Differential Geometry and Mathematical Physics: Part I. Manifolds, Lie Groups and Hamiltonian Systems

ISBN-13: 9789400753440 / Angielski / Twarda / 2012 / 762 str.

Gerd Rudolph; Matthias Schmidt

Opis

Starting from undergraduate level, this book systematically develops the basics of - Analysis on manifolds, Lie groups and G-manifolds (including equivariant dynamics) - Symplectic algebra and geometry, Hamiltonian systems, symmetries and reduction, - Integrable systems, Hamilton-Jacobi theory (including Morse families, the Maslov class and caustics). The first item is relevant for virtually all areas of mathematical physics, while the second item provides the basis of Hamiltonian mechanics. The last item introduces to important special areas. Necessary background knowledge on topology is provided in an appendix. The aim of this book is to enable the reader to access research monographs on more advanced topics. The style of this book is that of a mathematics textbook, with full proofs given in the text or as exercises. All material is illustrated by detailed examples, a number of which is taken up repeatedly for demonstrating how the methods evolve and interact.

Recenzje

From the reviews:

"The book is the first of two volumes on differential geometry and mathematical physics. The present volume deals with manifolds, Lie groups, symplectic geometry, Hamiltonian systems and Hamilton-Jacobi theory. ... There are several examples and exercises scattered throughout the book. The presentation of material is well organized and clear. The reading of the book gives real satisfaction and pleasure since it reveals deep interrelations between pure mathematics and theoretical physics." (Tomasz Rybicki, Mathematical Reviews, October, 2013)

Spis treści

1 Differentiable manifolds.-  2 Vector bundles.-  3 Vector fields.-  4 Differential forms.-  5 Lie groups.-  6 Lie group actions.-  7 Linear symplectic algebra.-  8 Symplectic geometry.-  9 Hamiltonian systems.-  10 Symmetries.- 11 Integrability.- 12 Hamilton-Jacobi theory.-  References

O autorze

Dr. Matthias Schmidt ist selbständiger Unternehmensberater für "Effektive Unternehmensethik" und Mitglied des Vorstands der Wirtschaftsjunioren Kaiserslautern.

Blurb

Starting from an undergraduate level, this book systematically develops the basics of

• Calculus on manifolds, vector bundles, vector fields and differential forms,

• Lie groups and Lie group actions,

• Linear symplectic algebra and symplectic geometry,

• Hamiltonian systems, symmetries and reduction, integrable systems and Hamilton-Jacobi theory.

The topics listed under the first item are relevant for virtually all areas of mathematical physics. The second and third items constitute the link between abstract calculus and the theory of Hamiltonian systems. The last item provides an introduction to various aspects of this theory, including Morse families, the Maslov class and caustics.

The book guides the reader from elementary differential geometry to advanced topics in the theory of Hamiltonian systems with the aim of making current research literature accessible. The style is that of a mathematical textbook,with full proofs given in the text or as exercises. The material is illustrated by numerous detailed examples, some of which are taken up several times for demonstrating how the methods evolve and interact.