Dynamical systems and geometric mechanics. An introduction
The author's goal in this book is to present a broad perspective on the fields of dynamical systems and geometric mechanics. He accomplishes this using an introductory approach that could be used for a course for graduate students or advanced undergraduates, and in principle completed in a single semester. The intended audience includes students in mathematics, physics and engineering. Previous course work experience in linear algebra and differential equations is highly desirable. Exposure to dynamical systems at the level of two-dimensional systems and phase planes would also be helpful. The author has attempt to make the book self-contained, but a fair degree of mathematical maturity would be necessary for the less-experienced reader. The book is divided into two parts. The first focuses on first-order autonomous differential equations in \(\mathbb{R}^n\); the author calls this ``dynamical systems''. The second part -- on ``geometric mechanics'' -- moves to first-order equations on smooth differentiable manifolds with their associated vector fields. The author includes a considerable number of applications part. Many of these pertain to mechanics. Topics in the first part include linear systems, nonlinear systems and their linearization, invariant manifolds (including the stable manifold and center manifold theorems), periodic orbits, bifurcations and chaos. The second part starts with background material on differentiable manifolds, then moves on to Lagrangian and Hamiltonian mechanics. Lie groups and Lie agebras are treated next with applications for rigid body mechanics. The final two chapters take up nonholonomic mechanics. The first one concentrates on quasi-velocities and moving frames, with an application to constrained optimal control. The second introduces fiber bundles and provides an application of rolling-without-slipping constraints. This book is exceptionally well-written. It also has an extensive and excellent bibliography.
scientific article; zbMATH DE number 6920778