# Geometric Mechanics on Riemannian Manifolds

## Applications to Partial Differential Equations

• Ovidiu Calin
• Der-Chen Chang Textbook

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

1. Front Matter
Pages i-xv
2. Pages 1-16
3. Pages 17-31
4. Pages 33-54
5. Pages 55-65
6. Pages 67-96
7. Pages 97-111
8. Pages 113-135
9. Pages 137-147
10. Pages 149-173
11. Pages 237-250
12. Pages 251-270
13. Back Matter
Pages 271-279

### Introduction

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.

### Keywords

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

#### Authors and affiliations

• Ovidiu Calin
• 1
• Der-Chen Chang
• 2
1. 1.Department of MathematicsEastern Michigan UniversityYpsilantiUSA
2. 2.Department of MathematicsGeorgetown UniversityWahington, DCUSA