Marcella Palese

Symmetries in finite order variational sequences

We refer to Krupkas variational sequence, i.e. the quotient of the de Rham sequence on a finite order jet space with respect to a ‘variationally trivial’ subsequence. Among the morphisms of the variational sequence there are the Euler-Lagrange operator and the Helmholtz operator.In this note we show that the Lie derivative operator passes to the quotient in the variational sequence. Then we define the variational Lie derivative as an operator on the sheaves of the variational sequence. Explicit representations of this operator give us some abstract versions of Noethers theorems, which can be interpreted in terms of conserved currents for Lagrangians and Euler-Lagrange morphisms.

Covariant gauge-natural conservation laws

When a gauge-natural invariant variational principle is assigned, to determine canonical covariant conservation laws, the vertical part of gauge-natural lifts of infinitesimal principal automorphisms—defining infinitesimal variations of sections of gauge-natural bundles—must satisfy generalized Jacobi equations for the gauge-natural invariant Lagrangian. Vice versa all vertical parts of gauge-natural lifts of infinitesimal principal automorphisms which are in the kernel of generalized Jacobi morphisms are generators of canonical covariant currents and superpotentials. In particular, only a few gauge-natural lifts can be considered as canonical generators of covariant gauge-natural physical charges.

Canonical Connections in Gauge-Natural Field Theories

We investigate canonical aspects concerning the relation between symmetries and conservation laws in gauge-natural field theories. In particular, we find that a canonical spinor connection can be selected by the simple requirement of the global existence of canonical superpotentials for the Lagrangian describing the coupling of gravitational and Fermionic fields. In fact, the naturality of a suitably defined variational Lagragian implies the existence of an associated energy-momentum conserved current. Such a current defines a Hamiltonian form in the corresponding phase space; we show that an associated Hamiltonian connection is canonically defined along the kernel of the generalized gauge-natural Jacobi morphism and uniquely characterizes the canonical spinor connection.

Nonlinear 2 + 1-Dimensional Field Equations from Incomplete Lie Algebra Structures

Abstract We show that the nonlinear (2+1)-dimensional Three-Wave Resonant Interaction equations, describing several important physical phenomena, can be generated starting from incomplete Lie algebras in the framework of multidimensional prolongation structures. We make use of an ansatz involving the structure equations of a principal prolongation connection induced by an admissible Backlund map.

Generalized Bianchi identities in gauge-natural field theories and the curvature of variational principles*

By resorting to Noethers Second Theorem, we relate the generalized Bianchi identities for Lagrangian field theories on gauge-natural bundles with the kernel of the associated gauge-natural Jacobi morphism. A suitable definition of the curvature of gauge-natural variational principles can be consequently formulated in terms of the Hamiltonian connection canonically associated with a generalized Lagrangian obtained by contracting field equations.

Variationally equivalent problems and variations of Noether currents

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.

Gauge-natural Noether currents and connection fields

We study geometric aspects concerned with symmetries and conserved quantities in gauge-natural invariant variational problems and investigate implications of the existence of a reductive split structure associated with canonical Lagrangian conserved quantities on gauge-natural bundles. In particular, we characterize the existence of covariant conserved quantities in terms of principal Cartan connections on gauge-natural prolongations.

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.

Lagrangian reductive structures on gauge-natural bundles

A reductive structure is associated here with the Lagrangian canonically defined conserved quantities on gauge-natural bundles. Infinitesimal parametrized transformations defined by the gauge-natural lift of infinitesimal principal automorphisms induce a variational sequence such that the generalized Jacobi morphism is naturally self-adjoint. As a consequence, its kernel defines a reductive split structure on the relevant underlying principal bundle.

Higgs fields on spinor gauge-natural bundles

We show that the Lie derivative of spinor fields is parametrized by Higgs fields defined by the kernel of a gauge-natural Jacobi morphism associated with the Einstein?Cartan? Dirac Lagrangian. In particular, the generalized Kosmann lift to the total bundle of the theory is constrained by variational Higgs fields on gauge-natural bundles.