Geometric Mechanics - Part Iii: Broken Symmetry And Composition Of Maps

Why study geometric mechanics? Beyond its mathematical depth and beauty, geometric mechanics provides a robust framework for exploring the geometric structures underpinning many dynamic systems crucial to physics. Principles of Geometric Mechanics is a textbook presented in a lecture notes format, providing precise definitions and practical examples. These notes have evolved over decades of the author's teaching and research, as well as insights from his other influential textbooks and surveys.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.Principles of Geometric Mechanics is designed to facilitate both course learning and individual study. With its focused notes, numerous examples, and exercises set in a modern context, it serves as a valuable resource for students, course instructors and researchers.