GEOMETRIC MECHANICS (PART III)

Geometric Mechanics: Part III is a textbook presented in a lecture notes format, providing precise definitions and practical examples across a series of 31 lectures that have been developed from the author's extensive experience of teaching and research. Geometric mechanics is an incredibly rich field of study: beyond its mathematical depth and beauty, it provides a robust framework for exploring the geometric structures underpinning many dynamical systems crucial to physics.

The first part introduces undergraduate mathematics and physics students to the applications of geometric mechanics in finite dimensional dynamical systems of ordinary differential equations. The second part covers the essential theory of manifolds and Lie groups to prepare senior undergraduates and graduate students for the modern applications of geometric mechanics. These applications are introduced in the third part, which delves into the geometric mechanics of partial differential equations that govern the dynamics of ideal continuum mechanics, including fluids and plasmas, at the cutting edge of current research.

This textbook is designed to facilitate both course learning and individual study. With focused notes, numerous examples, and nearly 200 exercises, it serves as a valuable resource for postgraduate students, course instructors, and researchers.

Contents:

• Basic Elements:
  • Introduction
  • Counterpoints between Mathematics and Physics
  • Particle Mechanics of Newton, Lagrange and Hamilton
  • Matrix Lie Groups and Lie Algebras
  • The Rigid Body in ℝ³
  • Broken Symmetry: Heavy Top Equations
  • Lagrangian and Hamiltonian Methods for Geometric Ray Optics in Translation-Invariant, Axisymmetric Material
  • Rigid Body Equations on SO(n)
  • Exercises: Inside the Geometric Mechanics Cube
• Geometric Mechanics on Manifolds:
  • Geometric Structure of Classical Mechanics
  • Introduction to Vector Fields
  • Derivatives of Differentiable Maps: The Tangent Lift
  • Lifted Actions and the Jacobi–Lie Bracket on Vector Fields
  • Lie Group Action on Its Tangent Bundle
  • Hamilton's Principle on Manifolds
  • Euler–Lagrange Equations on Manifolds
  • Momentum Maps
  • Hamiltonian Vector Fields and Differential Forms
  • More About Vector Fields and Differential Forms
  • Euler–Poincaré Reduction Theorem
  • EPDiff: An Euler–Poincaré Equation on the Diffeomorphisms
  • EPDiff Solution Behaviour in 1D
  • Diffeons: Singular Momentum Solutions of the EPDiff Equation for Geodesic Motion in Higher Dimensions
  • The Geometry of the Momentum Map
• Euler–Poincaré Framework of Continuum Partial Differential Equations:
  • Euler–Poincaré Framework of Fluid Dynamics
  • Euler–Poincaré Theory of Geophysical Fluid Dynamics
  • Five More Continuum Applications
  • Dispersive Shallow Water (DSW) Equations in 1D and 2D
  • Rotating Shallow Magnetised Water (RSW-MHD)
  • Incompressible 2D MHD Alfvén Wave Turbulence
  • Lüst Hall Magnetohydrodynamics
  • Appendix A Geometric Mechanics: Definitions and Topics

Readership: This book is suitable for adoption for courses in mathematics, physics and mathematical physics, as well as for related fields, including robotics, control theory, engineering, and computing. It is also an excellent resource for self study for advanced undergraduates and graduate students in these areas.