03882nam a22005175i 4500 978-0-8176-4421-5 DE-He213 20260521092025.0 cr nn 008mamaa 100301s2005 xxu| s |||| 0|eng d 9780817644215 99780817644215 10.1007/b138771 doi 516.36 23 Calin, Ovidiu. author. Geometric Mechanics on Riemannian Manifolds [electronic resource] : Applications to Partial Differential Equations / by Ovidiu Calin, Der-Chen Chang. Boston, MA : Birkhäuser Boston, 2005. XV, 278 p. online resource. text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda Applied and Numerical Harmonic Analysis 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. 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. 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. Chang, Der-Chen. author. SpringerLink (Online service) Springer eBooks Printed edition: 9780817643546 Applied and Numerical Harmonic Analysis http://dx.doi.org/10.1007/b138771 Ver el texto completo en las instalaciones del CICY ZDB-2-SMA ddc ER 35555 35555 0 0 ddc 0 0 LE CICY CICY EL 2025-10-06 0 516.36 2025-10-06 08:44:34 2025-10-06 ER