Noether's Theorem Research Articles

Conservation laws and path-independent integrals in mechanical-diffusion-electrochemical reaction coupling system

In this study, the conservation laws οf dissipative mechanical-diffusion-electrochemical reaction system are systematically obtained based on Noether's theorem. According to linear, irreversible thermodynamics, dissipative phenomena can be described by an irreversible force and an irreversible flow. Additionally, the Lagrange function, L and the generalized Hamilton least-action principle are proposed to be used to obtain the conservation integrals. A group of these integrals, including the J-, M-, and L-integrals, can be then obtained using the classical Noether approach for dissipative processes. The relation between the J-integral and the energy release rate is illustrated. The path-independence of the J-integral is then proven. The J-integral, derived based on Noether's theorem, is a line integral, contrary to the propositions of existing published works that describe it both as a line and an area integral. Herein, we prove that the outcomes are identical, and identify the physical meaning of the area integral, a concept that was not explained previously. To show that the J-integral can dominate the distribution of the corresponding field quantities, an example of a partial, stress-diffusion coupling process is disscussed.

Symmetries, Conservation Laws, and Noether's Theorem for Differential‐Difference Equations

AbstractThis paper mainly contributes to the extension of Noether's theorem to differential‐difference equations. For this purpose, we first investigate the prolongation formula for continuous symmetries, which makes a characteristic representation possible. The relations of symmetries, conservation laws, and the Fréchet derivative are also investigated. For nonvariational equations, because Noether's theorem is now available, the self‐adjointness method is adapted to the computation of conservation laws for differential‐difference equations. Several differential‐difference equations are investigated as illustrative examples, including the Toda lattice and semidiscretizations of the Korteweg–de Vries (KdV) equation. In particular, the Volterra equation is taken as a running example.

Variational calculus with conformable fractional derivatives

Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different necessary conditions of invariance are obtained. As particular cases, we prove fractional versions of Noether's symmetry theorem. Invariant conditions for fractional optimal control problems, using the Hamiltonian formalism, are also investigated. As an example of potential application in Physics, we show that with conformable derivatives it is possible to formulate an Action Principle for particles under frictional forces that is far simpler than the one obtained with classical fractional derivatives.

A Noether theorem for stochastic operators on Schatten classes

We show that a stochastic (Markov) operator S acting on a Schatten class C1 satisfies the Noether condition (i.e. S′(A)=A and S′(A2)=A2, where A∈C∞ is a Hermitian and bounded operator on a fixed separable and complex Hilbert space (H,〈⋅,⋅〉)), if and only if S(EA(G)XEA(G))=EA(G)S(X)EA(G) for any state X∈C1 and all Borel sets G⊆R, where EA(G) denotes the orthogonal projection coming from the spectral resolution A=∫σ(A)zEA(dz). Similar results are obtained for stochastic one-parameter continuous semigroups.

Noether symmetries and stability of ideal gas solutions in Galileon cosmology

A class of generalized Galileon cosmological models, which can be described by a point-like Lagrangian, is considered in order to utilize Noether's Theorem to determine conservation laws for the field equations. In the Friedmann-Lema\^itre-Robertson-Walker universe, the existence of a nontrivial conservation law indicates the integrability of the field equations. Due to the complexity of the latter, we apply the differential invariants approach in order to construct special power-law solutions and study their stability.

Special divisors on marked chains of cycles

We completely describe all Brill–Noether loci on metric graphs consisting of a chain of g cycles with arbitrary edge lengths, generalizing work of Cools, Draisma, Payne, and Robeva. The structure of these loci is determined by displacement tableaux on rectangular partitions, which we define. More generally, we fix a marked point on the rightmost cycle, and completely analyze the loci of divisor classes with specified ramification at the marked point, classifying them using displacement tableaux. Our results give a tropical proof of the generalized Brill–Noether theorem for general marked curves, and serve as a foundation for the analysis of general algebraic curves of fixed gonality.

Symmetries and singularities of the Szekeres system

The Szekeres system is studied with two methods for the determination of conservation laws. Specifically we apply the theory of group invariant transformations and the method of singularity analysis. We show that the Szekeres system admits a Lagrangian and the conservation laws that we find can be derived by the application of Noether's theorem. The stability for the special solutions of the Szekeres system is studied and it is related with the Left or Right Painlevé Series which describes the expansions.

