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Geometric Mechanics on Riemannian Manifolds
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Geometric Mechanics on Riemannian Manifolds

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

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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Caractéristiques techniques

  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

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