Noether Operators Research Articles

Noether, partial Noether operators and first integrals for a linear system

We obtain Noether and partial Noether operators corresponding to a Lagrangian and a partial Lagrangian for a system of two linear second-order ordinary differential equations (ODEs) with variable coefficients. The canonical form for a system of two second-order ordinary differential equations is invoked and a special case of this system is studied for both Noether and partial Noether operators. Then the first integrals with respect to Noether and partial Noether operators are obtained for the linear system under consideration. We show that the first integrals for both the Noether and partial Noether operators are the same. This can give rise to further studies on systems from a partial Lagrangian viewpoint as systems in general do not admit Lagrangians.

Partial Noether operators and first integrals via partial Lagrangians

AbstractThe notions of partial Lagrangians, partial Noether operators and partial Euler–Lagrange equations are used in the construction of first integrals for ordinary differential equations that need not be derivable from variational principles. We obtain a Noether‐like theorem that provides the first integral by means of a formula which has the same structure as the Noether integral. However, the invariance condition for the determination of the partial Noether operators is different as we have a partial Lagrangian and as a result partial Euler–Lagrange equations. Applications given include those that admit a standard Lagrangian such as the harmonic oscillator, modified Emden and Ermakov–Pinney equations and systems of two second‐order equations that do not have standard Lagrangians. Copyright © 2007 John Wiley & Sons, Ltd.

Regular elliptic boundary-value problem for a homogeneous equation in a two-sided improved scale of spaces

We study a regular elliptic boundary-value problem for a homogeneous differential equation in a bounded domain. We prove that the operator of this problem is a Fredholm (Noether) operator in a two-sided improved scale of functional Hilbert spaces. The elements of this scale are Hormander-Volevich-Paneyakh isotropic spaces. We establish an a priori estimate for a solution and investigate its regularity.

Fundamental Operator-Functions of Degenerate Differential and Difference-Differential Operators in Banach Spaces Which Have a Noether Operator in the Principal Part

Continuing the study by the first author, we construct fundamental operator-functions for some classes of degenerate differential operators (both ordinary and partial) and difference-differential operators in Banach spaces which have a Noether operator at the leading (time) derivative. To this end, we use the method of Jordan chains of Noether operators and the methods of distribution theory in Banach spaces.

Operators on generalized power series

Given a ring $C$ and a totally (resp. partially) ordered set of monomials $\mathfrak{M}$, Hahn (resp. Higman) defined the set of power series $C [[ \mathfrak{M} ]]$ with well-ordered (resp. Noetherian or well-quasi-ordered) support in $\mathfrak{M}$. This set $C [[ \mathfrak{M} ]]$ can usually be given a lot of additional structure: if $C$ is a field and $\mathfrak{M}$ a totally ordered group, then Hahn proved that $C [[ \mathfrak{M} ]]$ is a field. More recently, we have constructed fields of ``transseries'' of the form $C [[ \mathfrak{M} ]]$ on which we defined natural derivations and compositions. In this paper we develop an operator theory for generalized power series of the above form. We first study linear and multilinear operators. We next isolate a big class of so-called Noetherian operators $\Phi : C [[ \mathfrak{M} ]] \rightarrow C [[ \mathfrak{N}]]$, which include (when defined) summation, multiplication, differentiation, composition, etc. Our main result is the proof of an implicit function theorem for Noetherian operators. This theorem may be used to explicitly solve very general types of functional equations in generalized power series.

The Construction of Noetherian Operators

We give an elementary and constructive, purely algebraic proof for the existence of noetherian differential operators for primary submodules of finite-dimensional free modules over polynomial algebras. By means of these operators the submodule can be described by differential conditions on the associated characteristic variety. This important result and the terminology are due to V. P. Palamodov. However, his, L. Ehrenpreis' and later J.-E. Björk's proofs of the existence theorem use complicated algebraic and analytic techniques and are not constructive as far as we see. The idea to characterize primary ideals by their associated differential operators is due to W. Gröbner. But M. Noether's Fundamentalsatz is based on similar ideas and is obviously the origin of Palamodov's terminology.

Nonlinear boundary-value problems for systems of ordinary differential equations

We consider nonlinear boundary-value problems (with Noetherian operator in the linear part) for systems of ordinary differential equations in the neighborhood of generating solutions. By using the Lyapunov — Schmidt method, we establish conditions for the existence of solutions of these boundary-value problems and propose iteration algorithms for their construction.

Lie–Bäcklund and Noether Symmetries with Applications

New identities relating the Euler–Lagrange, Lie–Backlund and Noether operators are obtained. Some important results are shown to be consequences of these fundamental identities. Furthermore, we generalise an interesting example presented by Noether in her celebrated paper and prove that any Noether symmetry is equivalent to a strict Noether symmetry, i.e. a Noether symmetry with zero divergence. We then use the symmetry based results deduced from the new identities to construct Lagrangians for partial differential equations. In particular, we show how the knowledge of a symmetry and its corresponding conservation law of a given partial differential equation can be utilised to construct a Lagrangian for the equation. Several examples are given.

Boundary-value problems for systems of difference equations

Boundary-value problems for systems of difference equations with discrete argument whose linear part is the Noetherian operator are considered. The necessary and sufficient conditions of the solvability of difference boundary-value problems of this sort are obtained.

