3,670 research outputs found
Constructing towers with skeletons from open Lie algebras and integrability
We provide a given algebraic structure with the structure of an infinitesimal
algebraic skeleton. The necessary conditions for integrability of the absolute
parallelism of a tower with such a skeleton are dispersive nonlinear models and
related conservation laws given in the form of associated linear spectral
problems.Comment: misprints corrected, title changed, some remarks adde
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} -degree and -order
quotient sheaf on the fibered manifold \bY_{\zet} \times_{\bX} \mathfrak{K},
where is an appropriate subbundle of the vector bundle of
(prolongations of) infinitesimal right-invariant automorphisms . 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 obtained
contracting the Euler--Lagrange form of the original Lagrangian with
. 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 .Comment: 16 pages, abstract rewritten, body slightly revised, Proc. Winter
School "Geometry and Physics" (Srni,CZ 2005
On a class of polynomial Lagrangians
In the framework of finite order variational sequences a new class of
Lagrangians arises, namely, \emph{special} Lagrangians. These Lagrangians are
the horizontalization of forms on a jet space of lower order. We describe their
properties together with properties of related objects, such as
Poincar\'e--Cartan and Euler--Lagrange forms, momenta and momenta of generating
forms, a new geometric object arising in variational sequences. Finally, we
provide a simple but important example of special Lagrangian, namely the
Hilbert--Einstein Lagrangian.Comment: LaTeX2e, amsmath, diagrams, hyperref; 15 page
Some aspects of the homogeneous formalism in Field Theory and gauge invariance
We propose a suitable formulation of the Hamiltonian formalism for Field
Theory in terms of Hamiltonian connections and multisymplectic forms where a
composite fibered bundle, involving a line bundle, plays the role of an
extended configuration bundle. This new approach can be interpreted as a
suitable generalization to Field Theory of the homogeneous formalism for
Hamiltonian Mechanics. As an example of application, we obtain the expression
of a formal energy for a parametrized version of the Hilbert--Einstein
Lagrangian and we show that this quantity is conserved.Comment: 9 pages, slightly revised, to appear in Proc. Winter School "Geometry
and Physics", Srni (CZ) 200
Global Generalized Bianchi Identities for Invariant Variational Problems on Gauge-natural Bundles
We derive both {\em local} and {\em global} generalized {\em Bianchi
identities} for classical Lagrangian field theories on gauge-natural bundles.
We show that globally defined generalized Bianchi identities can be found
without the {\em a priori} introduction of a connection. The proof is based on
a {\em global} decomposition of the {\em variational Lie derivative} of the
generalized Euler--Lagrange morphism and the representation of the
corresponding generalized Jacobi morphism on gauge-natural bundles. In
particular, we show that {\em within} a gauge-natural invariant Lagrangian
variational principle, the gauge-natural lift of infinitesimal principal
automorphism {\em is not} intrinsically arbitrary. As a consequence the
existence of {\em canonical} global superpotentials for gauge-natural Noether
conserved currents is proved without resorting to additional structures.Comment: 24 pages, minor changes, misprints corrected, a misprint in the
coordinate expression of the Jacobi morphism corrected; final version to
appear in Arch. Math. (Brno
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
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
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