Symmetry and conservation laws for tuberculosis model

In this paper, three-dimensional system of the tuberculosis (TB) model is reduced into a two-dimensional first-order and one-dimensional second-order differential equations. We use the method of Jacobi last multiplier to construct linear Lagrangians of systems of two first-order ordinary differential equations and nonlinear Lagrangian of the corresponding single second-order equation. The Noether's theorem is used for determining conservation laws. We apply the techniques of symmetry analysis to a model to identify the combinations of parameters which lead to the possibility of the linearization of the system and provide the corresponding solutions.

Recursion operators and bi-Hamiltonian structure of the general heavenly equation

We discover two additional Lax pairs and three nonlocal recursion operators for symmetries of the general heavenly equation introduced by Doubrov and Ferapontov. Converting the equation to a two-component form, we obtain Lagrangian and Hamiltonian structures of the two-component general heavenly system. We study all point symmetries of the two-component system and, using the inverse Noether theorem in the Hamiltonian form, obtain all the integrals of motion corresponding to each variational (Noether) symmetry. We discover that in the two-component form we have only a single nonlocal recursion operator. Composing the recursion operator with the first Hamiltonian operator we obtain second Hamiltonian operator. We check the Jacobi identities for the second Hamiltonian operator and compatibility of the two Hamiltonian structures using P. Olver’s theory of functional multi-vectors. Our well-founded conjecture is that P. Olver’s method works fine for nonlocal operators. We show that the general heavenly equation in the two-component form is a bi-Hamiltonian system integrable in the sense of Magri. We demonstrate how to obtain nonlocal Hamiltonian flows generated by local Hamiltonians by using formal adjoint recursion operator.

Conservation laws for collisional and turbulent transport processes in toroidal plasmas with large mean flows

A novel gyrokinetic formulation is presented by including collisional effects into the Lagrangian variational principle to yield the governing equations for background and turbulent electromagnetic fields and gyrocenter distribution functions, which can simultaneously describe classical, neoclassical, and turbulent transport processes in toroidal plasmas with large toroidal flows on the order of the ion thermal velocity. Noether's theorem modified for collisional systems and the collision operator given in terms of Poisson brackets are applied to derivation of the particle, energy, and toroidal momentum balance equations in the conservative forms, which are desirable properties for long-time global transport simulation.

Brill–Noether theory for cyclic covers

We recall that the Brill–Noether Theorem gives necessary and sufficient conditions for the existence of a gdr. Here we consider a general n-fold, étale, cyclic cover p:C˜→C of a curve C of genus g and investigate for which numbers r,d a gdr exists on C˜. For r=1 this means computing the gonality of C˜. Using degeneration to a special singular example (containing a Castelnuovo canonical curve) and the theory of limit linear series for tree-like curves we show that the Plücker formula yields a necessary condition for the existence of a gdr which is only slightly weaker than the sufficient condition given by the results of Laksov and Kleimann [24], for all n,r,d.

Noether theorems and quantum anomalies

A quantum anomaly is the breaking of symmetry with respect to some transformations after the quantization of a classical Hamiltonian or Lagrangian system. It is shown that both the Noether theorems (including their infinite-dimensional versions) and the explanation of the origin of quantum anomalies can be obtained by using similar formulas for derivatives of functions whose values are measure in the former case and pseudomeasures in the latter.

The Prolongation Structures for the System of the Reaction-Diffusion Type

Equations of the reaction-diffusion type are very well known and have been extensively studied in many research areas. In this paper, the prolongation structures for the system of the reaction-diffusion type are investigated from theory of coverings. The realizations and the classifications of the one-dimensional coverings of the system are researched. And the corresponding conservation law of the one-dimensional Abelian coverings is concluded, which is closely connected with the symmetry of the system by Noether theorem.

Generalization of approximate partial Noether approach in phase space

The approximate partial Noether theorem proposed earlier for the ordinary differential equations (ODEs) (Naeem and Mahomed in Nonlinear Dyn 57(1–2):303–311, 2009) is generalized in phase space for the perturbed Hamiltonian-type systems. The notion of approximate partial Hamiltonian is developed. An approximate partial Hamiltonian gives rise to an approximate Hamiltonian-type perturbed dynamical system of first-order ODEs. An approximate Legendre-type transformation connects the approximate partial Lagrangian and what we term as approximate partial Hamiltonian. The formulas for approximate partial Hamiltonian operators determining equations and first integrals are provided explicitly. We have explained our approach with the help of simple illustrative example. Then, it is applied to establish the approximate first integrals, reductions and exact solutions of two perturbed cubically coupled Duffing–Van der Pol oscillators. Both resonant and nonresonant cases are considered.

