Lectures on Mechanics

By: Jerrold E. Marsden, J. W. S. Cassels (Editor), N. J. Hitchin (Editor)

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Paperback | 29 June 1992

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268 Pages

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The use of geometric methods in classical mechanics has proven to be a fruitful exercise, with the results being of wide application to physics and engineering. Here Professor Marsden concentrates on these geometric aspects, and especially on symmetry techniques. The main points he covers are: the stability of relative equilibria, which is analyzed using the block diagonalization technique; geometric phases, studied using the reduction and reconstruction technique; and bifurcation of relative equilibria and chaos in mechanical systems. A unifying theme for these points is provided by reduction theory, the associated mechanical connection and techniques from dynamical systems. These methods can be applied to many control and stabilization situations, and this is illustrated using rigid bodies with internal rotors, and the use of geometric phases in mechanical systems. To illustrate the above ideas and the power of geometric arguments, the author studies a variety of specific systems, including the double spherical pendulum and the classical rotating water molecule. This book, based on the 1991 LMS Invited Lectures, will be valued by pure and applied mathematicians, physicists and engineers who work in geometry, nonlinear dynamics, mechanics, and robotics.

Industry Reviews

"...centres around symmetry and symplectic quotients. Many examples are given illustrating the utility and relevance of symplectic quotients...readable and stimulating." Michael Atiyah, Bulletin of the American Mathematical Society "...The virtue [of this book] is in the breadth, relevance, and complexity of the examples treated..." Mathematical Reviews

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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
Table of Contents provided by Syndetics. All Rights Reserved.

Lectures on Mechanics

At a Glance

Paperback

Industry Reviews

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