Variationally Equivalent Problems and Variations of Noether Currents

Abstract

We consider systems of local variational problems defining non vanishing cohomolgy classes. 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 such a current be global.2000 MSC: 55N30, 55R10, 58A12, 58A20, 58E30, 70S10.

Variational Sequences, Representation Sequences and Applications in Physics

“…Furthermore, there always exists a closed form α ∈ Λ r n , e.g., a Lepage equivalent, which represents the cohomology class [[Ξ V ε]] dR and which projects onto Ξ V ε ∈ Λ r n /Θ r n . Thus, if σ is a critical section, J r σ * (α) = 0 and the corresponding class vanishes in H n dR (X); therefore, by our assumption on the nth cohomology groups [[Ξ V ε]] dR = 0 and there exist global Noether-Bessel-Hagen conserved currents for all generalized symmetries Ξ [28].…”

“…The variational Lie derivative can be seen as a local differential operator which acts on cohomology classes trivializing them [84]. This fact has important consequences for symmetries and conservation laws associated with local variational problems generating global Euler-Lagrange expressions; see, e.g., [7,25,28].…”

“…We call the (local) current ι − β ι a Noether-Bessel-Hagen current. Thus Noether-Bessel-Hagen currents ι − β ι are local currents associated with a generalized symmetry (conserved along critical sections); in [28] it was proved that a Noether-Bessel-Hagen current is variationally equivalent to a global current if and only if 0 = δ n−1 (…”

Topological obstructions in Lagrangian field theories, with an application to 3D Chern–Simons gauge theory

“…It is now of a certain importance (for the study of cohomological obstructions to the existence of global solutions) that, for local presentations of a variational problem, the corresponding conserved currents are, in principle, local currents, see e.g. [28,29,30,37]. However, symmetries of such local Lagrangians are also symmetries of the associated global Euler-Lagrange equations.…”

Cohomological obstructions in locally variational field theories

“…This suggest us that, if we know that the first variational derivative of a local presentation is closed, then the second variational derivative define a trivial cohomology class. In fact, the first variational derivative -with respect to symmetries of Euler-Lagrange expressions -of a local presentation is closed; therefore a trivial cohomology class is defined by the variational derivative of currents [9,10].…”