Overview
- Introduction to the calculus of variations which leads directly to current research topics
- Combines classical material with modern techniques and results
Part of the book series: Lectures in Mathematics. ETH Zürich (LM)
Access this book
Tax calculation will be finalised at checkout
Other ways to access
About this book
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.
Similar content being viewed by others
Keywords
Table of contents (3 chapters)
-
Front Matter
-
Back Matter
Authors and Affiliations
Bibliographic Information
Book Title: Selected Chapters in the Calculus of Variations
Authors: Jürgen Moser, Oliver Knill
Series Title: Lectures in Mathematics. ETH Zürich
DOI: https://doi.org/10.1007/978-3-0348-8057-2
Publisher: Birkhäuser Basel
-
eBook Packages: Springer Book Archive
Copyright Information: Springer Basel AG 2003
Softcover ISBN: 978-3-7643-2185-7Published: 23 May 2003
eBook ISBN: 978-3-0348-8057-2Published: 06 December 2012
Edition Number: 1
Number of Pages: VI, 134
Number of Illustrations: 11 b/w illustrations, 1 illustrations in colour
Topics: Calculus of Variations and Optimal Control; Optimization, Differential Geometry