Energy extremality in the presence of a black hole

We derive the so-called first law of black hole mechanics for variations about stationary black hole solutions to the Einstein - Maxwell equations in the absence of sources. That is, we prove that where the black hole parameters M, , A, , J, V and Q denote mass, surface gravity, horizon area, angular velocity of the horizon, angular momentum, electric potential of the horizon and charge respectively. The unvaried fields are those of a stationary, charged, rotating black hole and the variation is to an arbitrary `nearby' black hole which is not necessarily stationary. Our approach is four-dimensional in spirit and uses techniques involving action variations and Noether operators. We show that the above formula holds on any asymptotically flat spatial 3-slice which extends from an arbitrary cross section of the (future) horizon to spatial infinity. (Thus, the existence of a bifurcation surface is irrelevant to our demonstration. On the other hand, the derivation assumes without proof that the horizon possesses at least one of the following two (related) properties: (i) it cannot be destroyed by arbitrarily small perturbations of the metric and other fields which may be present, (ii) the expansion of the null geodesic generators of the perturbed horizon goes to zero in the distant future.)

Analysis of complete integrability of a system inverse to the nonlinear Benney-Kaup system

We study a system inverse to the nonlinear dynamical Benney-Kaup system. For this system, we prove the existence of an infinite hierarchy of functionally independent and involution conservation laws; a pair of implicative and Noetherian operators is constructed which enables one to write the system in a bi-Hamiltonian form; we also present the explicit form of the Lax operator.

Hamiltonian and recursion operators for two-dimensional scalar fields

The equation for the Noether operator is obtained. It gives the necessary conditions for complete integrability of the field equations. For several double-component models the Hamiltonian pairs and the recursion operators are presented.

The inversion of fractional integrals on a sphere

The purpose of the paper is to invert Riesz potentials and some other fractional integrals on then-dimensional spherical surface in ℝn+1 in the closed form. New descriptions of spaces of the fractional smoothness on a sphere are obtained in terms of spherical hypersingular integrals. It is shown that Riesz potentials of the ordersn,n + 2,n + 4, ... on a sphere are Noether operators and theird-characteristic depends on the radius of the sphere.

The Einstein pseudo-tensor and the flux integral for perturbed static space-times

The flux integral for axisymmetric polar perturbations of static vacuum space-times, derived in an earlier paper directly from the relevant linearized Einstein equations, is rederived with the aid of the Einstein pseudo-tensor by a simple algorism. A similar earlier effort with the aid of the Landau–Lifshitz pseudo-tensor failed. The success with the Einstein pseudo-tensor is due to its special distinguishing feature that its second variation retains its divergence-free property provided only the equations governing the static space-time and its linear perturbations are satisfied. When one seeks the corresponding flux integral for Einstein‒Maxwell space-times, the common procedure of including, together with the pseudo-tensor, the energy‒momentum tensor of the prevailing electromagnetic field fails. But, a prescription due to R. Sorkin, of including instead a suitably defined ‘Noether operator’, succeeds.

The gravitational-electromagnetic Noether operator and the second-order energy flux

The Noether operator for gravity is recalled and that for the electromagnetic field derived, its difference from the electromagnetic stress tensor being pointed out. It is then shown how the Noether operator’s defining equation leads, in the case of perturbations about a stationary solution, to a conserved energy current depending quadratically on the first-order perturbations alone. The formal background of the paper by Chandrasekhar & Ferrari is thereby clarified.

The Finite Hilbert Transform in ℒ2

AbstractIt is well known that the finite HILBERT transform T is a NOETHER (FREDHOLM) operator when considered as a map from ℒp into itself if 1 < p < 2 or 2 < p < ∞. When p = 2, the map T is not a NOETHER operator. We present two theorems which characterize the range of T in ℒ2 and, as immediate consequences, give simple expressions for its inverse.

On the index and the homotopic classification of Noetherian operators with a finite group of shifts

On the index and the homotopic classification of Noetherian operators with a finite group of shifts

A bilocal periodic problem for the Sturm-Liouville and Dirac operators and some applications to the theory of nonlinear dynamical systems. I

Isospectral problems for operator-valued Sturm-Liouville and Dirac differential expressions are considered. Within the framework of the gradient method, one establishes the complete integrability of the Lax associated nonlinear Hamiltonian systems with a bilocal implectic pair of Noetherian operators on a manifold of integral operators.

A new derivation of the quadrupole formula for angular momentum

A new derivation of the farfield quadrupole formula for radiated angular momentum is presented, based on the gravitational Noether operator. Those metrics for which the angular momentum flux is well defined are characterized, and it is shown that this is free of the supertranslation ambiguity in the Newtonian approximation.

Pseudotensors in asymptotically curvilinear coordinates

We show how to calculate pseudotensor-based conserved quantities for isolated systems in general relativity, in a way which allows an arbitrary asymptotic behavior of the coordinate system used. Our method is a generalization of that given by Persides [1], and allows the asymptotic evaluation of energy, momentum, and angular momentum in any coordinate system. We carry out the calculation for the Schutz-Sorkin gravitational Noether operator, which is a pseudotensorial operator on vector fields that reduces to the familiar pseudotensors for particular choices of the fields.