Selected Chapters in the Calculus of Variations:
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| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Basel
Birkhäuser Basel
2003
|
| Series: | Lectures in Mathematics. ETH Zürich
|
| Subjects: | |
| Online Access: | Volltext |
| Item Description: | 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 description 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 related and have the same mathematical foundation. We will not follow those approaches but will make a connection to classical results of Jacobi, Legendre, Weierstrass 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 between minimals and extremal fields. In this way, we will be led to Mather sets |
| Physical Description: | 1 Online-Ressource (VI, 134p) |
| ISBN: | 9783034880572 9783764321857 |
| DOI: | 10.1007/978-3-0348-8057-2 |
Staff View
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| 500 | |a 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 description 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 related and have the same mathematical foundation. We will not follow those approaches but will make a connection to classical results of Jacobi, Legendre, Weierstrass 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 between minimals and extremal fields. In this way, we will be led to Mather sets | ||
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| spelling | Moser, Jürgen Verfasser aut Selected Chapters in the Calculus of Variations by Jürgen Moser, Oliver Knill Basel Birkhäuser Basel 2003 1 Online-Ressource (VI, 134p) txt rdacontent c rdamedia cr rdacarrier Lectures in Mathematics. ETH Zürich 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 description 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 related and have the same mathematical foundation. We will not follow those approaches but will make a connection to classical results of Jacobi, Legendre, Weierstrass 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 between minimals and extremal fields. In this way, we will be led to Mather sets Mathematics Global differential geometry Mathematical optimization Calculus of Variations and Optimal Control; Optimization Differential Geometry Mathematik Geodätischer Fluss (DE-588)4156670-1 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 gnd rswk-swf Dynamisches System (DE-588)4013396-5 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 s Dynamisches System (DE-588)4013396-5 s Geodätischer Fluss (DE-588)4156670-1 s 1\p DE-604 Knill, Oliver Sonstige oth https://doi.org/10.1007/978-3-0348-8057-2 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Moser, Jürgen Selected Chapters in the Calculus of Variations Mathematics Global differential geometry Mathematical optimization Calculus of Variations and Optimal Control; Optimization Differential Geometry Mathematik Geodätischer Fluss (DE-588)4156670-1 gnd Variationsrechnung (DE-588)4062355-5 gnd Dynamisches System (DE-588)4013396-5 gnd |
| subject_GND | (DE-588)4156670-1 (DE-588)4062355-5 (DE-588)4013396-5 |
| title | Selected Chapters in the Calculus of Variations |
| title_auth | Selected Chapters in the Calculus of Variations |
| title_exact_search | Selected Chapters in the Calculus of Variations |
| title_full | Selected Chapters in the Calculus of Variations by Jürgen Moser, Oliver Knill |
| title_fullStr | Selected Chapters in the Calculus of Variations by Jürgen Moser, Oliver Knill |
| title_full_unstemmed | Selected Chapters in the Calculus of Variations by Jürgen Moser, Oliver Knill |
| title_short | Selected Chapters in the Calculus of Variations |
| title_sort | selected chapters in the calculus of variations |
| topic | Mathematics Global differential geometry Mathematical optimization Calculus of Variations and Optimal Control; Optimization Differential Geometry Mathematik Geodätischer Fluss (DE-588)4156670-1 gnd Variationsrechnung (DE-588)4062355-5 gnd Dynamisches System (DE-588)4013396-5 gnd |
| topic_facet | Mathematics Global differential geometry Mathematical optimization Calculus of Variations and Optimal Control; Optimization Differential Geometry Mathematik Geodätischer Fluss Variationsrechnung Dynamisches System |
| url | https://doi.org/10.1007/978-3-0348-8057-2 |
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