Theory of dynamical systems and the relations between classical and quantum mechanics

We give a review of some works where it is shown that certain quantum-like features are exhibited by classical systems. Two kinds of problems are considered. The first one concerns the specific heat of crystals (the so called Fermi-Pasta-Ulam problem), where a glassy behavior is observed, and the energy distribution is found to be of Planck-like type. The second kind of problems concerns the self-interaction of a charged particle with the electromagnetic field, where an analog of the tunnel effect is proven to exist, and moreover some nonlocal effects are exhibited, leading to a natural hidden variable theory which violates Bell's inequalities.