Showing papers on "Noether's theorem published in 1991"
TL;DR: In this article, the geometrical foundations of first order Lagrangian and Hamiltonian field theories are clarified by introducing in a systematic way multisymplectic manifolds, the field theoretical analogues of the symplectic structures used in Geometrical mechanics.
Abstract: The general purpose of this paper is to attempt to clarify the geometrical foundations of first order Lagrangian and Hamiltonian field theories by introducing in a systematic way multisymplectic manifolds, the field theoretical analogues of the symplectic structures used in geometrical mechanics. Much of the confusion surrounding such terms as gauge transformation and symmetry transformation as they are used in the context of Lagrangian theory is thereby eliminated, as we show. We discuss Noether's theorem for general symmetries of Lagrangian and Hamiltonian field theories. The cohomology associated to a group of symmetries of Hamiltonian or Lagrangian field theories is constructed and its relation with the structure of the current algebra is made apparent.
250 citations
TL;DR: In this article, a self-dual three-brane soliton solution of D=10 type IIB supergravity was shown to break half the supersymmetries, and saturate a Bogomol'nyi bound between the mass per unit volume and a conserved Noether charge.
207 citations
TL;DR: In this article, the authors consider formal groups of transformations on the space of differential and net variables and show that preservation of meaning of difference derivatives under transformations necessarily leads to Lie-Backlund group.
Abstract: We consider formal groups of transformations on the space of differential and net (finite-difference) variables. We show that preservation of meaning of difference derivatives under transformations necessarily leads to Lie-Backlund group. We derive formulas for extension to net variables and formulate criteria for preservation of uniformity and invariance of differences of the network and a test for the invariance of difference equations. With the help of formal Newton series we construct the ideal of the algebra of all Lie-Backlund operators on a uniform network which is used to derive tests for the conservatism of difference equations on the basis of a discrete analog of Noether's identity.
117 citations
TL;DR: In this paper, a new category of relativistic thermodynamic models is developed for the systematic representation of viscous conducting fluid media (allowing for several independent charged or neutral chemical constituents) using guidelines provided by Noether identities arising from a generalized variation procedure of convective type, and the specification of a particular model is determined just by giving the algebraic dependence a single "master function" on the relevant dynamical variables, which are supposed here to consist of an entropy current 4-vector and a set of particle current 4vectors corresponding to the various chemical constituents, together
Abstract: Using guidelines provided by Noether identities arising from a generalized variation procedure of convective type, a new (nonlinear and exactly self-consistent) category of relativistic thermodynamic models is developed for the systematic representation of viscous conducting fluid media (allowing for several independent charged or neutral chemical constituents). Apart from the provision of a set of dissipation coefficients of the usual (reactivity, resistivity, and viscosity) type, the specification of a particular model is determined just by giving the algebraic dependence a single ‘master function’, Λ say, on the relevant dynamical variables, which are supposed here to consist of an entropy current 4-vector and a set of particle current 4-vectors corresponding to the various chemical constituents, together with a set of symmetric (rank 3) viscosity tensors, which are considered as being dynamically independent of the corresponding current vectors except in the degenerate limit of linear viscosity. The master function is set up as a generalization of an ordinary lagrangian function, to which it reduces in the relevant non - dissipative limit, and, as in the conservative case, it is used for the construction of derived quantities in such a way that appropriate self-consistency conditions are satisfied as identities. In particular the relevant stress-momentum-energy tensor is obtained directly in terms of the independent variables and of their dynamical conjugates (whose role is hidden in the traditional approach as developed by Israel & Stewart), which are set of ordinary 4-momentum (not 4-momentum density) covectors associated with the independent currents, and a set of generalised Cauchy type strain (not strain - rate) tensors associated with the independent viscous stress contributions. The range of application of the category obtained in this way is intended to include that of the standard (Israel-Stewart) formalism to which it is expected to be effectively equivalent in the limit of sufficiently small deviations from thermodynamic equilibrium.
92 citations
TL;DR: The nonlinear O(3) σ-model in (2 + 1) dimensions, modified by the addition of a potential term, admits solutions of Q-ball type as discussed by the authors.
80 citations
59 citations
TL;DR: In this paper, it was shown that when p = 2, the HILBERT transform T is not a NOETHER operator when p < 2 or 2 < p < ∞.
Abstract: It 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.
