In the first case, the gravitational coupling is the standard Newton constant and the scalar field Lagrangian is simply added to the Ricci scalar. It can consist of $i)$ a kinetic term, $ii)$ a kinetic term and a self-interaction potential, or $iii)$ just an interaction potential. In the first case, the scalar field is nothing else but a cyclic variable and then it is related to a conserved quantity. The second case is the relevant one since, by a variational principle, it is possible to obtain a Klein-Gordon equation where the self-interacting potential $V(\phi)$ leads the dynamics. The third case means that the scalar field has no dynamics. When the coupling is the Newton constant, we are in the so-called [*Einstein frame*]{}.

In fact, as firstly pointed out by Brans and Dicke [@brans], a gravitational theory can be made more “Machian” by relaxing the hypothesis that the coupling is constant. They introduced a scalar field $\phi$ non-minimally coupled to the Ricci scalar $R$ and a kinetic term for such a scalar field into the gravitational action. The result was that the coupling was non-minimal and coordinate dependent. In other words, the gravitational interaction is assumed to change with distance and 