Geometry Of Mechanics

This book sets out to demonstrate, interpret, and analyse the geometrical structures underlying classical mechanics. Through exploring the applications of these structures in the context of dissipative, autonomous, and nonautonomous conservative dynamical systems, a series of insightful exercises are developed in order to both consolidate and clarify the theoretical concepts introduced.Geometry of Mechanics provides an informative exploration of the classic geometrical structures, including symplectic structure, Lagrangian and Hamiltonian formalisms, and the Riemannian structure of systems. Lesser-known frameworks are also investigated, such as the (Skinner-Rusk) unified Lagrangian-Hamiltonion formalism, the geometric proof of Lee Hwa Chung's invariance theorem, and a new geometric formulation of the Hamilton-Jacobi equation, among others. Although the primary focus of this exposition is upon the regular case, singular systems are also considered and explained where applicable.Each chapter concludes with a set of problems, some of which are intended to be solved solely by the application of results presented, while others contain instructive results complementary to those presented in the chapters, complete with comments, suggestions, and recommendations. Interested readers will also find an extensive and state of the art bibliography, including a great number of works produced in recent decades related to all topics explored.