About the Book
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. This represents a significant conceptual departure from more traditional approaches based on the use of local coordinates.
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. This geometric perspective natively avoids the difficulties associated with coordinate singularities (such as gimbal lock) that frequently plague standard analytical mechanics.
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 formulations. Understanding of the material is enhanced by numerous in-depth examples throughout the book, culminating in non-trivial applications involving multi-body systems.
Key Features
- Global Perspective: Emphasizes formulations that hold everywhere on the configuration manifold, bypassing the limitations of local coordinate charts.
- Geometric Mechanics Toolkit: Introduces essential differential geometric concepts (embedded manifolds, tangent/cotangent vectors, Lie groups) directly within the context of physical problems.
- Avoiding Singularities: Demonstrates how to write globally valid equations of motion that are immune to coordinate singularities, highly relevant for aerospace, robotics, and computational control.
- Pedagogical Structure: Written for a broad audience of mathematicians, engineers, and physicists. Basic background in differential geometry is helpful but not essential, as relevant concepts are built from the ground up.
- Computational Relevance: Lays the rigorous continuous-time foundation required for constructing structure-preserving geometric integrators and geometric control algorithms.