# Geometric Mechanics on Riemannian Manifolds

## Applications to Partial Differential Equations

Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

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Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

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.

Calculus of Variations Euler–Lagrange equation Fourier transform Minimal surface Potential differential geometry manifold partial differential equation

- DOI https://doi.org/10.1007/b138771
- Copyright Information Birkhäuser Boston 2005
- Publisher Name Birkhäuser Boston
- eBook Packages Mathematics and Statistics
- Print ISBN 978-0-8176-4354-6
- Online ISBN 978-0-8176-4421-5
- Series Print ISSN 2296-5009
- Series Online ISSN 2296-5017
- Buy this book on publisher's site