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© 2013

Differential Geometry and Mathematical Physics

Part I. Manifolds, Lie Groups and Hamiltonian Systems

Benefits

  • Provides profound yet compact knowledge in manifolds, tensor fields, differential forms, Lie groups, G-manifolds and symplectic algebra and geometry for theoretical physicists

  • Prepares the reader to access the research literature in Hamiltonian mechanics and related areas

  • Complete account to Marsden-Weinstein reduction, including the singular case

  • Detailed examples for all methods

Book

Part of the Theoretical and Mathematical Physics book series (TMP)

Table of contents

  1. Front Matter
    Pages I-XIII
  2. Gerd Rudolph, Matthias Schmidt
    Pages 1-51
  3. Gerd Rudolph, Matthias Schmidt
    Pages 53-92
  4. Gerd Rudolph, Matthias Schmidt
    Pages 93-164
  5. Gerd Rudolph, Matthias Schmidt
    Pages 165-217
  6. Gerd Rudolph, Matthias Schmidt
    Pages 219-267
  7. Gerd Rudolph, Matthias Schmidt
    Pages 269-314
  8. Gerd Rudolph, Matthias Schmidt
    Pages 315-352
  9. Gerd Rudolph, Matthias Schmidt
    Pages 353-425
  10. Gerd Rudolph, Matthias Schmidt
    Pages 427-490
  11. Gerd Rudolph, Matthias Schmidt
    Pages 491-567
  12. Gerd Rudolph, Matthias Schmidt
    Pages 569-640
  13. Gerd Rudolph, Matthias Schmidt
    Pages 641-727
  14. Back Matter
    Pages 729-759

About this book

Introduction

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.

Keywords

Analysis on Manifolds Differential Geometry Applied Hamilton-Jacobi Theory Hamiltonian Systems Integrable Systems Lie Groups Applied Manifold Symmetries and Reduction Symplectic Geometry Symplectic Reduction

Authors and affiliations

  1. 1.Institute for Theoretical PhysicsUniversity of LeipzigLeipzigGermany
  2. 2.Institute for Theoretical PhysicsUniversity of LeipzigLeipzigGermany

Bibliographic information

Industry Sectors
Energy, Utilities & Environment

Reviews

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)