usual fractional calculus. The use of non-holonomic variation equations allows us to consider field equations and equations of motion for a wide class of irreversible processes, dissipative and open systems, non-Lagrangian and non-Hamiltonian field theories and systems. In addition, the proposed GF action principle and the GF Noether theorem are generalized to equations containing general fractional integrals (GFI) in addition to general fractional derivatives (GFD). Examples of field equations with GFDs and GFIs are suggested. The energy–momentum tensor, orbital angular-momentum tensor and spin angular-momentum tensor are given for general fractional non-Lagrangian field theories. Examples of application of generalized first Noether’s theorem are suggested for scalar end vector fields of non-Lagrangian field theory.

Non-Abelian Constructivist Lagrangian

A whole Yang-Mills symmetry is proposed. A grouping physics is constituted. It consists in inserting a given Yang-Mills field Aaμ in a fields set {Aa μI } constituted by other fields families, I = 1, . . . , N. Each field becomes part of a whole. A set action physics happens preserving the Yang-Mills symmetry. However the usual properties of an isolated field are extended to antireductionist properties. An associative physics is formed. A Yang-Mills whole quantum system is constituted. A whole Yang-Mills physics isobtained. The quantum corresponding to a specific Aa μI field inserted in a whole develops features depending on thefields set {Aa μI } associativity. Properties established from a so-called constructivist gauge theory are identified. Usual YM interactions are enlarged to YM interrelationships. Classical equations are studied under set action. A Yang-Mills whole unity is constituted by a constructivist Lagrangian. The reductionist approach substituted by constructivism. Physics under set transformations. A cause and effect relationship is expressed based on whole unity. The whole is that moves to future. Minimal action principle, Noether theorem, Bianchi identities are derived. A fields set with diversity, interdependence, nonlinearity, chance is expressed.

Generalization of Noether Theorem and action principle for non-Lagrangian theories

A non-Lagrangian field theory is a theory whose field equations cannot be derived from the action principle. The action principle is described by holonomic variational equations. These equations can be used only for standard field theories, equations of which can be represented as Euler–Lagrange equations for some Lagrangian density. In the general case, one should use non-holonomic variational equations to obtain a wider class of field equations. In this paper, it is proposed to use the Sedov non-holonomic variational equation as a generalization of action principle. Examples of application of non-holonomic variational equations to derive various field equations are suggested. The standard Noether theorem is formulated for the standard (Lagrangian) field theory with an action functional and expresses the invariance of the Lagrangian with respect to some continuous group of transformations. In this paper, a generalization of Noether’s theorem for non-Lagrangian field theories is proposed and proved by using the non-holonomic variational equation. The expression of Noether current is described in the general case of non-Lagrangian field theory. The energy–momentum tensor, orbital angular-momentum tensor and spin angular-momentum tensor are given for non-Lagrangian field theories. Examples of application of generalized first Noether’s theorem are suggested for scalar end vector fields of non-Lagrangian field theory. An important result of this paper is the proof of possibility of existence of some analogues of dissipative structures in non-Lagrangian field theories and some properties of such structures are suggested.

A theorem of Gordan and Noether via Gorenstein rings

Gordan and Noether proved in their fundamental theorem that an hypersurface X=V(F)subseteq {{mathbb {P}}}^n with nle 3 is a cone if and only if F has vanishing hessian (i.e. the determinant of the Hessian matrix). They also showed that the statement is false if nge 4, by giving some counterexamples. Since their proof, several others have been proposed in the literature. In this paper we give a new one by using a different perspective which involves the study of standard Artinian Gorenstein {{mathbb {K}}}-algebras and the Lefschetz properties. As a further application of our setting, we prove that a standard Artinian Gorenstein algebra R={{mathbb {K}}}[x_0,dots ,x_4]/J with J generated by a regular sequence of quadrics has the strong Lefschetz property. In particular, this holds for Jacobian rings associated to smooth cubic threefolds.

