scholarly journals Variations by generalized symmetries of local Noether strong currents equivalent to global canonical Noether currents

2016 ◽  
Vol 24 (2) ◽  
pp. 125-135
Author(s):  
Marcella Palese
Keyword(s):  
Inverse Problem ◽  
Variational Problem ◽  
Cohomology Class ◽  
Lie Derivative ◽  
Local Problem ◽  
Variational Derivative ◽  
Field Equations ◽  
Invariance Properties ◽  
Generalized Symmetries ◽  
Variational Sequence

Abstract We will pose the inverse problem question within the Krupka variational sequence framework. In particular, the interplay of inverse problems with symmetry and invariance properties will be exploited considering that the cohomology class of the variational Lie derivative of an equivalence class of forms, closed in the variational sequence, is trivial. We will focalize on the case of symmetries of globally defined field equations which are only locally variational and prove that variations of local Noether strong currents are variationally equivalent to global canonical Noether currents. Variations, taken to be generalized symmetries and also belonging to the kernel of the second variational derivative of the local problem, generate canonical Noether currents - associated with variations of local Lagrangians - which in particular turn out to be conserved along any section. We also characterize the variation of the canonical Noether currents associated with a local variational problem.

Quadratic Lagrangians and gauge-invariance in covariant field theories

1960 ◽  
Vol 56 (3) ◽  
pp. 247-251 ◽  
Author(s):  
G. Stephenson
Keyword(s):  
General Relativity ◽  
Gauge Transformation ◽  
Gauge Invariance ◽  
Variational Principles ◽  
Scalar Function ◽  
Field Equations ◽  
General Group ◽  
Metric Tensor ◽  
Invariance Properties ◽  
Image Position

The idea of gauge-invariance in general relativity was first introduced by Weyl(1) who proposed that the field equations of gravitation should be invariant, not only under the general group of coordinate transformations, but also under the gauge-transformationwhere is the symmetric metric tensor, is the symmetric affine connexion and λ(x8) is an arbitrary scalar function of the coordinates. In this way it was possible to introduce into the theory a four-vector Ak which in consequence of (1·1) transformed assuch that the six-vector remained an invariant quantity under the gauge-transformation. It was Weyl's hope that by widening the invariance properties gauge-transformation. It was Weyl's hope that by widening the invariance properties of general relativity in this way the vector Ak and its associated six-vector Fik could be interpreted as representing the electromagnetic field. However, no obvious or unique way of doing this was found. More recently (see Stephenson (2,3) and Higgs (4)) gaugeinvariant variational principles formed from Lagrangians quadratic in the Riemann—Christoffel curvature tensor and its contractions have been discussed by performing the variations with respect to the symetric and symetric independently (following the palatini method).

Nonlinear integral equations for electromagnetic inverse problems

Geophysics ◽  
1987 ◽  
Vol 52 (9) ◽  
pp. 1297-1302 ◽  
Author(s):  
E. Gómez‐Treviño
Keyword(s):  
Electrical Conductivity ◽  
Inverse Problem ◽  
Integral Equations ◽  
Kernel Functions ◽  
Nonlinear Dependence ◽  
Field Equations ◽  
Nonlinear Integral Equations ◽  
Scaling Properties ◽  
Electromagnetic Problems ◽  
Special Cases

The scaling properties of Maxwell’s equations allow the existence of simple yet general nonlinear integral equations for electrical conductivity. These equations were developed in an attempt to reduce the generality of linearization to the exclusive scope of electromagnetic problems. The reduction is achieved when the principle of similitude for quasi‐static fields is imposed on linearized forms of the field equations. The combination leads to exact integral relations which represent a unifying framework for the general electromagnetic inverse problem. The equations are of the same form in both time and frequency domains and hold for all observations that scale as electric and magnetic fields do; direct current resistivity and magnetometric resistivity methods are considered as special cases. The kernel functions of the integral equations are closely related, through a normalization factor, to the Frechét kernels of the conventional equations obtained by linearization. Accordingly, the sensitivity functions play the role of weighting functions for electrical conductivity despite the nonlinear dependence of the model and the data. In terms of the integral equations, the inverse problem consists of extracting information about a distribution of conductivity from a given set of its spatial averages. The form of the new equations leads to the consideration of their numerical solution through an approximate knowledge of their kernel functions. The integral equation for magnetotelluric soundings illustrates this approach in a simple fashion.

scholarly journals FREE FIELD EQUATIONS FOR SPACE–TIME ALGEBRAS WITH TENSORIAL MOMENTUM

2002 ◽  
Vol 17 (21) ◽  
pp. 1393-1406 ◽  
Author(s):  
R. MANVELYAN ◽  
R. MKRTCHYAN
Keyword(s):  
Einstein Equation ◽  
Gauge Invariance ◽  
Free Field ◽  
Space Time ◽  
Field Equations ◽  
Invariance Properties ◽  
Diffeomorphism Invariance ◽  
Rank Tensor

Free field equations, with various spins, for space–time algebras with second-rank tensor (instead of the usual vector) momentum are constructed. Similar algebras are appearing in superstring/M theories. Special attention is paid to gauge invariance properties, in particular the spin-two equations with gauge invariance are constructed for dimensions 2+2 and 2+4, and the connection with Einstein equation and diffeomorphism invariance is established.

