  The Nordstr\"om-Vlasov system is a relativistic Lorentz invariant
generalization of the Vlasov-Poisson system in the gravitational case. The
asymptotic behavior of solutions and the non-linear stability of steady states
are investigated. It is shown that solutions of the Nordstr\"om-Vlasov system
with energy grater or equal to the mass satisfy a dispersion estimate in terms
of the conformal energy. When the energy is smaller than the mass, we prove
existence and non-linear (orbital) stability of a class of static solutions
(isotropic polytropes) against general perturbations. The proof of orbital
stability is based on a variational problem associated to the minimization of
the energy functional under suitable constraints.
