Newtonian-like gravity with variable $G$
  We propose a Lagrangian formulation for a varying $G$ Newtonian-like theory
inspired by the Brans-Dicke gravity. Rather than imposing an {\it ad hoc}
dependence for the gravitational coupling, as previously done in the
literature, in our proposal the running of $G$ emerges naturally from the
internal dynamical structure of the theory. We explore the features of the
resulting gravitational field for static and spherically symmetric mass
distributions as well as within the cosmological framework.
