Geometric Mechanics on Riemannian Manifolds - Applications to Partial Differential Equations

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.
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. The text is enriched with good examples and exercises at the end of every chapter. It is also an ideal resource for pure and applied mathematicians and theoretical physicists working in these areas.

List of contents

Introductory Chapter.- Laplace Operators on Riemannian Manifolds.- Lagrangian Formalism on Riemannian Manifolds.- Harmonic Maps from a Lagrangian Viewpoint.- Conservation Theorems.- Hamiltonian Formalism.- Hamilton-Jacobi Theory.- Minimal Hypersurfaces.- Radially Symmetric Spaces.- Fundamental Solutions for Heat Operators with Potentials.- Fundamental Solutions for Elliptic Operators.- Mechanical Curves.

Summary

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.

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. The text is enriched with good examples and exercises at the end of every chapter. It is also an ideal resource for pure and applied mathematicians and theoretical physicists working in these areas.

Additional text

"...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

Report

"...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

Product details

• Authors: Ovidi Calin, Ovidiu Calin, Der-Chen Chang, Der-Cheng Chang
• Publisher: Springer, Basel
• Languages: English
• Product format: Hardback
• Released: 05.01.2005
• EAN: 9780817643546
• ISBN: 978-0-8176-4354-6
• No. of pages: 278
• Weight: 575 g
• Illustrations: XVI, 278 p. 26 illus.
• Series: Applied and Numerical Harmonic Analysis; Applied and Numerical Harmonic Analysis; Applied and Numerical Harmonic
• Subject: Natural sciences, medicine, IT, technology > Mathematics > Analysis