50 citations
TL;DR: In this article, a general framework was described in which subgroups of the loop group AGIn(: act on the space of harmonic maps from S2 to Gln(:. This represents a simplification of the action considered by Zakharov-Mikhailov-Shabat [ZM, ZS] in that we take the contour for the Riemann-Hilbert problem to be a union of circles; however, it reduces the basic ingredient to the well-known Birkhoff decomposition of AGInS, and this facilitates a rigorous treatment
Abstract: We describe a general framework in which subgroups of the loop group AGIn(: act on the space of harmonic maps from S2 to Gln(: . This represents a simplification of the action considered by Zakharov-Mikhailov-Shabat [ZM, ZS] in that we take the contour for the Riemann-Hilbert problem to be a union of circles; however, it reduces the basic ingredient to the well-known Birkhoff decomposition of AGIn(S, and this facilitates a rigorous treatment. We give various concrete examples of the action, and use these to investigate a suggestion of Uhlenbeck [Uh] that a limiting version of such an action ("completion") gives rise to her fundamental process of "adding a uniton". It turns out that this does not occur, because completion preserves the energy of harmonic maps. However, in the special case of harmonic maps from S2 to complex projective space, we describe a modification of this completion procedure which does indeed reproduce "adding a uniton". One aspect of the theory of symmetry groups for diffierential equations is the idea of "proliferation of solutions". By this is meant that, starting with an obvious solution, application of symsmetries can lead to further (perhaps less obvious) solutions. In this paper we shall apply the general framework described by Zakharov et al. [ZM, ZS] to the harmonic map equation (the principal chiral or sigma model of mathematical physics). To put this into a wider context, we recall that a fundamental discovery in the study of the Korteweg-de Vries and related equations was that the space of solutions admits a certain infinite-dimensional symmetry group. Historically, the first manifestation of this (via Noether's theorem) was the appearance of infinitely many conservation laws. Later this led to an elegant algebraic description of solutions of the equation as the points of an orbit of a representation of an infinite-dimensional group. (See, for example, [Wi] for a brief description of some of these ideas.) The existence of previously unnoticed symmetries ("hidden symmetries") for other equations became an interesting possibility, and attempts have been made (so far with much less success than in the case of the KdV equation) to unearth them. One such example is the harmonic map equation for maps from a surface D into a compact Lie group G (or homogeneous space G/H ). The basic observation [Po, Uh, ZM, ZS] which introduces an infinite-dimensional group is that harmonic maps from D to G correspond to certain Received by the editors February 9, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 58E20. (r)1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page
49 citations
TL;DR: In this article, a new methodology is proposed with the aim of establishing conservation laws for dissipative systems described by partial differential equations, which is applicable only to systems governed by equations derivable variationally from a Lagrangian.
40 citations
TL;DR: In this paper, the generalized first noether theorem (GFNT) for a constrained Hamiltonian system and the generalized Noether identities (GNI) for systems with a non-invariant action integral are derived, which may be useful to analyse the Dirac constraint for such a system.
Abstract: For a canonical formalism with a higher-order derivative, the corresponding generalized first Noether theorem (GFNT) for a constrained Hamiltonian system and the generalized Noether identities (GNI) for a system with a non-invariant action integral are derived, which may be useful to analyse the Dirac constraint for such a system. Using the GFNT another example is given in which Dirac's conjecture fails; using the GNI the strong and weak conservation laws are deduced and it is pointed out that for certain variant systems there is also a Dirac constraint. Suppose that there are only first-class constraints (FCC) in a system, then an algorithm for the construction of a gauge generator is developed, once the Hamiltonian and the FCC of the system with a higher-order Lagrangian are given.
38 citations
TL;DR: In this article, the authors generalize M. Green's Explicit Noether-Lefschetz Theorem to the family of smooth complete intersection surfaces in the higher dimensional projective spaces.
Abstract: We generalize M. Green's Explicit Noether-Lefschetz Theorem to the family of smooth complete intersection surfaces in the higher dimensional projective spaces. Moreover, we give a new proof of the Density Theorem due to C. Ciliberto, J. Harris, and R. Miranda [5].
TL;DR: In this paper, the flux integral for axisymmetric polar perturbations of static vacuum space-times is derived with the aid of the Einstein pseudo-tensor by a simple algorism.
Abstract: 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.
TL;DR: In this article, the authors analyse Noether's Second Theorem from a geometric viewpoint using the concepts of vector fields and forms along tangent bundle projections, and show that the second theorem can be analyzed from a geometrical viewpoint.
Abstract: We analyse Noether's Second Theorem from a geometric viewpoint using the concepts of vector fields and forms along tangent bundle projections.
TL;DR: The concept of a section along a map is a fundamental concept within the framework of the geometrical description of classical mechanics as discussed by the authors, and it has been shown that the concept can be used to define basic objects in higher-order Lagrangian mechanics.
