  In this paper we shall review the equivalence between Palatini$-f(\mathcal
R)$ theories and Brans- Dicke (BD) theories at the level of action principles.
We shall define the Helmholtz Lagrangian associated to Palatini$-f(\mathcal R)$
theory and we will define some transformations which will be useful to recover
Einstein frame and Brans-Dicke frame. We shall see an explicit example of
matter field and we will discuss how the conformal factor affects the physical
quantities.
