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.