Editorial
Erratum to: Variations by generalized symmetries of local Noether strong currents equivalent to global canonical Noether currents
We correct misprints in a formula in the last sentence at the end of page 129; the first paragraph of subsection 4.1; misprints at the end of page 132 and in Proposition 1 at page 133 of the paper ‘Variations by generalized symmetries of local Noether strong currents equivalent to global canonical Noether currents’, Communications in Mathematics 24 (2016), 125-135. DOI: 10.1515/cm-2016-0009
Variations by generalized symmetries of local Noether strong currents equivalent to global canonical Noether currents
We will pose the inverse problem question within the Krupka variational sequence framework. In particular, the interplay of inverse problems with symmetry and invariance properties will be exploited considering that the cohomology class of the variational Lie derivative of an equivalence class of forms, closed in the variational sequence, is trivial. We will focalize on the case of symmetries of globally defined field equations which are only locally variational and prove that variations of local...
I fondamenti epistemologici della Relatività Generale e la sua «eredità matematica»
Si analizzano alcuni aspetti fondazionali della Relatività Generale dal punto di vista epistemologico, dando una particolare rilevanza al fatto che non ha più senso parlare di tempo durante il quale le dinamiche si svolgono, né di spazio in cui le dinamiche hanno luogo. La Relatività Generale afferma infatti l'identificazione tra spazio-tempo – che è una varietà metrica e dunque un'entità a priori non dinamica – e la materia – che soggiace ad equazioni differenziali ed è dunque un'entità dinamica...
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...
Global generalized Bianchi identities for invariant variational problems on gauge-natural bundles
We derive both local and global generalized Bianchi identities for classical Lagrangian field theories on gauge-natural bundles. We show that globally defined generalized Bianchi identities can be found without the a priori introduction of a connection. The proof is based on a global decomposition of the 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...
Symmetries in finite order variational sequences
We refer to Krupka’s 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...
Second variational derivative of local variational problems and conservation laws
We consider cohomology defined by a system of local Lagrangian and investigate under which conditions the variational Lie derivative of associated local currents is a system of conserved currents. The answer to such a question involves Jacobi equations for the local system. Furthermore, we recall that it was shown by Krupka et al. that the invariance of a closed Helmholtz form of a dynamical form is equivalent with local variationality of the Lie derivative of the dynamical form; we remark that...
Gauge-natural field theories and Noether theorems: canonical covariant conserved currents
Summary: We specialize in a new way the second Noether theorem for gauge-natural field theories by relating it to the Jacobi morphism and show that it plays a fundamental role in the derivation of canonical covariant conserved quantities. In particular we show that Bergmann-Bianchi identities for such theories hold true covariantly and canonically only along solutions of generalized gauge-natural Jacobi equations. Vice versa, all vertical parts of gauge-natural lifts of infinitesimal principal automorphisms...
On a class of polynomial Lagrangians
Generalized Jacobi morphisms in variational sequences
Summary: We provide a geometric interpretation of generalized Jacobi morphisms in the framework of finite order variational sequences. Jacobi morphisms arise classically as an outcome of an invariant decomposition of the second variation of a Lagrangian. Here they are characterized in the context of generalized Lagrangian symmetries in terms of variational Lie derivatives of generalized Euler-Lagrange morphisms. We introduce the variational vertical derivative and stress its link with the classical...
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