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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