is also true.

The physical meaning of such a result is that systems with vanishing energy are conformally invariant at the level of equations of motion.

Conformally equivalent Lagrangians in the scalar field Cosmology
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Let us discuss now the conformal equivalence of Lagrangians for scalar fields in a general Riemannian space of four dimensions (see also [@allemandi]). The field equations in the scalar-tensor cosmology can be derived from two different variational principles. In the first case we consider a scalar field $\phi$ which is minimally coupled to gravity and the equations of motion follow from the action $$S_{M}=\int d\tau dx^{3}\sqrt{-g}\left[  R+\frac{\varepsilon}{2}g_{ij}\phi
^{;i}\phi^{;j}-V\left(  \phi\right)  \right]  \label{CLN.11}$$ where $\varepsilon=\pm 1$ defines quintessence or phantom field cosmology respectively.

In the second case we assume a scalar field $\psi$ (different from the minimally coupled scalar field $\phi$) which interacts with the gravitational field (non minimal coupling) and the corresponding action is given by $$S_{NM}=\int d\tau dx^{3}\sqrt{-\bar{g}}\left[  F\left(  \psi\right)  \bar
{R}+\frac{\varepsilon}{2}\bar{g}^{ij}\psi_{;i}\psi_{;j}-\bar{V}\left(
\psi\right)  \right]  \label{CL.12.0}$$ where $F(\psi)$ is the coupling function between the gravitational and the scalar field $\psi$ respectively. Below we pose the following proposition.

***Theorem:*** *The field equations for a non minimally coupled scalar field $\psi$ with Lagrangian $\bar{L}\left(  \tau,x^{k},\dot
{x}^{k}\right)  $ and 