0.1 Introduction These lecture notes describe a new development in the
calculus of variations which is called Aubry-Mather-Theory. The starting
point for the theoretical physicist Aubry was a model for the descrip-
tion of the motion of electrons in a two-dimensional crystal. Aubry
investigated a related discrete variational problem and the
corresponding minimal solutions. On the other hand, Mather started with
a specific class of area-preserving annulus mappings, the so-called
monotone twist maps. These maps appear in mechanics as Poincare maps.
Such maps were studied by Birkhoff during the 1920s in several papers.
In 1982, Mather succeeded to make essential progress in this field and
to prove the existence of a class of closed invariant subsets which are
now called Mather sets. His existence theorem is based again on a
variational principle. Although these two investigations have different
motivations, they are closely re- lated and have the same mathematical
foundation. We will not follow those ap- proaches but will make a
connection to classical results of Jacobi, Legendre, Weier- strass and
others from the 19th century. Therefore in Chapter I, we will put
together the results of the classical theory which are the most
important for us. The notion of extremal fields will be most relevant.
In Chapter II we will investigate variational problems on the
2-dimensional torus. We will look at the corresponding global minimals
as well as at the relation be- tween minimals and extremal fields. In
this way, we will be led to Mather sets.