Introduction to Dynamical Systems and Geometric Mechanics
by Jared M. Maruskin
360 pages, 120 illustrations.
published April 2012.
ISBN: 978-0-9850627-0-5 (hardcover)
About this Book
This book provides a comprehensive introduction to the fields of dynamical systems and geometric mechanics in a single volume. The exposition is concise and clear, and is supplemented with a number of illuminating examples and exercises. The text offers an ample amount of flexibility: core sections from both parts of the text may be reasonably digested in a single-semester introductory graduate-level course, or the text may be covered in more detail over a two-semester course. The text also discusses several new results that have yet to appear in any other textbook, and each chapter concludes with an application that can serve as a springboard project for further investigation, projects, or in-class discussion.
In the first part of the text, we discuss 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 text 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 final chapters cover Lagrangian and Hamiltonian mechanics from a modern geometric perspective, mechanics on Lie groups, and nonholonomic mechanics via both moving frames and fiber bundle decompositions
Table of Contents (Back to Top)
Part I: Dynamical Systems
Chapter 1 Linear Systems
1.1 Eigenvector Approach
1.2 Matrix Exponentials
1.3 Matrix Representation of Solutions
1.4 Stability Theory
1.5 Fundamental Matrix Solutions
1.6 Nonhomogeneous and Nonautonomous Solutions
1.7 Application: Linear Control Theory
Chapter 2 Linearization of Trajectories
2.1 Introduction and Numerical Simulation
2.2 Linearization of Trajectories
2.3 Stability of Trajectories
2.4 Lyapunov Exponents
2.5 Linearization and Stability of Fixed Points
2.6 Dynamical Systems in Mechanics
2.7 Application: Elementary Astrodynamics
2.8 Application: Planar Circular Restricted Three Body Problem
Chapter 3 Invariant Manifolds
3.1 Asymptotic Behavior of Trajectories
3.2 Invariant Manifolds in R^n
3.3 The Stable Manifold Theorem
3.4 The Contraction Mapping Theorem
3.5 The Graph Transform Method
3.6 The Central Manifold Theorem
3.7 Application: Stability in Rigid-Body Dynamics
Chapter 4 Periodic Orbits
4.1 Summation Notation
4.2 Poincaré Maps
4.3 Poincaré Reduction of the State-Transition Matrix
4.4 Invariant Manifolds of Periodic Orbits
4.5 Families of Periodic Orbits
4.6 Floquet Theory
4.7 Application: Periodic Orbit Families in the Hill Problem
Chapter 5 Bifurcations and Chaos
5.1 The Poincaré-Bendixson Theorem
5.2 Bifurcation and Hysteresis
5.3 Period-Doubling Bifurcations
5.4 Chaos
5.5 Application: Billiards
Part II: Geometric Mechanics
Chapter 6 Differentiable Manifolds
6.1 Differentiable Manifolds
6.2 Vectors on Manifolds
6.3 Mappings
6.4 Vector Fields and Flows
6.5 Jacobi-Lie Bracket
6.6 Differential Forms
6.7 Riemannian Geometry
6.8 Application: The Foucault Pendulum
Chapter 7 Lagrangian Mechanics
7.1 Hamilton's Principle
7.2 Variations of Curves and Virtual Displacements
7.3 Euler-Lagrange Equation
7.4 Distributions and Frobenius' Theorem
7.5 Nonholonomic Mechanics
7.6 Application: Nöther's Theorem
Chapter 8 Hamiltonian Mechanics
8.1 The Legendre Transformation
8.2 Hamilton's Equations of Motion
8.3 Hamiltonian Vector Fields and Conserved Quantities
8.4 Routh's Equations
8.5 Symplectic Manifolds
8.6 Symplectic Invariants
8.7 Application: Optimal Control and Pontryagin's Principle
8.8 Application: Symplectic Probability Propagation
Chapter 9 Lie Groups and Rigid Body Mechanics
9.1 Lie Groups and Their Lie Algebras
9.2 Left Translations and Adjoints
9.3 Euler-Poincare Equations
9.4 Application: Rigid Body Mechanics
9.5 Application: Linearization of Hamiltonian Systems
Chapter 10 Moving Frames and Nonholonomic Mechanics
10.1 Quasivelocities and Moving Frames
10.2 A Lie Algebra Bundle
10.3 Maggi's Equation
10.4 Hamel's Equation
10.5 Relation Between the Hamel and Euler-Poincare Equations
10.6 Application: Constrained Optimal Control
Chapter 11 Fiber Bundles and Nonholonomic Mechanics
11.1 Fiber Bundles
11.2 The Transpositional Relation and Suslov's Principle
11.3 Voronets' Equation
11.4 Combined Hamel-Suslov Approach
11.5 Application: Rolling-Without-Slipping Constraints
Ebook Edition (Back to Top)
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ISBN 978-0-9850627-1-2 (ebook)