Constraints, symmetry transformations and conserved charges for massless Abelian 3-form theory

We demonstrate the existence of the first-class constraints on the massless Abelian 3-form theory which generate the classical gauge symmetry transformations for this theory in any arbitrary D-dimension of spacetime. We write down the explicit expression for the generator in terms of these first-class constraints. Using the celebrated Noether theorem, corresponding to the gauge symmetry transformations, we derive the Noether conserved current and conserved charge. The latter is connected with the first-class constraints of the theory in a subtle manner as we demonstrate clearly in our present investigation. We comment on the first-class constraints within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism where the conserved (anti-)BRST charges are the generalizations of the above generator for the classical gauge symmetry transformation. The standard Noether conserved (anti-)BRST charges are found to be non-nilpotent. We derive the nilpotent versions of the (anti-)BRST charges. One of the interesting observations of our present endeavor is the result that only the nilpotent versions of the conserved (anti-)BRST charges lead to the annihilation of the physical states by the operator form of the first-class constraints at the quantum level which is consistent with the Dirac quantization condition for the systems that are endowed with any kind of constraints. We comment on the existence of the Curci-Ferrari (CF) type restrictions from different theoretical angles, too.

About Lie algebra classification, conservation laws, and invariant solutions for the relativistic fluid sphere equation

The optimal generating operators for the relativistic fluid sphere equation have been derived. We have characterized all invariant solutions of this equation using these operators. Furthermore, we have introduced variational symmetries and their corresponding conservation laws, employing both Noether's theorem and Ibragimov's method. Finally, we have classified the Lie algebra associated with the given equation.

Lagrangian Reduction by Stages in Field Theory

We propose a category of bundles in order to perform Lagrangian reduction by stages in covariant Field Theory. This category plays an analogous role to Lagrange–Poincaré bundles in Lagrangian reduction by stages in Mechanics and includes both jet bundles and reduced covariant configuration spaces. Furthermore, we analyze the resulting reconstruction condition and formulate the Noether theorem in this context. Finally, a model of a molecular strand with rotors is seen as an application of this theoretical frame.

A Pointed Weak Energy Conservation Law via Noether's Theorem

A Pointed Weak Energy Conservation Law via Noether's Theorem

A global invariant and Hasse invariants at finite or real primes

In their 2019 paper, Lee and Park presented a formula for the arithmetic Chern–Simons invariant. This formula gives a relation between this invariant and the local Hasse invariants at certain finite primes. Given a number field having real embeddings, we present alternative formulas to give relations between the arithmetic Chern–Simons invariant and the local Hasse invariants at certain primes including all real ones. As an application, we propose a new problem which concerns the existence of a certain 3-cocycle. If the answer to this problem is positive, the obtained statement is an analogue of the Albert–Brauer–Hasse–Noether theorem.

Existence of a variational principle for PDEs with symmetries and current conservation

It is well-known from Noether's theorem that symmetries of a variational functional lead to conservation laws of the corresponding Euler-Lagrange equation. We reverse this statement and prove that a differential equation, which satisfies sufficiently many symmetries and corresponding conservation laws, leads to a variational functional, whose Euler-Lagrange equation is the given differential equation. Sufficiently many symmetries means that the set of symmetry vector fields satisfy span{Vp:V∈Sym}=TpE, that is, they span the tangent space TpE at each point p of a fiber bundle E, which describes the dependent- and independent coordinates. Higher order coordinates are described by the jet bundle JkE and we require no span-assumptions on TJkE. Our main theorem states that Noether's theorem can be reversed in this sense for second order differential equations, or more precisely, for so-called second order source forms on J2E, which are required to write the differential equation as a weak formulation (every Euler-Lagrange equation is derived from a first variation, that is, from a weak formulation). Counter examples show that our theorem is sharp.

Thermodynamic Entropy as a Noether Invariant from Contact Geometry.

We use a formulation of Noether's theorem for contact Hamiltonian systems to derive a relation between the thermodynamic entropy and the Noether invariant associated with time-translational symmetry. In the particular case of thermostatted systems at equilibrium, we show that the total entropy of the system plus the reservoir are conserved as a consequence thereof. Our results contribute to understanding thermodynamic entropy from a geometric point of view.