  Conformally related metrics and Lagrangians are considered in the context of
scalar-tensor gravity cosmology. After the discussion of the problem, we pose a
lemma in which we show that the field equations of two conformally related
Lagrangians are also conformally related if and only if the corresponding
Hamiltonian vanishes. Then we prove that to every non-minimally coupled scalar
field, we may associate a unique minimally coupled scalar field in a
conformally related space with an appropriate potential. The latter result
implies that the field equations of a non-minimally coupled scalar field are
the same at the conformal level with the field equations of the minimally
coupled scalar field. This fact is relevant in order to select physical
variables among conformally equivalent systems. Finally, we find that the above
propositions can be extended to a general Riemannian space of n-dimensions.
