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1. Classical Continuum Mechanics. Anisotropic elasticity is a field where geometrical methods are commonly used. Recent researches were conducted about symmetry groups of second grade materials, for which the response depends on the second deformation gradient. Moreover, we investigated linear functions defined on the space of anisotropic elasticty tensors which are not invariant under the action of the group of rotations but become so when their domain is restricted to a suitable subspace. A more complex problem in which we are interested is the construction of an integrity basis for the set of anisotropic elasticity tensors under the action of the rotation group. 2. Relativistic Continuum Mechanics. This line of research tries to understand some topics in the physics of general relativistic gravitational collapse. We developed a new approach to problems of existence/non existence of solutions of the differential equations describing radial null geodesics at singular spacetime points. Sufficient conditions of non-existence have been published by some of us in recent years as well as conditions for existence. These results have been successfully applied to the relevant cases, such as perfect fluid collapse. The future research will be devoted to understand as much as possible the complete picture in the spherical case with special emphasis on the collapse (of great interest in many fields, such as cosmology) of interacting scalar fields. We shall be involved as well in a first ongoing tentative to go beyond spherical symmetry (and therefore, to study phenomena such as possible emission of gravitational waves from naked singularities) using the Gerlach-Sengupta formalism of dynamical perturbation of spherically symmetric fields. A line of research on Cultural Astronomy is also cultivated, in connection with the Center for Cultural Heritage of the Politecnico. 3. Geometry of Hamiltonian Systems. In the framework of integrable Hamiltonian systems, it was recently proved by F. Treves that the conserved functionals of KdV hierarchy can be characterized as the functionals vanishing when evaluated on certain formal Laurent series. The research related with the vanishing residue condition of Treves investigates a connection with other known features of the soliton equations, in order to clarify the possible essential uniqueness of the Treves series and to apply the vanishing residue technique to all KdV-type hierarchies and other soliton hierarchies. A known classical result about perturbations of integrable dynamical systems is given by the so-called averaging principle: given an integrable system with action-angle coordinates and a small perturbation of it, the solution of the perturbed system is close to the solution of the averaged system, obtained from the original one by averaging over the angles. It is essential to evaluate the error introduced and, since this estimate is often only of qualitative type, in view of applications it is desirable to construct an approach of fully quantitative type. Another research is carried out about various aspects of the Kepler problem, including its regularization at collision, both with vector and hypercomplex methods. The “doubling technique” adopted for this investigations has opened a door in the literature for computer oriented researches.
Maria Dina Vivarelli