Geometric Mechanics on Riemannian Manifolds: Applications to Partial Differential Equations (Applied and Numerical Harmonic Analysis)

Calin, Ovidiu

ISBN 10: 0817643540 / ISBN 13: 9780817643546

Editore: Birkh�user, 2004

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Titolo: Geometric Mechanics on Riemannian Manifolds:...

Casa editrice: Birkh�user

Data di pubblicazione: 2004

Legatura: Rilegato

Condizione: Good

Descrizione articolo

Riassunto:

*  A geometric approach to problems in physics, many of which cannot be solved by any other methods

* Text is enriched with good examples and exercises at the end of every chapter

* Fine for a course or seminar directed at grad and adv. undergrad students interested in elliptic and hyperbolic differential equations, differential geometry, calculus of variations, quantum mechanics, and physics

Dalla quarta di copertina:

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 EulerLagrange 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; HamiltonJacobi 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.