Local variational problems and conservation laws

Abstract

We investigate globality properties of conserved currents associated with local variational problems admitting global Euler-Lagrange morphisms. We show that the obstruction to the existence of a global conserved current is the difference of two conceptually independent cohomology classes: one coming from using the symmetries of the EulerLagrange morphism and the other from the system of local Noether currents.

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.…”

Variationally Equivalent Problems and Variations of Noether Currents

“…It is a well known fact that Ξ being a generalized symmetry implies that E n (Ξ V ⌋η) = 0, thus locally Ξ V ⌋η = d H ν i , then there exists a 0-cocycle ν i , defined by µ ν = Ξ V ⌋η λ ≡ d H ν i . Notice that d(Ξ V ⌋η λ ) = 0, but in general δ n (Ξ V ⌋η λ ) = 0 [11]. Along critical sections this implies the conservation law…”

“…which are closed in the complex and define a non trivial cohomology class, admit a system of local Lagrangians, one for each open set in a suitable covering, which satisfy certain relations among them. Global projectable vector field on a jet fiber manifold which are symmetries of dynamical forms, in particular of locally variational dynamical forms, and corresponding formulations of Noether theorem II can be considered in order to determine obstructions to the globality of associated conserved quantities [11]. Analogously, the concept of global (and local) variationally trivial Lagrangians and, in general, of variationally trivial currents (i.e.…”

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

“…Since we assume η λ i to be closed, Theorem 3 reduces (case q = n + 1) to L Ξ η λ i = E n (Ξ V ⌋η λ i ), and if j r Ξ is such that L Ξ η λ i = 0, then E n (Ξ V ⌋η λ i ) = 0; therefore, locally we have Ξ V ⌋η λ i = d H ν i . Notice that, although Ξ V ⌋η λ i is global, in general it defines a non trivial cohomology class [4]; it is clear that ν i is a (local) current which is conserved on-shell (i.e. along critical sections).…”