Extended gravity

We shall show equivalence between Palatini-f(ℛ) theories and Brans–Dicke (BD) theories at the level of action principles in generic dimension with generic matter coupling. We do that by introducing the Helmholtz Lagrangian associated to Palatini-f(ℛ) theory and then performing frame transformations in order to recover Einstein frame and BD frame. This clarifies the relation among different formulations and the transformations among different frames. Additionally, it defines a formulation a lá Palatini for the BD theory which is dynamically equivalent to metric BD (unlike the standard Palatini-formulation of metric BD theory which are not dynamically equivalent). In conclusion, we discuss interpretation of extended theories of gravitation and perspectives.