Applications to Partial Differential Equations Calin, Ovidiu. creator author. Chang, Der-Chen. author. SpringerLink (Online service) text xxu 2005 monographic eng

electronic

electronic resource

access XV, 278 p. online resource. 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. 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. by Ovidiu Calin, Der-Chen Chang. MATHEMATICS HARMONIC ANALYSIS DIFFERENTIAL EQUATIONS, PARTIAL GLOBAL DIFFERENTIAL GEOMETRY MATHEMATICAL PHYSICS MATHEMATICS DIFFERENTIAL GEOMETRY PARTIAL DIFFERENTIAL EQUATIONS MATHEMATICAL METHODS IN PHYSICS ABSTRACT HARMONIC ANALYSIS APPLICATIONS OF MATHEMATICS 516.36 9780817644215 99780817644215 http://dx.doi.org/10.1007/b138771 http://dx.doi.org/10.1007/b138771 100301 20260521092025.0 978-0-8176-4421-5