Geometric Mechanics on Riemannian Manifolds: Applications to Partial Differential Equations

ISBN-13: 9780817643546 / Angielski / Twarda / 2004 / 278 str.

Ovidiu Calin; Der-Chen Chang; O. Calin

Opis

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, Schrodinger's, Einstein's and Newton's equations, and others. 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, which have the advantage of providing quantitative or at least qualitative descriptions of operators, many of which cannot be treated by other methods, are introduced. 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 variational approach to mechanical curves. The te

Recenzje

"...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

Spis treści

* Preface * Introductory Chapter * Laplace Operator 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 * Bibliography * Index

Blurb

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.