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Lectures on Mechanics : London Mathematical Society Lecture Note Series - Jerrold E. Marsden

Lectures on Mechanics

London Mathematical Society Lecture Note Series


Published: 29th June 1992
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The use of geometric methods in classical mechanics has proven fruitful, with wide applications in physics and engineering. In this book, Professor Marsden concentrates on these geometric aspects, 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.

"...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

Prefacep. ix
Introductionp. 1
The Classical Water Molecule and the Ozone Moleculep. 1
Hamiltonian Formulationp. 3
Geometry, Symmetry, and Reductionp. 9
Stabilityp. 12
Geometric Phasesp. 16
The Rotation Group and the Poincare Spherep. 23
A Crash Course in Geometric Mechanicsp. 27
Symplectic and Poisson Manifoldsp. 27
The Flow of a Hamiltonian Vector Fieldp. 29
Cotangent Bundlesp. 29
Lagrangian Mechanicsp. 31
Lie-Poisson Structuresp. 32
The Rigid Bodyp. 33
Momentum Mapsp. 34
Reductionp. 36
Singularities and Symmetryp. 39
A Particle in a Magnetic Fieldp. 40
Cotangent Bundle Reductionp. 43
Mechanical G-systemsp. 43
The Classical Water Moleculep. 46
The Mechanical Connectionp. 50
The Geometry and Dynamics of Cotangent Bundle Reductionp. 54
Examplesp. 59
Lagrangian Reductionp. 66
Coupling to a Lie groupp. 72
Relative Equilibriap. 77
Relative Equilibria on Symplectic Manifoldsp. 77
Cotangent Relative Equilibriap. 79
Examplesp. 82
The Rigid Bodyp. 87
The Energy-Momentum Methodp. 93
The General Techniquep. 93
Example: The Rigid Bodyp. 97
Block Diagonalizationp. 101
The Normal Form for the Symplectic Structurep. 107
Stability of Relative Equilibria for the Double Spherical Pendulump. 110
Geometric Phasesp. 115
A Simple Examplep. 115
Reconstructionp. 117
Cotangent Bundle Phases--a Special Casep. 119
Cotangent Bundles--General Casep. 120
Rigid Body Phasesp. 122
Moving Systemsp. 125
The Bead on the Rotating Hoopp. 127
Stabilization and Controlp. 131
The Rigid Body with Internal Rotorsp. 131
The Hamiltonian Structure with Feedback Controlsp. 132
Feedback Stabilization of a Rigid Body with a Single Rotorp. 134
Phase Shiftsp. 137
The Kaluza-Klein Description of Charged Particlesp. 141
Optimal Control and Yang-Mills Particlesp. 144
Discrete reductionp. 147
Fixed Point Sets and Discrete Reductionp. 149
Cotangent Bundlesp. 155
Examplesp. 157
Sub-Block Diagonalization with Discrete Symmetryp. 162
Discrete Reduction of Dual Pairsp. 166
Mechanical Integratorsp. 171
Definitions and Examplesp. 171
Limitations on Mechanical Integratorsp. 175
Symplectic Integrators and Generating Functionsp. 177
Symmetric Symplectic Algorithms Conserve Jp. 178
Energy-Momentum Algorithmsp. 180
The Lie-Poisson Hamilton-Jacobi Equationp. 182
Example: The Free Rigid Bodyp. 186
Variational Considerationsp. 187
Hamiltonian Bifurcationp. 189
Some Introductory Examplesp. 189
The Role of Symmetryp. 196
The One to One Resonance and Dual Pairsp. 202
Bifurcations in the Double Spherical Pendulump. 204
Continuous Symmetry Groups and Solution Space Singularitiesp. 205
The Poincare-Melnikov Methodp. 207
The Role of Dissipationp. 217
Referencesp. 225
Indexp. 250
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521428446
ISBN-10: 0521428440
Series: London Mathematical Society Lecture Note Series
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 268
Published: 29th June 1992
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 22.8 x 15.2  x 1.5
Weight (kg): 0.41