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Introduction (by J. Moser translation from the german by O. Knill )

These lecture notes describe a new development in the calculus of variations called Aubry-Mather-Theory . The starting point for the theoretical physicist Aubry was the description of the motion of electrons in a two-dimensional crystal in terms of a simple model. To do so, he investigated a discrete variational problem and the corresponding minimals.

On the other hand, Mather started from a specific class of area-preserving annulus mappings, so called monotone twist maps . These maps appear in mechanics as Poincaré maps. Such maps were studied by Birkhoff during the 1920's in several basic papers. Mather succeeded in 1982 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.

Evenso these two investigations have different motivations, they are closely related and have the same mathematical foundation. In the following, we will now not follow those approaches but will make a connection to classical results of Jacobi, Legendre, Weierstrass and others from the 19'th century. Therefore in chapter~I, we will put together the results of the classical theory which are the for us most important. The notion of extremal fields will be the most relevant one in the following.

In chapter~II we investigate variational problems on the 2-dimensional torus and the corresponding global minimals as well as the relation between minimals and extremal fields. In this way, we will be guided to Mather sets. Finally, in chapter~III, we will learn the connection with monotone twist maps, which was the starting point for Mather's theory. We will so come to a discrete variational problem which is the basis for Aubry's investigations.

This theory has additionally interesting applications in differential geometry, namely for the geodesic flow on two-dimensional surfaces, especially on the torus. In this context the minimal geodesics as investigated by Morse and Hedlund (1932) play a distinguished role.

As Bangert has shown, the theories of Aubry and Mather lead to new results for the geodesic flow on the two-dimensional torus. The restriction to two dimensions is essential as the example in the last section of these lecture notes shows. These differential geometric questions are treated at the end of the third chapter.

The beautiful survey article of Bangert should be at hand with these lecture notes.

Our description aims less towards generality as rather to show the relations of newer developments with classical notions with the extremal fields. Especially, the Mather sets appear like this as 'generalized extremal fields'.

For the production of these lecture notes O. Knill assisted me, to whom I want to express my thanks.

Züurich, September 1988, J. Moser

On these lecture notes

These lectures were given by J. Moser in the spring of 1988 at the ETH Zürich. The students were in the 6.-8'th semester (3'th-4'th year of a 4 year curriculum). There were however also PhD students (graduate students) and visitors of the FIM (research institute at the ETH) in the auditorium.

In the last 12 years since the event the research on this special topic in the calculus of variations has made some progress. A few hints to the literature are attached in an appendix. Because important questions are still open, these lecture notes might maybe be of more than historical value.

In March 2000, I had stumbled over old floppy diskettes which contained the lecture notes which I had written in the summer of 1998 using the text processor 'Signum' on an Atary ST. J. Moser had looked carefully through the lecture notes in September 1988. Because the text editor was pixel based and is now obsolete, the typesetting had to be done new in LaTeX. The original has not been changed except for small, mostly stylistic or typographical corrections.

Austin, TX, June 2000, O. Knill
Cambridge, MA, April 2002, (small corrections during english translation)

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