  We study a classical model of Helium atom in which, in addition to the
Coulomb forces, the radiation reaction forces are taken into account. This
modification brings in the model a new qualitative feature of a global
character. Indeed, as pointed out by Dirac, in any model of classical
electrodynamics of point particles involving radiation reaction one has to
eliminate, from the a priori conceivable solutions of the problem, those
corresponding to the emission of an infinite amount of energy. We show that the
Dirac prescription solves a problem of inconsistency plaguing all available
models which neglect radiation reaction, namely, the fact that in all such
models most initial data lead to a spontaneous breakdown of the atom. A further
modification is that the system thus acquires a peculiar form of dissipation.
In particular, this makes attractive an invariant manifold of special physical
interest, the zero--dipole manifold, that corresponds to motions in which no
energy is radiated away (in the dipole approximation). We finally study
numerically the invariant measure naturally induced by the time--evolution on
such a manifold, and this corresponds to studying the formation process of the
atom. Indications are given that such a measure may be singular with respect to
that of Lebesgue.
