Extended theories of gravitation

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