Introduction to Dynamical Systems and Geometric Mechanics provides a comprehensive tour of two fields that are intimately entwined: dynamical systems is the study of the behavior of physical systems that may be described by a set of nonlinear first-order ordinary differential equations in Euclidean space, whereas geometric mechanics explore similar systems that instead evolve on differentiable manifolds. The first part discusses the linearization and stability of trajectories and fixed points, invariant manifold theory, periodic orbits, Poincaré maps, Floquet theory, the Poincaré-Bendixson theorem, bifurcations, and chaos. The second part of the book begins with a self-contained chapter on differential geometry that introduces notions of manifolds, mappings, vector fields, the Jacobi-Lie bracket, and differential forms. Dynamical Systems and Geometric Mechanics: An Introduction, 2nd edition 4 Contents 6 Preface 10 Theme 10 Map of the Book 11 Background and References 11 Acknowledgments 12 Part I: Dynamical Systems 14 1 Linear Systems 16 1.1 Eigenvector Approach 16 1.2 Matrix Exponentials 24 1.3 Matrix Representation of Solutions 28 1.4 Stability Theory 37 1.5 Fundamental Matrix Solutions 41 1.6 Nonhomogeneous and Nonautonomous Systems 48 1.7 Application: Linear Control Theory 54 2 Linearization of Trajectories 58 2.1 Introduction and Numerical Simulation 58 2.2 Linearization of Trajectories 60 2.3 Stability of Trajectories 63 2.4 Lyapunov Exponents 64 2.5 Linearization and Stability of Fixed Points 69 2.6 Dynamical Systems in Mechanics 79 2.7 Application: Elementary Astrodynamics 81 2.8 Application: Planar Circular Restricted Three-Body Problem 84 3 Invariant Manifolds 94 3.1 Asymptotic Behavior of Trajectories 94 3.2 Invariant Manifolds in Rn 98 3.3 Stable Manifold Theorem 101 3.4 Contraction Mapping Theorem 105 3.5 Graph Transform Method 108 3.6 Center Manifold Theory 114 3.7 Application: Stability in Rigid-Body Dynamics 118 4 Periodic Orbits 124 4.1 Summation Notation 124 4.2 Poincaré Maps 125 4.3 Poincaré Reduction of the State-Transition Matrix 129 4.4 Invariant Manifolds of Periodic Orbits 132 4.5 Families of Periodic Orbits 134 4.6 Floquet Theory 135 4.7 Application: Periodic Orbit Families in the Hill Problem 139 5 Bifurcations and Chaos 144 5.1 Poincaré–Bendixson Theorem 144 5.2 Bifurcation and Hysteresis 147 5.3 Period Doubling Bifurcations 151 5.4 Chaos 154 5.5 Application: Billiards 156 Part II: Geometric Mechanics 162 6 Differentiable Manifolds 164 6.1 Differentiable Manifolds 164 6.2 Vectors on Manifolds 166 6.3 Mappings 171 6.4 Vector Fields and Flows 172 6.5 Jacobi–Lie Bracket 174 6.6 Differential Forms 180 6.7 Riemannian Geometry 183 6.8 Application: The Foucault Pendulum 190 6.9 Application: General Relativity 192 7 Lagrangian Mechanics 196 7.1 Hamilton’s Principle 196 7.2 Variations of Curves and Virtual Displacements 199 7.3 Euler–Lagrange Equation 201 7.4 Distributions and Frobenius’ Theorem 205 7.5 Mechanical Systems with Holonomic Constraints 208 7.6 Nonholonomic Mechanics 209 7.7 Application: Nöther’s Theorem 221 8 Hamiltonian Mechanics 226 8.1 Legendre Transform 226 8.2 Hamilton’s Equations of Motion 229 8.3 Hamiltonian Vector Fields and Conserved Quantities 231 8.4 Routh’s Equations 235 8.5 Symplectic Manifolds 237 8.6 Symplectic Invariants 242 8.7 Application: Optimal Control and Pontryagin’s Principle 247 8.8 Application: Symplectic Probability Propagation 253 9 Lie Groups and Rigid-Body Mechanics 256 9.1 Lie Groups and Their Lie Algebras 256 9.2 Left Translations and Adjoints 261 9.3 Euler–Poincaré Equation 266 9.4 Application: Rigid-Body Mechanics 269 9.5 Application: Linearization of Hamiltonian Systems 280 10 Moving Frames and Nonholonomic Mechanics 284 10.1 Quasivelocities and Moving Frames 284 10.2 A Lie Algebra Bundle 287 10.3 Maggi’s Equation 290 10.4 Hamel’s Equation 292 10.5 Relation between the Hamel and Euler–Poincaré Equations 302 10.6 Application: Constrained Optimal Control 304 11 Fiber Bundles and Nonholonomic Mechanics 310 11.1 Fiber Bundles 311 11.2 The Transpositional Relation and Suslov’s Principle 315 11.3 Voronets’ Equation 318 11.4 Combined Hamel–Suslov Approach 322 11.5 Application: Rolling-Without-Slipping Constraints 326 Bibliography 334 Index 346 Back Matter 350
Introduction to Dynamical Systems and Geometric Mechanics provides a comprehensive tour of two fields that are intimately entwined: dynamical systems is the study of the behavior of physical systems that may be described by a set of nonlinear first-order ordinary differential equations in Euclidean space, whereas geometric mechanics explore similar systems that instead evolve on differentiable manifolds.
The first part discusses the linearization and stability of trajectories and fixed points, invariant manifold theory, periodic orbits, Poincaré maps, Floquet theory, the Poincaré-Bendixson theorem, bifurcations, and chaos. The second part of the book begins with a self-contained chapter on differential geometry that introduces notions of manifolds, mappings, vector fields, the Jacobi-Lie bracket, and differential forms.
The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer
