Noether equations and conservation laws

The purpose of the paper is to present a rigorous derivation of the relation between conservation laws and transformations leaving invariant the action integral. The (space-)time development of a physical system is represented by a cross section of a product bundleM. A Lagrange function is defined as a mapping where is the bundle space of the first jet extension ofM. A symmetry transformation is defined as a bundle automorphism ofM, carrying solutions of the Euler-Lagrange equation into solutions of the same equation. An important class of symmetry transformations is that of generalized invariant transformations: they are defined by specifying their action on the Euler-Lagrange equation. The generators of generalized invariant transformations are solutions of a system of linear, homogeneous partial differential equation (Noether equations). The set of all solutions of these equations has a natural structure of Lie algebra. In a simple manner, the Noether equations give rise to differential conservation laws.Supported by Air Force Office of Scientifie Research and Aeronautical Research Laboratories.On leave of absence from the Institute of Theoretical Physics, Warsaw University, Warsaw, Poland.     

Nonlocal symmetries and conservation laws of the Sinh-Gordon equation

Nonlocal symmetries of the (1+1)-dimensional Sinh-Gordon (ShG) equation are obtained by requiring it, together with its Bäcklund transformation (BT), to be form invariant under the infinitesimal transformation. Naturally, the spectrum parameter in the BT enters the nonlocal symmetries, and thus through the parameter expansion procedure, infinitely many nonlocal symmetries of the ShG equation can be generated accordingly. Making advantages of the consistent conditions introduced when solving the nonlocal symmetires, some new nonlocal conservation laws of the ShG equation related to the nonlocal symmetries are obtained straightforwardly. Finally, taking the nonlocal symmetries as symmetry constraint conditions imposing on the BT, some new finite and infinite dimensional nonlinear systems are constructed.     

Higher-order symmetries and conservation laws of multi-dimensional Gordon-type equations

In this paper a class of multi-dimensional Gordon-type equations are analysed using a multiplier and homotopy approach to construct conservation laws. The main focus is the analysis of the classical versions of the Gordon-type equations and obtaining higher-order variational symmetries and corresponding conserved quantities. The results are extended to the multi-dimensional Gordon-type equations with the two-dimensional Klein–Gordon equation in particular yielding interesting results.     

Dynamical symmetries and conservation laws for Korteweg-de Vries equation

The KdV-equation in two space time dimensions with the set of rapidly decreasing test functions as initial conditions is treated in the setting of nonlinear group and Lie algebra representations. The topological properties of the direct and inverse scattering mappings are discussed in detail.The algebra of continuous constants of motion turns out to be generated as in the linear case by three constants of motion and an extension of a representation of the e₂ Lie algebra on space-time symmetries to its enveloping algebra. The integrability of these representations is studied.It is further proved that the “moment problem” does not have a unique solution in this setting.The existence of noncommutative algebras of smooth time independent constants of motion is pointed out.     

Energy conservation laws and antimatter

In field theory, an energy-momentum tensor fails to be conserved if internal symmetries are broken. This revised version was published online in August 2006 with corrections to the Cover Date.     

The symmetries and conservation laws of some Gordon-type equations in Milne space-time

In this letter, the Lie point symmetries of a class of Gordon-type wave equations that arise in the Milne space-time are presented and analysed. Using the Lie point symmetries, it is showed how to reduce Gordon-type wave equations using the method of invariants, and to obtain exact solutions corresponding to some boundary values. The Noether point symmetries and conservation laws are obtained for the Klein–Gordon equation in one case. Finally, the existence of higherorder variational symmetries of a projection of the Klein–Gordon equation is investigated using the multiplier approach.     

On the correspondence between symmetries and conservation laws of evolution equations

There exists a well-defined correspondence between symmetries and conservation laws if an evolution equation admits a Hamiltonian formulation. We discuss whether the Hamiltonian structure is necessary for the correspondence.     

Integral versus local conservation laws in metric theories of gravity

Previously we considered extensions of metric theories of gravitation with non-zero divergence of the energy-momentum tensor. We confirm here that lack of local conservation laws does not exclude, a priori, existence of integral conservation laws.     

Explicit formulas for symmetries and conservation laws of the Kadomtsev-Petviashvili equation

Two nonlocal recursion operators are given, which yield explicit formulas for infinite hierarchies of symmetry generators and conservation laws for the two-dimensional Korteweg-de Vries equation. It is shown that the constants of the motion are in involution and that the symmetries commute.     

Projective symmetries and laws of conservation in the K-spaces determined by gravitational fields

The symmetries of equations of motion for probe bodies (projective symmetries) and the corresponding laws of conservation in the K-spaces determined by the gravitational fields of type (3) are studied. The results define all mechanical and field laws of conservation in the foregoing gravitational fields resulting from projective symmetries, in particular, from isometries and homotheties. The metric ansatzes found can be used for construction of new exact solutions to the Einstein equations and for examination of their large-scale (geodesic) structure. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 30–37, April, 2008.     

Infinitesimal symmetries and conservation laws of the DNLSE hierarchy and the Noether's theorem

The hierarchy of integrable nonlinear equations associated with the quadratic bundle is considered. The expressions for the solution of linearization of these equations and their conservation law in the terms of solutions of corresponding Lax pairs are found. It is shown for the first member of the hierarchy that the conservation law is connected with the solution of linearized equation due to the Noether's theorem. The local hierarchy and three nonlocal ones of the infinitesimal symmetries and conservation laws explicitly expressed through the variables of the nonlinear equations are derived.     

