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We explicate some epistemological implications of stationary principles and in particular of Noether Theorems. Noether's contribution to the problem of covariance, in fact, is epistemologically relevant, since it moves the attention from equations to conservation laws., Comment: 6 pages, misprints corrected, reference list updated, MSC added; presented at the XXXII Int. Congress of the Italian Society of Historians of Physics and Astronomy (SISFA), Rome 2012

We study cosmological models derived from higher-order Gauss-Bonnet gravity $F(R,G)$ by using the Lagrange multiplier approach without assuming the presence of additional fields with the exception of standard perfect fluid matter. The presence of Lagrange multipliers reduces the number of allowed solutions. We need to introduce compatibility conditions of the FRW equations, which impose strict restrictions on the metric or require the introduction of additional exotic matter. Several classes of $F(R,G)$ models are generated and discussed., Comment: 8 pages, to appear in Astro. Space Sci

Proceedings of the second edition of the international Workshop "Dynamics and Thermodynamics of Blackholes and Naked Singularities" (Department of Mathematics of the Politecnico of Milano from May 10-12, 2007., Comment: Links added

We review Extended Theories of Gravity in metric and Palatini formalism pointing out their cosmological and astrophysical application. The aim is to propose an alternative approach to solve the puzzles connected to dark components., Comment: 44 pages, 11 figures

Starting from a characterization of admissible Cheataev and vakonomic variations in a field theory with constraints we show how the so called parametrized variational calculus can help to derive the vakonomic and the non-holonomic field equations. We present an example in field theory where the non-holonomic method proved to be unphysical.

We study higher--order variational derivatives of a generic second--order Lagrangian ${\cal L}={\cal L}(x,\phi,\partial\phi,\partial^2\phi)$ and in this context we discuss the Jacobi equation ensuing from the second variation of the action. We exhibit the different integrations by parts which may be performed to obtain the Jacobi equation and we show that there is a particular integration by parts which is invariant. We introduce two new Lagrangians, ${\cal L}_1$ and ${\cal L}_2$, associated to the first and second--order deformations of the original Lagrangian ${\cal L}_0$ respectively; they are in fact the first elements of a whole hierarchy of Lagrangians derived from ${\cal L}_0$. In terms of these Lagrangians we are able to establish simple relations between the variational derivatives of different orders of a given Lagrangian. We then show that the Jacobi equations of ${\cal L}_0$ may be obtained as variational equations, so that the Euler--Lagrange and the Jacobi equations are obtained from a single variational principle based on the first--order variation ${\cal L}_1$ of the Lagrangian. We can furthermore introduce an associated energy--momentum tensor ${{\cal H}^\mu}_\nu$ which turns out to be a conserved quantity if ${\cal L}_0$ is independent of space--time variables., Comment: This article was originally published in Differential Geometry and Applications, Proc. Conf., Aug. 28 - Sept. 1, 1995, Brno, Czech Republic (Masaryk University, Brno, 1996), pp. 353-372. Since it is hardly available we post it here for the interested audience

Variational principles for field theories where variations of fields are restricted along a parametrization are considered. In particular, gauge-natural parametrized variational problems are defined as those in which both the Lagrangian and the parametrization are gauge covariant and some further conditions is satisfied in order to formulate a N\"other theorem that links horizontal and gauge symmetries to the relative conservation laws (generalizing what Fern{\'a}ndez, Garc{\'{\i}}a and Rodrigo did in some recent papers). The case of vakonomic constraints in field theory is also studied within the framework of parametrized variational problems, defining and comparing two different concepts of criticality of a section, one arising directly from the vakonomic schema, the other making use of an adapted parametrization. The general theory is then applied to the case of hydrodynamics of a charged fluid coupled with its gravitational and electromagnetic field. A variational formulation including conserved currents and superpotentials is given that turns out to be computationally much easier than the standard one.

Proceedings of the international Workshop on ``Dynamics and Thermodynamics of Blackholes and Naked Singularities``, that took place at the Department of Mathematics of the Politecnico of Milano from 13 to 15 May 2004.

Alternative theories of gravity have been recently studied in connection with their cosmological applications, both in the Palatini and in the metric formalism. The aim of this paper is to propose a theoretical framework (in the Palatini formalism) to test these theories at the solar system level and possibly at the galactic scales. We exactly solve field equations in vacuum and find the corresponding corrections to the standard general relativistic gravitational field. On the other hand, approximate solutions are found in matter cases starting from a Lagrangian which depends on a phenomenological parameter. Both in the vacuum case and in the matter case the deviations from General Relativity are controlled by parameters that provide the Post-Newtonian corrections which prove to be in good agreement with solar system experiments., Comment: 17 pages, no figures

The current accelerated universe could be produced by modified gravitational dynamics as it can be seen in particular in its Palatini formulation. We analyze here a specific non-linear gravity-scalar system in the first order Palatini formalism which leads to a FRW cosmology different from the purely metric one. It is shown that the emerging FRW cosmology may lead either to an effective quintessence phase (cosmic speed-up) or to an effective phantom phase. Moreover, the already known gravity assisted dark energy dominance occurs also in the first order formalism. Finally, it is shown that a dynamical theory able to resolve the cosmological constant problem exists also in this formalism, in close parallel with the standard metric formulation., Comment: 21 pages, LaTeX file, no figures. Replaced version to be published on Phys. Rev. D

