The Newtonian limit of metric gravity theories with quadratic
  Lagrangians
  The Newtonian limit of fourth-order gravity is worked out discussing its
viability with respect to the standard results of General Relativity. We
investigate the limit in the metric approach which, with respect to the
Palatini formulation, has been much less studied in the recent literature, due
to the higher-order of the field equations. In addition, we refrain from
exploiting the formal equivalence of higher-order theories considering the
analogy with specific scalar-tensor theories, i.e. we work in the so-called
Jordan frame in order to avoid possible misleading interpretations of the
results. Explicit solutions are provided for several different types of
Lagrangians containing powers of the Ricci scalar as well as combinations of
the other curvature invariants. In particular, we develop the Green function
method for fourth-order theories in order to find out solutions. Finally, the
consistency of the results with respect to General Relativity is discussed.
