the paper is as follows. In section II, we discuss the issue of conformal transformations considering its historical development and connection with physical theories. From our point of view, this section is essential in order to fix the problem showing the urgency to discriminate physical variables among the conformally related models. The basic theoretical elements of the conformally related metric are presented in section III, where we also introduce the concept of conformally related Lagrangians and prove a lemma which shows that the field equations for two conformally related Lagrangian are also conformally related if the corresponding Hamiltonian vanishes. In section IV, we discuss the conformal equivalence of Lagrangians for scalar fields in a Riemannian space of dimension $4$ (for an extension see appendix A). In particular we enunciate a theorem which proves that the field equations for a non-minimally coupled scalar field are the same at the conformal level with the field equations of the minimally coupled scalar field. The necessity to preserve Einstein’s equations in the context of Friedmann-Lemaître-Robertson-Walker (FRLW) space-time leads us to apply, in section V, the current general analysis to the scalar field (quintessence or phantom) flat FLRW cosmologies. Finally, we draw our main conclusions 