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

Abstract

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.

Variational Sequences, Representation Sequences and Applications in Physics

“…We have the on shell conservation laws d H ( λι − β ι ) = 0. It can be proved that λι − L J s Ξ µ ι = λι−d H µι is not only closed but also exact on shell (for a detailed discussion see, e.g., [11]). In the following we shall give an example of application concerning symmetries of Jacobi type equations for a Chern-Simons 3D gauge theory.…”

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

“…We shall consider global projectable vector field on a jet fiber manifold which are symmetries of dynamical forms, in particular of locally variational dynamical forms. It is clear the relevant role played by the variational Lie derivative, a differential operator acting on equivalence classes of variational forms in the variational sequence [9], [25], by which Noether theorems can be formulated. In particular, variations of currents can be recognized in this approach.…”

“…As well known Noether Theorems relate symmetries of a variational problem to conserved quantities. In [9], [25] we formulated the Noether Theorems in terms of variational Lie derivatives of classes of forms modulo the contact structure.…”

“…where i = j r Ξ V p d V λi + ξ λ i is the usual canonical Noether current; the Noether-Bessel-Hagen current (λ i , Ξ) − β i is a local object and it is conserved along the solutions of Euler-Lagrange equations (local conservation law) [9], [17], [18], [30].…”

The Noether–Bessel-Hagen symmetry approach for dynamical systems

“…Our aim, in this paper, is to analyze Noether-type symmetries of dynamical systems with external forces, which leave invariant the corresponding equations of motion, known as the Noether-Bessel-Hagen symmetries [3]; for symmetries in local variational theories see monographs by Kossmann-Schwarzbach [4] and Krupka [5], and recent papers, e.g. [6][7][8][9][10][11][12]. As we will show, this is a straightforward generalization of the Noether Approach that can be extremely useful in several areas of physics like mechanics, field theory, cosmology and, in general, dynamical systems.…”