Geometric Mechanics on Riemannian Manifolds

Applications to partial Differential Equations

Ovidiu Calin, Der-Chen Chang - Collection Applied and Numerical Harmonic Analysis

278 pages, parution le 27/01/2005

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Résumé

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 Schrodinger'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-Jacob! theory; harmonic functions, maps, and geodesies; 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.

Sommaire

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

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	PAPIER
Éditeur(s)	Birkhäuser
Auteur(s)	Ovidiu Calin, Der-Chen Chang
Collection	Applied and Numerical Harmonic Analysis
Parution	27/01/2005
Nb. de pages	278
Format	16 x 24
Couverture	Relié
Poids	576g
Intérieur	Noir et Blanc
EAN13	9780817643546
ISBN13	978-0-8176-4354-6