| Preface | p. ix |
| Introduction | p. 1 |
| The Classical Water Molecule and the Ozone Molecule | p. 1 |
| Hamiltonian Formulation | p. 3 |
| Geometry, Symmetry, and Reduction | p. 9 |
| Stability | p. 12 |
| Geometric Phases | p. 16 |
| The Rotation Group and the Poincare Sphere | p. 23 |
| A Crash Course in Geometric Mechanics | p. 27 |
| Symplectic and Poisson Manifolds | p. 27 |
| The Flow of a Hamiltonian Vector Field | p. 29 |
| Cotangent Bundles | p. 29 |
| Lagrangian Mechanics | p. 31 |
| Lie-Poisson Structures | p. 32 |
| The Rigid Body | p. 33 |
| Momentum Maps | p. 34 |
| Reduction | p. 36 |
| Singularities and Symmetry | p. 39 |
| A Particle in a Magnetic Field | p. 40 |
| Cotangent Bundle Reduction | p. 43 |
| Mechanical G-systems | p. 43 |
| The Classical Water Molecule | p. 46 |
| The Mechanical Connection | p. 50 |
| The Geometry and Dynamics of Cotangent Bundle Reduction | p. 54 |
| Examples | p. 59 |
| Lagrangian Reduction | p. 66 |
| Coupling to a Lie group | p. 72 |
| Relative Equilibria | p. 77 |
| Relative Equilibria on Symplectic Manifolds | p. 77 |
| Cotangent Relative Equilibria | p. 79 |
| Examples | p. 82 |
| The Rigid Body | p. 87 |
| The Energy-Momentum Method | p. 93 |
| The General Technique | p. 93 |
| Example: The Rigid Body | p. 97 |
| Block Diagonalization | p. 101 |
| The Normal Form for the Symplectic Structure | p. 107 |
| Stability of Relative Equilibria for the Double Spherical Pendulum | p. 110 |
| Geometric Phases | p. 115 |
| A Simple Example | p. 115 |
| Reconstruction | p. 117 |
| Cotangent Bundle Phases--a Special Case | p. 119 |
| Cotangent Bundles--General Case | p. 120 |
| Rigid Body Phases | p. 122 |
| Moving Systems | p. 125 |
| The Bead on the Rotating Hoop | p. 127 |
| Stabilization and Control | p. 131 |
| The Rigid Body with Internal Rotors | p. 131 |
| The Hamiltonian Structure with Feedback Controls | p. 132 |
| Feedback Stabilization of a Rigid Body with a Single Rotor | p. 134 |
| Phase Shifts | p. 137 |
| The Kaluza-Klein Description of Charged Particles | p. 141 |
| Optimal Control and Yang-Mills Particles | p. 144 |
| Discrete reduction | p. 147 |
| Fixed Point Sets and Discrete Reduction | p. 149 |
| Cotangent Bundles | p. 155 |
| Examples | p. 157 |
| Sub-Block Diagonalization with Discrete Symmetry | p. 162 |
| Discrete Reduction of Dual Pairs | p. 166 |
| Mechanical Integrators | p. 171 |
| Definitions and Examples | p. 171 |
| Limitations on Mechanical Integrators | p. 175 |
| Symplectic Integrators and Generating Functions | p. 177 |
| Symmetric Symplectic Algorithms Conserve J | p. 178 |
| Energy-Momentum Algorithms | p. 180 |
| The Lie-Poisson Hamilton-Jacobi Equation | p. 182 |
| Example: The Free Rigid Body | p. 186 |
| Variational Considerations | p. 187 |
| Hamiltonian Bifurcation | p. 189 |
| Some Introductory Examples | p. 189 |
| The Role of Symmetry | p. 196 |
| The One to One Resonance and Dual Pairs | p. 202 |
| Bifurcations in the Double Spherical Pendulum | p. 204 |
| Continuous Symmetry Groups and Solution Space Singularities | p. 205 |
| The Poincare-Melnikov Method | p. 207 |
| The Role of Dissipation | p. 217 |
| References | p. 225 |
| Index | p. 250 |
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