Applied and Numerical Harmonic Analysis

Geometric Mechanics on Riemannian Manifolds

Applications to Partial Differential Equations

Authors: Calin, Ovidiu, Chang, Der-Chen

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About this Textbook

Differential geometry techniques have very useful and important applications in partial differential equations and quantum mechanics. This work presents a purely geometric treatment of problems in physics involving quantum harmonic oscillators, quartic oscillators, minimal surfaces, and Schrödinger's, Einstein's and Newton's equations. Historically, problems in these areas were approached using the Fourier transform or path integrals, although in some cases (e.g., the case of quartic oscillators) these methods do not work. New geometric methods are introduced in the work that have the advantage of providing quantitative or at least qualitative descriptions of operators, many of which cannot be treated by other methods. And, conservation laws of the Euler–Lagrange equations are employed to solve the equations of motion qualitatively when quantitative analysis is not possible.

Main topics include: Lagrangian formalism on Riemannian manifolds; energy momentum tensor and conservation laws; Hamiltonian formalism; Hamilton–Jacobi theory; harmonic functions, maps, and geodesics; fundamental solutions for heat operators with potential; and a variational approach to mechanical curves. The text is enriched with good examples and exercises at the end of every chapter.

Geometric Mechanics on Riemannian Manifolds is a fine text for a course or seminar directed at graduate and advanced undergraduate students interested in elliptic and hyperbolic differential equations, differential geometry, calculus of variations, quantum mechanics, and physics. It is also an ideal resource for pure and applied mathematicians and theoretical physicists working in these areas.

Reviews

"...This book, which contains some very interesting ideas and results, is primarily oriented towards graduate or advanced undergraduate students in mathematics and theoretical physics with interests in differential geometry, the calculus of variations and the study of PDE's, as well as in classical and quantum mechanics. In addition, for more experienced researchers in these fields, it may be a useful resource, written in a style that makes it easily accessible to a wide audience..." --- Mathematical Reviews

"The differential operators which are treated in the book are among the most important, not only in the theory of partial differential equation, but they appear naturally in geometry, mechanics or theoretical physics (especially quantum mechanics). Thus, the book should be of interest for anyone working in these fields, from advanced undergraduate students to experts. The book is written in a very pedagogical manner and does not assume many prerequisites, therefore it is quite appropriate to be used for special courses or for self-study. I have to mention that all chapters end with a number of well-chosen exercises that will imporve the understanding of the material and, also, that there are a lot of worked examples that will serve the same purpose." ---Mathematics Vol. L, No. 4

"The book is well written and contains a wealth of material. The authors make a concerted effort to simplify proofs taken from many sources [so] researchers will readily fin dthe infromations they seek, while students can develop their skills by filling in details of proofs, as well as by using the problem sets that end each chapter. The book provides the reader with an in-depth introduction to a rich and rapidly developing research area that has already produced remarkable results.

This book contains old and new basic results from a significant part of the modern theory of partial differential equations on Riemannian manifolds. All results are presented in an elementary way. Only a basic knowledge of basic functional analysis, mechanics and analysis is assumed. The book is well written and contains a wealth of material …. To conclude, this book provides the reader with an in-depth introduction to a rich and rapidly developing research area that has already produced remarkable results." ---Zentralblatt MATH


Table of contents (3 chapters)

Buy this book

eBook 66,99 €
price for Spain (gross)
  • ISBN 978-0-8176-4421-5
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover 84,19 €
price for Spain (gross)
  • ISBN 978-0-8176-4354-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
  • The final prices may differ from the prices shown due to specifics of VAT rules
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Bibliographic Information

Bibliographic Information
Book Title
Geometric Mechanics on Riemannian Manifolds
Book Subtitle
Applications to Partial Differential Equations
Authors
Series Title
Applied and Numerical Harmonic Analysis
Copyright
2005
Publisher
Birkhäuser Basel
Copyright Holder
Birkhäuser Boston
eBook ISBN
978-0-8176-4421-5
DOI
10.1007/b138771
Hardcover ISBN
978-0-8176-4354-6
Series ISSN
2296-5009
Edition Number
1
Number of Pages
XVI, 278
Number of Illustrations and Tables
26 b/w illustrations
Topics