Dynamical Systems: Differential Geometric Approach to Symmetry and Reduction

In their discussion of the subject of classical mechanics, the authors of this book use a new and stimulating approach which involves looking at dynamical systems from the viewpoint of differential geometry. They discuss the reduction of these systems, and the role played by symmetry and invariance in such reductions. Central to their approach is the view that symmetry is a tool to be applied to a model system after it has been built or discovered, rather than forming the foundation of such a system. The book is divided into two parts. The first is introductory, dealing with the foundations of mechanics, and describing the construction of a model system from the data available. The second part concentrates on invariance, symmetry and reduction; significantly, it discusses the importance of understanding local (Lie Algebra) and global (Lie Group) symmetry within the framework of the reduction of dynamical systems. 'Digressions' throughout the text explain the mathematical principles behind the concepts described.

Table of Contents:
Preface; FOUNDATIONS OF MECHANICS: Introduction to Part I; A Digression on Manifolds and Diffeomorphisms; Construction of Q: From Observables to the Configuration Space Manifold; Time and Transformations on Time; A Digression on Calculus on Manifolds; From Trajectories to the Linear Field; Lifting to a Carrier Space: Canonical Lifting; A Digression on Sub-manifolds and Smooth Maps; Transformations on TQ; Integrating the Dynamics on TQ: Hamiltonian and Lagrangian Formalisms; From the Tangent Bundle to the Cotangent Bundle; The Canonical Hamiltonian Formalism on TQ; Equivalent Lagrangians and Hamiltonians; Other Carrier Spaces: Action-angle Variables and the Hamilton-Jacobi Method; The Noether Theorem; REDUCTION, ACTIONS OF GROUPS AND ALGEBRAS: Reduction; Introduction to Part II; Linear Dynamical Systems: A Prelude to Reduction; A Digression on Foliations and Distributions; Reduction of Dynamical Systems; Through Regular Foliations; Foliation of Symplectic Manifolds and Reduction of Hamiltonian Systems; Algebra and Group Actions; A Digression on Lie Algebras and their Actions on Manifolds; Actions of Lie Algebras on Symplectic Manifolds; A Digression on Lie Groups and their Action on Manifolds; Actions of Lie Groups on Symplectic Manifolds; Parallelizable Manifolds, Dynamics on Lie Groups; Examples and Applications; Conclusion; References; Further Reading; Index. .pa