Introduction to Dynamical Systems and Geometric Mechanics


hardcover, 360 p.  published 2012.

This comprehensive text offers a tour de force through dynamical systems and geometric mechanics.  Constituting the first meeting of these closely related topics under a single cover, this text can be used as a one or two semester graduate-level course introducing students to these wonderful fields.  The book culminates with two chapters on a unified geometric treatment of nonholonomic mechanics, including both moving-frame and fiber-bundle descriptions of mechanics.  The author shares many illuminating insights on the details of dynamical systems and geometric mechanics that will be appreciated by the novice and expert alike.  Each chapter concludes with one or more application that can be used as a springboard for student investigation or class projects.  This text is bound to make a handsome addition to any personal collection.

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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

Additional Information

Weight 1.55 lbs
Dimensions 6.14 x 9.21 x 0.875 in


publication date

April, 2012.


xiv + 360


hardcover, Single-User eBook License


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