Abstract: We show that the concept of a section along a map is a fundamental concept within the framework of the geometrical description of classical mechanics We review the higher-order Lagrangian mechanics formulation, and simpler redefinitions of basic objects appear in a natural way As an application, Noether's theorem for higher-order Lagrangian mechanics admitting a converse is developed
TL;DR: In this article, the generalized quasi-symmetry of the infinitesimal transformation of the group of transformations G t for the given dynamical systems is introduced, and Noether's theorem and its inverse of nonconservative systems subject to first-ordernonlinear nonholonomic constraints are obtained.
Abstract: Noether's theorem and its inverse of nonconservative systems subject to first-ordernonlinear nonholonomic constraints are obtained by introducing the generalized quasi-symmetry of the infinitesimal transformation of the group of transformations G t for thegiven dynamical systems.
TL;DR: In this article, the generalized first and second Noether theorems (Noether identities) were generalized to a constrained hypersurface in phase space, and a preliminary application of the generalized Noether identities (GNI) to nonrelativistic charged particles in an electromagnetic field was shown to obtain electric charge conservation.
Abstract: We generalize the first and second Noether theorems (Noether identities) to a constrained system in phase space. As an example, the conservation law deriving from Lagrange's formalism cannot be obtained fromHE via the generalized first Noether theorem (GFNT); Dirac's conjecture regarding secondary first-class constraints (SFCC) is invalid in this example. A preliminary application of the generalized Noether identities (GNI) to nonrelativistic charged particles in an electromagnetic field shows that on the constrained hypersurface in phase space one obtains electric charge conservation. This conservation law is valid whether Dirac's conjecture holds true or not.
TL;DR: In this paper, the variational principles of Jourdain's form of nonlinear nonholonomic nonpotential system in noninertial reference frame are established, the generalized Noether's theorem of the system above is presented and proved, and the conserved quantities of system are studied.
Abstract: The new Lagrangian of the relative motion of mechanical system is constructed, the variational principles of Jourdain's form of nonlinear nonholonomic nonpotential system in noninertial reference frame are established, the generalized Noether's theorem of the system above is presented and proved, and the conserved quantities of system are studied.
TL;DR: In this article, the Noether operator's defining equation leads, in the case of perturbations about a stationary solution, to a conserved energy current depending quadratically on first-order perturbation alone.
Abstract: 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.
TL;DR: In this article, the invariance identities of Rund [14] involving the Lagrangian and the generators of the infinitesimal Lie group are used for writing down the first integrals of the generalized Emden-Fowler equation via the Noether theorem.
Abstract: After formulating the alternate potential principle for the nonlinear differential equation corresponding to the generalised Emden-Fowler equation, the invariance identities of Rund [14] involving the Lagrangian and the generators of the infinitesimal Lie group are used for writing down the first integrals of the said equation via the Noether theorem. Further, for physical realisable forms of the parameters involved and through repeated application of invariance under the transformation obtained, a number of exact solutions are arrived at both for the Emden-Fowler equation and classical Emden equations. A comparative study with Bluman-Cole and scale-invariant techniques reveals quite a number of remarkable features of the techniques used here.
TL;DR: In this paper, the authors consider quantum effective actions for arbitrary models possessing an infinite-dimensional group G of Noether symmetries and derive functional differential equations for the effective action whose exact solution is found to be given by the geometric action on a coadjoint orbit of the (central extended) Noether group G. As a particular application, they show that the light-cone quantized toroidal membrane is explicitly given by geometric co-orbit action of the group of area-preserving diffeomorphisms on torus.
TL;DR: For a system of two coupled nonlinear Schrodinger equations and the corresponding Madelung fluid equations, the similarity transformation and reductions to ordinary differential equations were calculated in this article.
Abstract: For a system of two coupled nonlinear Schrodinger equations and the corresponding Madelung fluid equations we calculate the similarity transformation and reductions to ordinary differential equations. The symmetries can be used to classify several types of solutions. Besides a system of coupled Painleve II equations we obtain a two dimensional quartic hamiltonian system. Using Noether's theorem and performing a Painleve-test we identify those parameters which lead to regular motion on tori. Using the results of the P-test we obtain heretofore undiscovered integrals of motion for the Madelung fluid by direct calculations. Investigating the dynamics of the system for parameters that are different from those obtained by the Painleve-test we calculate numerically the surface of section (KAM-tori) and study the transition into chaotic behaviour.