Hamiltonian dynamics and Noether symmetries in Extended Gravity Cosmology

We discuss the Hamiltonian dynamics for cosmologies coming from Extended Theories of Gravity. In particular, minisuperspace models are taken into account searching for Noether symmetries. The existence of conserved quantities gives selection rule to recover classical behavior in cosmic evolution according to the so-called Hartle criterion, which allows one to select correlated regions in the configuration space of dynamical variables. We show that such a statement works for general classes of Extended Theories of Gravity and is conformally preserved. Furthermore, the presence of Noether symmetries allows a straightforward classification of singularities that represent the points where the symmetry is broken. Examples for non-minimally coupled and higher-order models are discussed.     

Self-acceleration and matter content in bicosmology from Noether symmetries

In bigravity, when taking into account the potential existence of matter fields minimally coupled to the second gravitation sector, the dynamics of our Universe depends on some matter that cannot be observed in a direct way. In this paper, we assume the existence of a Noether symmetry in bigravity cosmologies in order to constrain the dynamics of that matter. By imposing this assumption we obtain cosmological models with interesting phenomenology. In fact, considering that our universe is filled with standard matter and radiation, we show that the existence of a Noether symmetry implies that either the dynamics of the second sector decouples, being the model equivalent to general relativity (GR), or the cosmological evolution of our universe tends to a de Sitter state with the vacuum energy in it given by the conserved quantity associated with the symmetry. The physical consequences of the genuine bigravity models obtained are briefly discussed. We also point out that the first model, which is equivalent to GR, may be favored due to the potential appearance of instabilities in the second model.     

The geometric nature of Lie and Noether symmetries

It is shown that the Lie and the Noether symmetries of the equations of motion of a dynamical system whose equations of motion in a Riemannian space are of the form [(x)\ddot]ⁱ+G_jkⁱ[(x)\dot]^j[(x)\dot] ^k+f(xⁱ)=0{\ddot{x}^{i}+\Gamma_{jk}^{i}\dot{x}^{j}\dot{x} ^{k}+f(x^{i})=0} where f(x ⁱ ) is an arbitrary function of its argument, are generated from the Lie algebra of special projective collineations and the homothetic algebra of the space respectively. Therefore the computation of Lie and Noether symmetries of a given dynamical system in these cases is reduced to the problem of computation of the special projective algebra of the space. It is noted that the Lie and Noether symmetry vectors are common to all dynamical systems moving in the same background space. The selection of the vectors which are Lie/Noether symmetries for a given dynamical system is done by means of a set of differential conditions involving the vectors and the potential function defining the dynamical system. The general results are applied to a number of different applications concerning (a) The motion in Euclidean space under the action of a general central potential (b) The motion in a space of constant curvature (c) The determination of the Lie and the Noether symmetries of class A Bianchi type hypersurface orthogonal spacetimes filled with a scalar field minimally coupled to gravity (d) The analytic computation of the Bianchi I metric when the scalar field has an exponential potential.     

Lie and Noether symmetries of geodesic equations and collineations

The Lie symmetries of the geodesic equations in a Riemannian space are computed in terms of the special projective group and its degenerates (affine vectors, homothetic vector and Killing vectors) of the metric. The Noether symmetries of the same equations are given in terms of the homothetic and the Killing vectors of the metric. It is shown that the geodesic equations in a Riemannian space admit three linear first integrals and two quadratic first integrals. We apply the results in the case of Einstein spaces, the Schwarzschild spacetime and the Friedman Robertson Walker spacetime. In each case the Lie and the Noether symmetries are computed explicitly together with the corresponding linear and quadratic first integrals.     

The variational symmetries and conservation laws in classical theory of Heisenberg (anti)ferromagnet

The non-linear partial differential equations describing the spin dynamics of Heisenberg ferro- and antiferromagnet are studied by Lie transformation group method. The generators of the admitted variational Lie symmetry groups are derived and conservation laws for the conserved currents are found via Noether's theorem.     

Infinitely many nonlocal symmetries and nonlocal conservation laws of the integrable modified KdV-sine-Gordon equation

Nonlocal symmetries related to the Bäcklund transformation (BT) for the modified KdV-sine-Gordon (mKdV-SG) equation are obtained by requiring the mKdV-SG equation and its BT form invariant under the infinitesimal transformations. Then through the parameter expansion procedure, an infinite number of new nonlocal symmetries and new nonlocal conservation laws related to the nonlocal symmetries are derived. Finally, several new finite and infinite dimensional nonlinear systems are presented by applying the nonlocal symmetries as symmetry constraint conditions on the BT.     

A note on Noether symmetries and conformal Killing vectors

A conjecture was stated in Hussain et al. (Gen Relativ Grav 41:2399, 2009), that the conformal Killing vectors form a subalgebra of the symmetries of the Lagrangian that minimizes arc length, for any spacetime. Here, a counter example is constructed to demonstrate that the above statement is not true in general for spacetimes of non-zero curvature.     

Structure properties and Noether symmetries for super-long elastic slender rod

DNA is a nucleic acid molecule with double-helical structures that are special symmetrical structures attracting great attention of numerous researchers. The super-long elastic slender rod, an important structural model of DNA and other long-train molecules, is a useful tool in analysing the symmetrical properties and the stabilities of DNA. This paper studies the structural properties of a super-long elastic slender rod as a structural model of DNA by using Kirchhoff＇s analogue technique and presents the Noether symmetries of the model by using the method of infinitesimal transformation. Baaed on Kirchhoff＇s analogue it analyses the generalized Hamilton canonical equations. The infinitesimal transfornaationa with rcspect to the radial coordinnte, the gonarnlizod coordinates, and the Cluasi-momenta of 5he model are introduced. The Noether gymmetries and conserved qugntities of the model are obtained.