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| Preface | |
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| Introduction | |
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| The Classical Water Molecule and the Ozone Molecule | |
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| Hamiltonian Formulation | |
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| Geometry, Symmetry, and Reduction | |
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| Stability | |
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| Geometric Phases | |
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| The Rotation Group and the Poincare Sphere | |
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| A Crash Course in Geometric Mechanics | |
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| Symplectic and Poisson Manifolds | |
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| The Flow of a Hamiltonian Vector Field | |
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| Cotangent Bundles | |
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| Lagrangian Mechanics | |
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| Lie-Poisson Structures | |
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| The Rigid Body | |
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| Momentum Maps | |
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| Reduction | |
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| Singularities and Symmetry | |
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| A Particle in a Magnetic Field | |
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| Cotangent Bundle Reduction | |
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| Mechanical G-systems | |
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| The Classical Water Molecule | |
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| The Mechanical Connection | |
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| The Geometry and Dynamics of Cotangent Bundle Reduction | |
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| Examples | |
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| Lagrangian Reduction | |
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| Coupling to a Lie group | |
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| Relative Equilibria | |
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| Relative Equilibria on Symplectic Manifolds | |
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| Cotangent Relative Equilibria | |
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| Examples | |
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| The Rigid Body | |
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| The Energy-Momentum Method | |
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| The General Technique | |
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| Example: The Rigid Body | |
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| Block Diagonalization | |
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| The Normal Form for the Symplectic Structure | |
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| Stability of Relative Equilibria for the Double Spherical Pendulum | |
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| Geometric Phases | |
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| A Simple Example | |
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| Reconstruction | |
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| Cotangent Bundle Phases--a Special Case | |
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| Cotangent Bundles--General Case | |
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| Rigid Body Phases | |
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| Moving Systems | |
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| The Bead on the Rotating Hoop | |
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| Stabilization and Control | |
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| The Rigid Body with Internal Rotors | |
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| The Hamiltonian Structure with Feedback Controls | |
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| Feedback Stabilization of a Rigid Body with a Single Rotor | |
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| Phase Shifts | |
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| The Kaluza-Klein Description of Charged Particles | |
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| Optimal Control and Yang-Mills Particles | |
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| Discrete reduction | |
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| Fixed Point Sets and Discrete Reduction | |
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| Cotangent Bundles | |
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| Examples | |
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| Sub-Block Diagonalization with Discrete Symmetry | |
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| Discrete Reduction of Dual Pairs | |
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| Mechanical Integrators | |
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| Definitions and Examples | |
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| Limitations on Mechanical Integrators | |
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| Symplectic Integrators and Generating Functions | |
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| Symmetric Symplectic Algorithms Conserve J | |
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| Energy-Momentum Algorithms | |
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| The Lie-Poisson Hamilton-Jacobi Equation | |
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| Example: The Free Rigid Body | |
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| Variational Considerations | |
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| Hamiltonian Bifurcation | |
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| Some Introductory Examples | |
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| The Role of Symmetry | |
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| The One to One Resonance and Dual Pairs | |
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| Bifurcations in the Double Spherical Pendulum | |
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| Continuous Symmetry Groups and Solution Space Singularities | |
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| The Poincare-Melnikov Method | |
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| The Role of Dissipation | |
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| References | |
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| Index | |