The variational formulation of an inverse problem for multidimensional nonlinear time-dependent Schrödinger equation

2016 ◽  
Vol 24 (5) ◽  
Author(s):  
Nigar Yıldırım Aksoy
Keyword(s):  
Inverse Problem ◽  
Schrödinger Equation ◽  
Variational Problem ◽  
Existence And Uniqueness ◽  
Variational Formulation ◽  
Schrodinger Equation ◽  
Time Dependent ◽  
Unknown Coefficient ◽  
Necessary Condition ◽  
Nonlinear Part

AbstractIn this paper, an inverse problem of determining the unknown coefficient of a multidimensional nonlinear time-dependent Schrödinger equation that has a complex number at nonlinear part is considered. The inverse problem is reformulated as a variational one which aims to minimize the observation functional. This paper presents existence and uniqueness theorems of solutions of the constituted variational problem, the gradient of the observation functional and a necessary condition for the solution of the variational problem.

scholarly journals Variational derivatives in locally Lagrangian field theories and Noether–Bessel-Hagen currents

2016 ◽  
Vol 13 (08) ◽  
pp. 1650067 ◽  
Author(s):  
Francesco Cattafi ◽  
Marcella Palese ◽  
Ekkehart Winterroth
Keyword(s):  
Conserved Quantity ◽  
Field Theories ◽  
Lie Derivative ◽  
Generalized Symmetry ◽  
Variational Sequence ◽  
Variational Derivatives ◽  
Noether Current

The variational Lie derivative of classes of forms in the Krupka’s variational sequence is defined as a variational Cartan formula at any degree, in particular for degrees lesser than the dimension of the basis manifold. As an example of application, we determine the condition for a Noether–Bessel-Hagen current, associated with a generalized symmetry, to be variationally equivalent to a Noether current for an invariant Lagrangian. We show that, if it exists, this Noether current is exact on-shell and generates a canonical conserved quantity.

scholarly journals Numerical solution for an inverse variational problem

2021 ◽  
Author(s):  
A. I. Garralda-Guillem ◽  
P. Montiel López
Keyword(s):  
Inverse Problem ◽  
Variational Problem ◽  
Numerical Solution ◽  
Numerical Method ◽  
Numerical Approximation ◽  
Existence Of A Solution ◽  
Numerical Example

AbstractIn the present work, firstly, we use a minimax equality to prove the existence of a solution to a certain system of varitional equations providing a numerical approximation of such a solution. Then, we propose a numerical method to solve a collage-type inverse problem associated with the corresponding system, and illustrate the behaviour of the method with a numerical example.

COHOMOLOGY OF LAGRANGE COMPLEXES INVARIANT UNDER PSEUDOGROUPS OF LOCAL TRANSFORMATIONS

2007 ◽  
Vol 04 (04) ◽  
pp. 669-705 ◽  
Author(s):  
ANDREA SPIRO
Keyword(s):  
Inverse Problem ◽  
Calculus Of Variations ◽  
Configuration Space ◽  
Cohomology Class ◽  
Cohomology Groups ◽  
Lagrange Equations ◽  
De Rham Cohomology ◽  
Base Manifold ◽  
De Rham

The inverse problem of the Calculus of Variations for Lagrangians and Euler–Lagrange equations invariant under a pseudogroup [Formula: see text] of local transformations of the base manifold is considered. Exploiting some ideas of Krupka, a theorem is proved showing that, if the configuration space consists of sections of tensor bundles or of local maps of a manifold into another, then such inverse problem is solvable whenever a certain cohomology class of [Formula: see text]-invariant forms on the configuration space is vanishing. In addition, for a few pseudogroups, the cohomology groups considered in the main result are explicitly determined in terms of the de Rham cohomology of the configuration space.

scholarly journals VARIATIONALLY EQUIVALENT PROBLEMS AND VARIATIONS OF NOETHER CURRENTS

2012 ◽  
Vol 10 (01) ◽  
pp. 1220024 ◽  
Author(s):  
MAURO FRANCAVIGLIA ◽  
MARCELLA PALESE ◽  
EKKEHART WINTERROTH
Keyword(s):  
Variational Problems ◽  
Local System ◽  
Variational Derivative ◽  
Invariance Properties ◽  
Symmetry Properties ◽  
Generalized Symmetry ◽  
Conserved Current ◽  
The First Variation ◽  
Equivalent Problems ◽  
A Current

We consider systems of local variational problems defining nonvanishing cohomology classes. Symmetry properties of the Euler–Lagrange expressions play a fundamental role since they introduce a cohomology class which adds up to Noether currents; they are related with invariance properties of the first variation, thus with the vanishing of a second variational derivative. In particular, we prove that the conserved current associated with a generalized symmetry, assumed to be also a symmetry of the variation of the corresponding local inverse problem, is variationally equivalent to the variation of the strong Noether current for the corresponding local system of Lagrangians. This current is conserved and a sufficient condition will be identified in order that such a current be global.

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