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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 | |