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A geometrical description of micropolar mixtures theory for N components including matter diffusion is analyzed from a thermodynamical viewpoint with internal variables. This theory introduces one independent rotation for each material element so that the resulting stress and strain tensors are generally non-symmetric. In order to consider dissipative effects due to rotation and to thermodiffusion we introduce a set of N independent scalar internal variables, which we denote by γA; the set of N independent vectorial internal variables defined by their gradients is indicated as ΛA. The behavior of the mixture is therefore described by a state space W including the classical variables (i.e. mass density, mass concentration, temperature and so on) together with internal variables and with their first gradients satisfying an evolution equation. General thermodynamical restrictions and residual dissipation inequalities are obtained by Clausius-Duhem inequality. Finally, by using geometric technique due to Maugin, the heat equation is derived in the first and second form. [ABSTRACT FROM AUTHOR]

We shall show that although Palatini -theories are equivalent to Brans-Dicke theories, still the first pass the Mercury precession of perihelia test, while the second do not. We argue that the two models are not physically equivalent due to different assumptions about free fall. We shall also go through perihelia test without fixing a conformal gauge (clocks or rulers) in order to highlight what can be measured in a conformal invariant way and what cannot. We shall argue that the conformal gauge is broken by choosing a definition of clock, rulers or, equivalently, of masses. [ABSTRACT FROM AUTHOR]

We consider systems of local variational problems defining nonvanishing cohomology classes. Symmetry properties of the Euler-Lagrange expressions play a fundamental role since they introduce a cohomology class which adds up to Noether currents; they are related with invariance properties of the first variation, thus with the vanishing of a second variational derivative. 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 that such a current be global. [ABSTRACT FROM AUTHOR]

In a previous paper we have sketched a geometrical model for ferromagnetic crystals within a geometrized framework of simple materials with internal variables. Using a model of magnetizable media investigated by Maugin and coworkers, in the state vector the magnetization vector together with its space gradient (subjected to an evolution equation depending on both internal and external magnetic field) were included. In this paper we deepen the study of this geometrical model. In particular, from the exploitation of Clausius-Duhem inequality the entropy production and the state laws are calculated explicitly and starting from one appropriate dynamical system the expression for the entropy function and necessary conditions for its existence are got. Finally, using a Maugin's technique, the heat equation in the first and second form is derived. [ABSTRACT FROM AUTHOR]

In a previous paper we outlined a geometric model for the thermodynamical description of ferrimagnetic crystals with a non-Euclidean structure. Applying a geometrization technique based on a model for magnetizable deformable media earlier introduced by Maugin, starting from an appropriate dynamical system on the fiber bundle of processes for simple material elements of these media, the expressions of the entropy function and the entropy 1-form were obtained. In this contribution we deepen the study of this geometrical model. We give a detailed description of the media under consideration, we introduce the transformation induced by the process and, applying the closure conditions for the entropy 1-form, we work out the necessary conditions for the existence of the entropy function. The derivation of the entropy 1-form is the starting point to introduce an extended thermodynamical phase space. Finally, from the exploitation of the Clausius-Duhem inequality, we derive, using Maugin's techniques, the residual dissipation inequality and the heat equation in the first and second form. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We calculate the relative conserved currents, superpotentials and conserved quantities between two homogeneous and isotropic universes. In particular, we prove that their relative "energy" (defined as the conserved quantity associated to cosmic time coordinate translations for a comoving observer) is vanishing and so are the other conserved quantities related to a Lie subalgebra of vector fields isomorphic to the Poincaré algebra. These quantities are also conserved in time. We also find a relative conserved quantity for such a kind of solution which is conserved in time though non-vanishing. This example provides at least two insights in the theory of conserved quantities in General Relativity. First, the contribution of the cosmological matter fluid to the conserved quantities is carefully studied and proved to be vanishing. Second, we explicitly show that our superpotential (that happens to coincide with the so-called KBL potential although it is generated differently) provides strong conservation laws under much weaker hypotheses than the ones usually required. In particular, the symmetry generator is not needed to be Killing (nor Killing of the background, nor asymptotically Killing), the prescription is quasi-local and it works fine in a finite region too and no matching condition on the boundary is required. [ABSTRACT FROM AUTHOR]

This work is focused on spherically symmetric space-times. More precisely, geometric and structural properties of spatially spherical shells of a dust universe are analyzed in detail considering recent results of our research. Moreover, exact solutions, obtained for constant Ricci principal curvatures, are inferred and qualitatively analyzed through suitable classic analogies. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We show that different topologies of a space-time manifold and different signatures of its metric can be encompassed into a single Lagrangian formalism, provided one adopts the first-order (Palatini) formulation and relies on nonlinear Lagrangians, that were earlier shown to produce, in the generic case, universality of Einstein field equations and of Komar's energy-momentum complex as well. An example in Relativistic Cosmology is provided. [ABSTRACT FROM AUTHOR]

In the framework of Digital Art applied to Scientific Visualization we have realized a short video together with an interactive multimedia aimed at visualizing and diffusing the main concepts of Special Relativity. This note makes a short presentation of it. [ABSTRACT FROM AUTHOR]

This paper is based on lectures given in Ladek-Zdroj, Poland during the 42nd edition of the Karpac winter school on Theoretical Physics. The aim of these lectures was to give an introduction to a rigorous, mathematically well based, variational approach to alternative theories of Gravity, in view of their very interesting recent applications in the framework of Cosmology. We discuss alternative theories of Gravity both in the metric, purely affine and Palatini formalism, stressing differences and analogies both from a physical and mathematical viewpoint. Moreover we give and introduction to almost all the alternative theories of Gravity, that have been recently studied in view of their physical interest (higher order theories of Gravity, including Lovelock and Chern-Simons Lagrangians). Finally we skip to one of the most striking recent applications of alternative theories of Gravity in the framework of cosmological models: alternative theories of Gravity have been proven to provide a consistent model for accelerating universes. [ABSTRACT FROM AUTHOR]

A preface for the March 2007 issue of the "International Journal of Geometric Methods in Modern Physics" is presented.