The Variational Approach to Alternative Theories of Gravity

This paper is based on lectures given in Ladek-Zdroj, Poland during the 42nd edition of the Karpac winter school on Theoretical Physics. The aim of these lectures was to give an introduction to a rigorous, mathematically well based, variational approach to alternative theories of Gravity, in view of their very interesting recent applications in the framework of Cosmology. We discuss alternative theories of Gravity both in the metric, purely affine and Palatini formalism, stressing differences and analogies both from a physical and mathematical viewpoint. Moreover we give and introduction to almost all the alternative theories of Gravity, that have been recently studied in view of their physical interest (higher order theories of Gravity, including Lovelock and Chern-Simons Lagrangians). Finally we skip to one of the most striking recent applications of alternative theories of Gravity in the framework of cosmological models: alternative theories of Gravity have been proven to provide a consistent model for accelerating universes.