Variational Lie derivative and cohomology classes

We relate cohomology defined by a system of local Lagrangian with the cohomology class of the system of local variational Lie derivative, which is in turn a local variational problem; we show that the latter cohomology class is zero, since the variational Lie derivative `trivializes' cohomology classes defined by variational forms. As a consequence, conservation laws associated with symmetries ensuring the vanishing of the second variational derivative of a local variational problem are globally defined.Comment: 7 pages, misprints in Corollary 2 and a misleading in the abstract and the introduction corrected, XIX International Fall Workshop on Geometry and Physic

Noether identities in Einstein--Dirac theory and the Lie derivative of spinor fields

We characterize the Lie derivative of spinor fields from a variational point of view by resorting to the theory of the Lie derivative of sections of gauge-natural bundles. Noether identities from the gauge-natural invariance of the first variational derivative of the Einstein(--Cartan)--Dirac Lagrangian provide restrictions on the Lie derivative of fields.Comment: 11 pages, completely rewritten, contains an example of application to the coupling of gravity with spinors; in v4 misprints correcte

Gauge-natural field theories and Noether Theorems: canonical covariant conserved currents

Recently we found that canonical gauge-natural superpotentials are obtained as global sections of the {\em reduced} $(n-2)$-degree and $(2s-1)$-order quotient sheaf on the fibered manifold \bY_{\zet} \times_{\bX} \mathfrak{K}, where $\mathfrak{K}$ is an appropriate subbundle of the vector bundle of (prolongations of) infinitesimal right-invariant automorphisms $\bar{\Xi}$. In this paper, we provide an alternative proof of the fact that the naturality property \cL_{j_{s}\bar{\Xi}_{H}}\omega (\lambda, \mathfrak{K})=0 holds true for the {\em new} Lagrangian $\omega (\lambda, \mathfrak{K})$ obtained contracting the Euler--Lagrange form of the original Lagrangian with $\bar{\Xi}_{V}\in \mathfrak{K}$. We use as fundamental tools an invariant decomposition formula of vertical morphisms due to Kol\'a\v{r} and the theory of iterated Lie derivatives of sections of fibered bundles. As a consequence, we recover the existence of a canonical generalized energy--momentum conserved tensor density associated with $\omega (\lambda, \mathfrak{K})$.Comment: 16 pages, abstract rewritten, body slightly revised, Proc. Winter School "Geometry and Physics" (Srni,CZ 2005

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

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.Comment: 20 page

Variational Sequences, Representation Sequences and Applications in Physics

This paper is a review containing new original results on the finite order variational sequence and its different representations with emphasis on applications in the theory of variational symmetries and conservation laws in physics

Particle-like, dyx-coaxial and trix-coaxial Lie algebra structures for a multi-dimensional continuous Toda type system

We prove that with a $(2+1)$-dimensional Toda type system are associated algebraic skeletons which are (compatible assemblings) of particle-like Lie algebras of dyons and triadons type. We obtain trix-coaxial and dyx-coaxial Lie algebra structures for the system from algebraic skeletons of some particular choice for compatible associated absolute parallelisms. In particular, by a first choice of the absolute parallelism, we associate with the $(2+1)$-dimensional Toda type system a trix-coaxial Lie algebra structure made of two (compatible) base triadons constituting a $2$-catena. Furthermore, by a second choice of the absolute parallelism, we associate a dyx-coaxial Lie algebra structure made of two (compatible) base dyons, as well as particle-like Lie algebra structures made of single $3$-dyons. Some explicit examples of applications such as conservation laws related to special solutions, and an inverse spectral problem are worked out.Comment: 24 pages, slight modifications, a few original references adde