Hamilton-Jacobi Methods in Fields, Particles and Information Geometry

The central point around which this thesis has been developed is the investigation of geometrical structures which are present in some theories of increasing interest in physics. In particular attention has been focused on information theory and quantum mechanics, where the systematic use of specific coordinate systems makes extremely difficult a proper geometrical interpretation of their contents. A guiding principle in this investigation has been the search for analogies with situations where the role of tensorial structures is better understood, first of all the realm of Lagrangian and Hamiltonian mechanics. Interestingly a unifying feature of all this investigation has been Hamilton-Jacobi theory and particularly its relationship with the definition of generating functions of canonical transformations.