TL;DR: In this article, the authors apply the Noether theorem with generahsed symmetries for discussing the integrability of the Swnglng Atwood Machine (SAM) model.
Abstract: In this work we apply the Noether theorem with generahsed symmetries for discussing the integrability of the Swnglng Atwood Machine (SAM) model. We analyse also the limitations of this procedure and compare it wth the Yoshida method. In the last years several analytical procedures have been applied in tile study of tile integrability of dy~lamical systems: the dkect method for the identification of invariants, the Painlev6 test, Melnikov's method, Ziglin's theorem, the analysis of the symmetries of the system, etc. In a recent paper published in this journal, lbfillaro et aL ill wed some results obtained by Yoshida [2] for discl~ssing the domain of integrability for a mo-dimensional Hamiltonian system: the so-called Swinging Atwood Machine (SAM). By the Liouville theorem, a two-dimensional Hamiltonian system will be integrable if exists a second conserved quantity (invariant) in addition to the energy. In this letter we apply a method based on the generalised Noetherian symmetries [3] for finding the integrable case and its assochted symmetries, for the SAM problem. We compare the two methods, Yoshida's theorem and the symmetry analysis, and discus their joint utilisation.
TL;DR: In this article, the Lagrangian formalism for time-dependent systems is developed using vector fields defined in the extended tangent bundle T(Q×R) and the Noether theorem for the extended Lagrangians L is considered.
Abstract: The Lagrangian formalism for time‐dependent systems is developed using vector fields defined in the extended tangent bundle T(Q×R) The definition of the time‐independent extended Lagrangian function L associated to a time‐dependent Lagrangian L is given and then the techniques of symplectic mechanics are used to prove some properties of the equations determining the Euler–Lagrange vector field The relations between the time‐independent and the time‐dependent Helmholtz conditions are analyzed from a geometric perspective and, finally, the Noether theorem for the extended Lagrangian L is considered
01 Jan 1991
TL;DR: In this article, Liu Duan and Zhang Jie-fang respectively studied the motion of constant mass mechanical system in inertial reference frame, generalized the wall-known Noether's
Abstract: In recent years, Liu Duan and Zhang Jie-fang respectively studied the motion of constant mass mechanical system in inertial reference frame, generalized the wall-known Noether’s
TL;DR: In this paper, the authors investigated the structure of the time-dependent Rayleigh-Ritz equations with potential simulations of heavy-ion collisions in mind and showed that the equations have a canonical (hamiltonian) structure, and the relation between Noether's theorem and Ehrenfest's theorem is discussed.
TL;DR: In this paper, the authors show that a naive gauging of such symmetries by coupling to gravity or W-gravity does not work if there are central charges, and two successful ways of gauging are presented.
TL;DR: In this paper, the connection between the conservation laws of differential-difference schemes in the mechanics of continuous media, in Lagrangian variables, on the one hand, and transformation groups on the other, is presented.
Abstract: A study is presented of the connection between the conservation laws of differential-difference schemes (differential with respect to time, difference with respect to space) in the mechanics of continuous media, in Lagrangian variables, on the one hand, and transformation groups, on the other. Noether's Theorem is generalized to a class of differential-difference schemes that possess an equivalent variational formulation. It is shown that a necessary and sufficient condition for this class to have a conservation law is invariance of the extremal values of a spatially discrete variational functional. Schemes for gas dynamics and incompressible liquids are considered as examples. Examples of computations are given.
TL;DR: In this paper, a Noethericity criterion and an index formula for singular integral operators with a Carleman shift and continuous coefficients in the spaces Lp (i < p < ~) were obtained.
Abstract: In [i] we have obtained a Noethericity criterion and an index formula for singular integral operators with a Carleman shift and with continuous coefficients in the spaces Lp (i < p < ~). In a series of publications of other authors (see, e.g., [2]), these results have been extended to the case of coefficients having discontinuities of the first kind. The purpose of this note is the construction of the Noether theory of singular integral operators with a Carleman shift and with coefficients having discontinuities of the second kind.
TL;DR: In this article, two methods for the construction of alternative Lagrangians to the standard Lagrangian L=T−V where V is a central potential are presented: the direct integration of the Hessian matrix and the use of the Noether symmetries of L.
Abstract: Two methods for the construction of alternative Lagrangians to the standard Lagrangian L=T−V where V is a central potential are presented: the direct integration of the Hessian matrix and the use of the Noether symmetries of L. The Lagrangians obtained are of the so‐called nonlinear homogenous (NLH) type and the study is centered in those Lagrangians not preserving the spherical symmetry of L. Finally, the associated Hamiltonian formalism is presented.