My main interest is the physics of multidimensional models of Kaluza-Klein. A fisrt line of research was the analysis of the ADM splitting of the KK 5D model, in order to achieve the Hamiltonian formulation of the dynamics and get insight in the understanding of gauge symmetry generation. The ADM slicing of KK model, and its physical meaning, is not obvious due to the existence of two possible procedures; starting by a 5D metric we can implement the KK paradigm and then apply a 3+1 ADM splitting; or, we can apply a 4+1 ADM splitting as first step, and then implement the KK paradigm. Outcomes appear to be different because the final set of variables seem to be not equivalent. A detailed analysys of the transformations laws and the tetradic picture of our variables allows us to establish conversion formulas between these two set of variables; once the Lagrangians resulting from these two procedures are recasted in the same set of variables is possibile to recognize that the only difference relies in surface term; then we conclude that we are dealing with equivalent dynamics and with a unique well defined Hamiltonian. Hence, ADM splitting is provided to commute with KK reduction. Thus, the proof of commutation ensures us that ADM splitting is compatible with KK model and allows us to compute the Hamiltonian. Moreover, Hamiltonian formulation, together with conversion formulas, shows clearly how the time component of the electromagnetic field (a gauge Lagrangian constraint of the dynamics) is given by combination of the geometrical Lagrangian multipliers of the 5D gravitational field. This result enforces the physical meaning of KK model and its estension to multidimensional models appears
interesting. Other lines of research are related to the presence of the extra scalar field (the scale factor of the extra dimension )in our dynamics and its role in the dynamics of particles and fields. Starting by the analysys of the 5D geodesic motion is possible, after dimensional reduction, to restore the usual electrodynamics. The charge-mass ratio, for a test particle, is recognized as a combination of a kinematical constant of motion and the scalar field. A large, slow varying, scalar field allows us to avoid the problem of planckian mass and to take into account the variation of fundamental constant over cosmological scales of time. At the same time the scalar field affects the behaviour of the metric fields, expecially near the singularity. More generally, there ares some reasons to think that this field can be an alternative time variable in the relational point of view. This work is in progress and seems interesting to deal with the introduction of matter fields and with a detailed cosmological solution; finally is worth noting that the existence of a compactified extra dimension sets the existence of a cut-off; thus is very interesting to address to our model the framework of non-commutative geometry.