The debate on the physical relevance of conformal transformations can be faced by taking the Palatini approach into account to gravitational theories. We show that conformal transformations are not only a mathematical tool to disentangle gravitational and matter degrees of freedom (passing from the Jordan frame to the Einstein frame) but they acquire a physical meaning considering the bi-metric structure of Palatini approach which allows to distinguish between spacetime structure and geodesic structure. Examples of higher-order and non-minimally coupled theories are worked out and relevant cosmological solutions in Einstein frame and Jordan frames are discussed showing that also the interpretation of cosmological observations can drastically change depending on the adopted frame.

The evidence of the acceleration of universe at present time has lead to investigate modified theories of gravity and alternative theories of gravity, which are able to explain acceleration from a theoretical viewpoint without the need of introducing dark energy. In this paper we study alternative gravitational theories defined by Lagrangians which depend on general functions of the Ricci scalar invariant in minimal interaction with matter, in view of their possible cosmological applications. Structural equations for the spacetimes described by such theories are solved and the corresponding field equations are investigated in the Palatini formalism, which prevents instability problems. Particular examples of these theories are also shown to provide, under suitable hypotheses, a coherent theoretical explanation of earlier results concerning the present acceleration of the universe and cosmological inflation. We suggest moreover a new possible Lagrangian, depending on the inverse of sinh(R), which gives an explanation to the present acceleration of the universe., Comment: 23 pages, Revtex4 file

We consider the geometric formulation of the Hamiltonian formalism for field theory in terms of {\em Hamiltonian connections} and {\em multisymplectic forms}. In this framework the covariant Hamilton equations for Mechanics and field theory are defined in terms of multisymplectic $(n+2)$--forms, where $n$ is the dimension of the basis manifold, together with connections on the configuration bundle. We provide a new geometric Hamiltonian description of field theory, based on the introduction of a suitable {\em composite fibered bundle} which plays the role of an {\em extended configuration bundle}. Instead of fibrations over an $n$--dimensional base manifold $\bX$, we consider {\em fibrations over a line bundle $\Tht$ fibered over $\bX$}. The concepts of {\em extended Legendre bundle}, {\em Hamiltonian connection}, {\em Hamiltonian form} and {\em covariant Hamilton equations} are introduced and put in relation with the corresponding standard concepts in the polymomentum approach to field theory., Comment: 9 pages, VIII Int. Conf. Diff. Geom. Appl. (Opava 2001); O. Kowalski et al. eds., misprints corrected

Two-spinor formalism for Einstein Lagrangian is developed. The gravitational field is regarded as a composite object derived from soldering forms. Our formalism is geometrically and globally well-defined and may be used in virtually any 4m-dimensional manifold with arbitrary signature as well as without any stringent topological requirement on space-time, such as parallelizability. Interactions and feedbacks between gravity and spinor fields are considered. As is well known, the Hilbert-Einstein Lagrangian is second order also when expressed in terms of soldering forms. A covariant splitting is then analysed leading to a first order Lagrangian which is recognized to play a fundamental role in the theory of conserved quantities. The splitting and thence the first order Lagrangian depend on a reference spin connection which is physically interpreted as setting the zero level for conserved quantities. A complete and detailed treatment of conserved quantities is then presented., Comment: 16 pages, Plain TEX

We criticize and generalize some properties of Noether charges presented in a paper by V. Iyer and R. M. Wald and their application to entropy of black holes. The first law of black holes thermodynamics is proven for any gauge-natural field theory. As an application charged Kerr-Newman solutions are considered. As a further example we consider a (1+2) black hole solution.

We apply a global and geometrically well-defined formalism for spinor-dilaton-gravity to two-dimensional manifolds. We discuss the general formalism and focus attention on some particular choices of the dilatonic potential. For constant dilatonic potential the model turns out to be completely solvable and the general solution is found. For linear and exponential dilatonic potentials we present the class of exact solutions with a Killing vector., Comment: 21 pages, LaTeX, minor changes in text and format, final version to appear in Classical and Quantum Gravity

auto-ignore This paper has been withdrawn by the authors., Comment: 21 pages, Plain TEX, a few remarks added

A new formalism for spinors on curved spaces is developed in the framework of variational calculus on fibre bundles. The theory has the same structure of a gauge theory and describes the interaction between the gravitational field and spinors. An appropriate gauge structure is also given to General Relativity, replacing the metric field with spin frames. Finally, conserved quantities and superpotentials are calculated under a general covariant form., Comment: 18 pages, Plain TEX, revision, explicit expression for superpotential has been added

Relying on the general theory of Lie derivatives a new geometric definition of Lie derivative for general spinor fields is given, more general than Kosmann's one. It is shown that for particular infinitesimal lifts, i.e. for Kosmann vector fields, our definition coincides with the definition given by Kosmann more than 20 years ago., Comment: 10 pages, Plain TEX