Variational principles and symmetries on fibered multisymplectic manifolds

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.

Conservation of generalized momentum maps in mechanical optimal control problems with symmetry

SummaryIn this paper, a structure‐preserving direct method for the optimal control of mechanical systems is developed. The new method accommodates a large class of one‐step integrators for the underlying state equations. The state equations under consideration govern the motion of affine Hamiltonian control systems. If the optimal control problem has symmetry, associated generalized momentum maps are conserved along an optimal path. This is in accordance with an extension of Noether's theorem to the realm of optimal control problems. In the present work, we focus on optimal control problems with rotational symmetries. The newly proposed direct approach is capable of exactly conserving generalized momentum maps associated with rotational symmetries of the optimal control problem. This is true for a variety of one‐step integrators used for the discretization of the state equations. Examples are the one‐step theta method, a partitioned variant of the theta method, and energy‐momentum (EM) consistent integrators. Numerical investigations confirm the theoretical findings. Copyright © 2016 John Wiley & Sons, Ltd.

Tropical independence, II: The maximal rank conjecture for quadrics

Building on our earlier results on tropical independence and shapes of divisors in tropical linear series, we give a tropical proof of the maximal rank conjecture for quadrics. We also prove a tropical analogue of Max Noether's theorem on quadrics containing a canonically embedded curve, and state a combinatorial conjecture about tropical independence on chains of loops that implies the maximal rank conjecture for algebraic curves.

Note on a Noetherian conservation law and its corresponding general class of nonlinear second‐order ordinary differential equations

By virtue of their fundamental nature and intrinsic elegance, there has long been a great interest in searching for the conservation laws of the problem at hand. In such field of research, the famous Noether's theorem has been recognized as a highlight. The objective of our note is to provide the reader with an original contribution to the study of Noetherian conservation laws. Namely, we aim at connecting a general class of nonlinear second‐order ordinary differential equations with a Noetherian conservation law. Our analysis will be considered within the known structure of Noether's theorem. An aspect to be here emphasized is that, particularly, we will use the following simplifying assumptions: the invariance condition holds absolutely, the transformation generators do not depend on the derivative of the generalized coordinate with respect to the independent variable. We will present our contribution via a theorem and a corollary. First, our theorem will establish a general condition to obtain Noetherian conservation law for a certain type of variational problem. Then, the corollary will demonstrate that such Noetherian conservation law is a first integral for a general class of nonlinear second‐order ordinary differential equations. This class of differential equations means the Lagrange's equation directly resulting from that type of variational problem. Last, a case of the Emden‐Fowler equation will be presented as a testing example.

Symmetry group and group representations associated with the thermodynamic covariance principle.

The main objective of this work [previously appeared in literature, the thermodynamical field theory (TFT)] is to determine the nonlinear closure equations (i.e., the flux-force relations) valid for thermodynamic systems out of Onsager's region. The TFT rests upon the concept of equivalence between thermodynamic systems. More precisely, the equivalent character of two alternative descriptions of a thermodynamic system is ensured if, and only if, the two sets of thermodynamic forces are linked with each other by the so-called thermodynamic coordinate transformations (TCT). In this work, we describe the Lie group and the group representations associated to the TCT. The TCT guarantee the validity of the so-called thermodynamic covariance principle (TCP): The nonlinear closure equations, i.e., the flux-force relations, everywhere and in particular outside the Onsager region, must be covariant under TCT. In other terms, the fundamental laws of thermodynamics should be manifestly covariant under transformations between the admissible thermodynamic forces, i.e., under TCT. The TCP ensures the validity of the fundamental theorems for systems far from equilibrium. The symmetry properties of a physical system are intimately related to the conservation laws characterizing that system. Noether's theorem gives a precise description of this relation. We derive the conserved (thermodynamic) currents and, as an example of calculation, a system out of equilibrium (tokamak plasmas) where the validity of TCP imposed at the level of the kinetic equations is also analyzed.

Noether symmetries of vacuum classes of pp-waves and the wave equation

The Noether symmetry algebras admitted by wave equations on plane-fronted gravitational waves with parallel rays are determined. We apply the classification of different metric functions to determine generators for the wave equation, and also adopt Noether's theorem to derive conserved forms. For the possible cases considered, there exist symmetry groups with dimensions two, three, five, six and eight. These symmetry groups contain the homothetic symmetries of the spacetime.