Selected chapters in the calculus of variations

These lecture notes present a new development in the calculus of variations known as Aubry-Mather Theory. The work of Aubry began with a model describing electron motion in a two-dimensional crystal, where he explored a related discrete variational problem and its minimal solutions. Mather, on the other hand, focused on area-preserving annulus mappings, specifically monotone twist maps, which are relevant in mechanics as Poincaré maps. Mather made significant strides in 1982, proving the existence of closed invariant subsets called Mather sets, based on a variational principle. Despite different motivations, Aubry's and Mather's investigations share a common mathematical foundation. This text will not follow their approaches but will connect to classical results from the 19th century by Jacobi, Legendre, Weierstrass, and others. Chapter I compiles essential classical theory results, emphasizing the notion of extremal fields. Chapter II explores variational problems on the 2-dimensional torus, examining global minimals and the relationship between minimals and extremal fields, ultimately leading to the concept of Mather sets.