Selected Chapters in the Calculus of Variations

Häftad, Engelska, 2003

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These lecture notes describe the Aubry-Mather-Theory within the calculus of variations. The text consists of the translated original lectures of Jurgen Moser and a bibliographic appendix with comments on the current state of the art in this field of interest. Students will find a rapid introduction to the calculus of variations, leading to modern dynamical systems theory. Differential geometric applications are discussed, in particular billiards and minimal geodesics on the two-dimensional torus. Many exercises and open questions make this book a valuable resource for both teaching and research.

Produktinformation

Utgivningsdatum2003-05-23
Mått170 x 240 x 12 mm
Vikt264 g
FormatHäftad
SpråkEngelska
SerieLectures in Mathematics. ETH Zürich
Antal sidor134
FörlagBirkhauser Verlag AG
ISBN9783764321857

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1 One-dimensional variational problems.- 1.1 Regularity of the minimals.- 1.2 Examples.- 1.3 The accessory variational problem.- 1.4 Extremal fields for n=1.- 1.5 The Hamiltonian formulation.- 1.6 Exercises to Chapter 1.- 2 Extremal fields and global minimals.- 2.1 Global extremal fields.- 2.2 An existence theorem.- 2.3 Properties of global minimals.- 2.4 A priori estimates and a compactness property.- 2.5 Ma for irrational a, Mather sets.- 2.6 Ma for rational a.- 2.7 Exercises to chapter II.- 3 Discrete Systems, Applications.- 3.1 Monotone twist maps.- 3.2 A discrete variational problem.- 3.3 Three examples.- 3.4 A second variational problem.- 3.5 Minimal geodesics on T2.- 3.6 Hedlund’s metric on T3.- 3.7 Exercises to chapter III.- A Remarks on the literature.- Additional Bibliography.