  We show that the standard Fermi--Pasta--Ulam system, with a suitable choice
for the interparticle potential, constitutes a model for glasses, and indeed an
extremely simple and manageable one. Indeed, it allows one to describe the
landscape of the minima of the potential energy and to deal concretely with any
one of them, determining the spectrum of frequencies and the normal modes. A
relevant role is played by the harmonic energy $\mathcal E$ relative to a given
minimum, i.e., the expansion of the Hamiltonian about the minimum up to second
order. Indeed we find that there exists an energy threshold in $\mathcal E$
such that below it the harmonic energy $\mathcal E$ appears to be an
approximate integral of motion for the whole observation time. Consequently,
the system remains trapped near the minimum, in what may be called a vitreous
or glassy state. Instead, for larger values of $\mathcal E$ the system rather
quickly relaxes to a final equilibrium state. Moreover we find that the
vitreous states present peculiar statistical behaviors, still involving the
harmonic energy $\mathcal E$. Indeed, the vitreous states are described by a
Gibbs distribution with an effective Hamiltonian close to $\mathcal E$ and with
a suitable effective inverse temperature. The final equilibrium state presents
instead statistical properties which are in very good agreement with the Gibbs
distribution relative to the full Hamiltonian of the system.
