Persistence of regular motions for nearly integrable Hamiltonian systems in the thermodynamic limit

A review is given of the studies aimed at extending to the thermodynamic limit stability results of Nekhoroshev type for nearly integrable Hamiltonian systems. The physical relevance of such an extension, i. e., of proving the persistence of regular (or ordered) motions in that limit, is also discussed. This is made in connection both with the old Fermi – Pasta – Ulam problem, which gave origin to such discussions, and with the optical spectral lines, the existence of which was recently proven to be possible in classical models, just in virtue of such a persistence.