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Preface

Introduction / 1：

The Classical Water Molecule and the Ozone Molecule / 1.1：

Hamiltonian Formulation / 1.2：

Geometry, Symmetry, and Reduction / 1.3：

Stability / 1.4：

Geometric Phases / 1.5：

The Rotation Group and the Poincare Sphere / 1.6：

A Crash Course in Geometric Mechanics / 2：

Symplectic and Poisson Manifolds / 2.1：

The Flow of a Hamiltonian Vector Field / 2.2：

Cotangent Bundles / 2.3：

Lagrangian Mechanics / 2.4：

Lie-Poisson Structures / 2.5：

The Rigid Body / 2.6：

Momentum Maps / 2.7：

Reduction / 2.8：

Singularities and Symmetry / 2.9：

A Particle in a Magnetic Field / 2.10：

Cotangent Bundle Reduction / 3：

Mechanical G-systems / 3.1：

The Classical Water Molecule / 3.2：

The Mechanical Connection / 3.3：

The Geometry and Dynamics of Cotangent Bundle Reduction / 3.4：

Examples / 3.5：

Lagrangian Reduction / 3.6：

Coupling to a Lie group / 3.7：

Relative Equilibria / 4：

Relative Equilibria on Symplectic Manifolds / 4.1：

Cotangent Relative Equilibria / 4.2：

The Energy-Momentum Method / 4.3：

The General Technique / 5.1：

Example: The Rigid Body / 5.2：

Block Diagonalization / 5.3：

The Normal Form for the Symplectic Structure / 5.4：

Stability of Relative Equilibria for the Double Spherical Pendulum / 5.5：

A Simple Example / 6：

Reconstruction / 6.2：

Cotangent Bundle Phases--a Special Case / 6.3：

Cotangent Bundles--General Case / 6.4：

Rigid Body Phases / 6.5：

Moving Systems / 6.6：

The Bead on the Rotating Hoop / 6.7：

Stabilization and Control / 7：

The Rigid Body with Internal Rotors / 7.1：

The Hamiltonian Structure with Feedback Controls / 7.2：

Feedback Stabilization of a Rigid Body with a Single Rotor / 7.3：

Phase Shifts / 7.4：

The Kaluza-Klein Description of Charged Particles / 7.5：

Optimal Control and Yang-Mills Particles / 7.6：

Discrete reduction / 8：

Fixed Point Sets and Discrete Reduction / 8.1：

Sub-Block Diagonalization with Discrete Symmetry / 8.2：

Discrete Reduction of Dual Pairs / 8.5：

Mechanical Integrators / 9：

Definitions and Examples / 9.1：

Limitations on Mechanical Integrators / 9.2：

Symplectic Integrators and Generating Functions / 9.3：

Symmetric Symplectic Algorithms Conserve J / 9.4：

Energy-Momentum Algorithms / 9.5：

The Lie-Poisson Hamilton-Jacobi Equation / 9.6：

Example: The Free Rigid Body / 9.7：

Variational Considerations / 9.8：

Hamiltonian Bifurcation / 10：

Some Introductory Examples / 10.1：

The Role of Symmetry / 10.2：

The One to One Resonance and Dual Pairs / 10.3：

Bifurcations in the Double Spherical Pendulum / 10.4：

Continuous Symmetry Groups and Solution Space Singularities / 10.5：

The Poincare-Melnikov Method / 10.6：

The Role of Dissipation / 10.7：

References

Index

Preface

Introduction / 1：

The Classical Water Molecule and the Ozone Molecule / 1.1：

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