Lectures on mechanics

Preface......Page 6
The Classical Water Molecule and the Ozone Molecule......Page 11
Lagrangian and Hamiltonian Formulation......Page 13
The Rigid Body......Page 15
Geometry, Symmetry and Reduction......Page 22
Stability......Page 25
Geometric Phases......Page 29
The Rotation Group and the Poincaré Sphere......Page 36
Symplectic and Poisson Manifolds......Page 39
The Flow of a Hamiltonian Vector Field......Page 41
Cotangent Bundles......Page 42
Lagrangian Mechanics......Page 43
Lie--Poisson Structures and the Rigid Body......Page 44
The Euler--Poincaré Equations......Page 47
Momentum Maps......Page 49
Symplectic and Poisson Reduction......Page 52
Singularities and Symmetry......Page 55
A Particle in a Magnetic Field......Page 56
Tangent and Cotangent Bundle Reduction......Page 59
Mechanical G-systems......Page 60
The Classical Water Molecule......Page 62
The Mechanical Connection......Page 66
The Geometry and Dynamics of Cotangent Bundle Reduction......Page 71
Examples......Page 75
Lagrangian Reduction and the Routhian......Page 82
The Reduced Euler--Lagrange Equations......Page 87
Coupling to a Lie group......Page 89
Relative Equilibria on Symplectic Manifolds......Page 95
Cotangent Relative Equilibria......Page 98
Examples......Page 100
The Rigid Body......Page 105
The General Technique......Page 111
Example: The Rigid Body......Page 115
Block Diagonalization......Page 119
The Normal Form for the Symplectic Structure......Page 124
Stability of Relative Equilibria for the Double Spherical Pendulum......Page 127
A Simple Example......Page 131
Reconstruction......Page 133
Cotangent Bundle Phases---a Special Case......Page 135
Cotangent Bundles---General Case......Page 136
Rigid Body Phases......Page 138
Moving Systems......Page 140
The Bead on the Rotating Hoop......Page 142
The Rigid Body with Internal Rotors......Page 145
The Hamiltonian Structure with Feedback Controls......Page 146
Feedback Stabilization of a Rigid Body with a Single Rotor......Page 148
Phase Shifts......Page 151
The Kaluza--Klein Description of Charged Particles......Page 155
Optimal Control and Yang--Mills Particles......Page 158
Discrete Reduction......Page 161
Fixed Point Sets and Discrete Reduction......Page 163
Cotangent Bundles......Page 169
Examples......Page 171
Sub-Block Diagonalization with Discrete Symmetry......Page 175
Discrete Reduction of Dual Pairs......Page 178
Definitions and Examples......Page 183
Limitations on Mechanical Integrators......Page 187
Symplectic Integrators and Generating Functions......Page 189
Symmetric Symplectic Algorithms Conserve  J......Page 190
Energy--Momentum Algorithms......Page 192
The Lie--Poisson Hamilton--Jacobi Equation......Page 194
Example: The Free Rigid Body......Page 197
PDE Extensions......Page 198
Some Introductory Examples......Page 201
The Role of Symmetry......Page 208
The One-to-One Resonance and Dual Pairs......Page 213
Bifurcations in the Double Spherical Pendulum......Page 215
Continuous Symmetry Groups and Solution Space Singularities......Page 216
The Poincaré--Melnikov Method......Page 217
The Role of Dissipation......Page 227
Double Bracket Dissipation......Page 233
Bibliography......Page 237
Index......Page 256