Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds: A Geometric Approach to Modeling and Analysis (Interaction of Mechanics and Mathematics)

This book provides an accessible introduction to the variational formulation of Lagrangian and Hamiltonian mechanics, with a novel emphasis on global descriptions of the dynamics, which is a significant conceptual departure from more traditional approaches based on the use of local coordinates on the configuration manifold. In particular, we introduce a general methodology for obtaining globally valid equations of motion on configuration manifolds that are Lie groups, homogeneous spaces, and embedded manifolds, thereby avoiding the difficulties associated with coordinate singularities. The material is presented in an approachable fashion by considering concrete configuration manifolds of increasing complexity, which then motivates and naturally leads to the more general formulation that follows. Understanding of the material is enhanced by numerous in-depth examples throughout the book, culminating in non-trivial applications involving multi-body systems. This book is written for a general audience of mathematicians, engineers, and physicists with a basic knowledge of mechanics. Some basic background in differential geometry is helpful, but not essential, as the relevant concepts are introduced in the book, thereby making the material accessible to a broad audience, and suitable for either self-study or as the basis for a graduate course in applied mathematics, engineering, or physics. Read more... Abstract: This book provides an accessible introduction to the variational formulation of Lagrangian and Hamiltonian mechanics, with a novel emphasis on global descriptions of the dynamics, which is a significant conceptual departure from more traditional approaches based on the use of local coordinates on the configuration manifold. In particular, we introduce a general methodology for obtaining globally valid equations of motion on configuration manifolds that are Lie groups, homogeneous spaces, and embedded manifolds, thereby avoiding the difficulties associated with coordinate singularities. The material is presented in an approachable fashion by considering concrete configuration manifolds of increasing complexity, which then motivates and naturally leads to the more general formulation that follows. Understanding of the material is enhanced by numerous in-depth examples throughout the book, culminating in non-trivial applications involving multi-body systems. This book is written for a general audience of mathematicians, engineers, and physicists with a basic knowledge of mechanics. Some basic background in differential geometry is helpful, but not essential, as the relevant concepts are introduced in the book, thereby making the material accessible to a broad audience, and suitable for either self-study or as the basis for a graduate course in applied mathematics, engineering, or physics Front Matter ....Pages i-xxvii Mathematical Background (Taeyoung Lee, Melvin Leok, N. Harris McClamroch)....Pages 1-51 Kinematics (Taeyoung Lee, Melvin Leok, N. Harris McClamroch)....Pages 53-88 Classical Lagrangian and Hamiltonian Dynamics (Taeyoung Lee, Melvin Leok, N. Harris McClamroch)....Pages 89-129 Lagrangian and Hamiltonian Dynamics on \((\mathsf{S}^{1})^{n}\) (Taeyoung Lee, Melvin Leok, N. Harris McClamroch)....Pages 131-206 Lagrangian and Hamiltonian Dynamics on \((\mathsf{S}^{2})^{n}\) (Taeyoung Lee, Melvin Leok, N. Harris McClamroch)....Pages 207-271 Lagrangian and Hamiltonian Dynamics on \(\mathsf{SO(3)}\) (Taeyoung Lee, Melvin Leok, N. Harris McClamroch)....Pages 273-311 Lagrangian and Hamiltonian Dynamics on \(\mathsf{SE(3)}\) (Taeyoung Lee, Melvin Leok, N. Harris McClamroch)....Pages 313-346 Lagrangian and Hamiltonian Dynamics on Manifolds (Taeyoung Lee, Melvin Leok, N. Harris McClamroch)....Pages 347-398 Rigid and Multi-Body Systems (Taeyoung Lee, Melvin Leok, N. Harris McClamroch)....Pages 399-484 Deformable Multi-Body Systems (Taeyoung Lee, Melvin Leok, N. Harris McClamroch)....Pages 485-520 Back Matter ....Pages 521-539