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Lectures on Mechanics

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ISBN-10: 0521428440

ISBN-13: 9780521428446

Edition: 1992

Authors: Jerrold E. Marsden, J. W. S. Cassels, N. J. Hitchin

List price: $79.99
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Description:

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…    
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Book details

List price: $79.99
Copyright year: 1992
Publisher: Cambridge University Press
Publication date: 4/30/1992
Binding: Paperback
Pages: 268
Size: 6.25" wide x 9.00" long x 0.75" tall
Weight: 0.792
Language: English

Preface
Introduction
The Classical Water Molecule and the Ozone Molecule
Hamiltonian Formulation
Geometry, Symmetry, and Reduction
Stability
Geometric Phases
The Rotation Group and the Poincare Sphere
A Crash Course in Geometric Mechanics
Symplectic and Poisson Manifolds
The Flow of a Hamiltonian Vector Field
Cotangent Bundles
Lagrangian Mechanics
Lie-Poisson Structures
The Rigid Body
Momentum Maps
Reduction
Singularities and Symmetry
A Particle in a Magnetic Field
Cotangent Bundle Reduction
Mechanical G-systems
The Classical Water Molecule
The Mechanical Connection
The Geometry and Dynamics of Cotangent Bundle Reduction
Examples
Lagrangian Reduction
Coupling to a Lie group
Relative Equilibria
Relative Equilibria on Symplectic Manifolds
Cotangent Relative Equilibria
Examples
The Rigid Body
The Energy-Momentum Method
The General Technique
Example: The Rigid Body
Block Diagonalization
The Normal Form for the Symplectic Structure
Stability of Relative Equilibria for the Double Spherical Pendulum
Geometric Phases
A Simple Example
Reconstruction
Cotangent Bundle Phases--a Special Case
Cotangent Bundles--General Case
Rigid Body Phases
Moving Systems
The Bead on the Rotating Hoop
Stabilization and Control
The Rigid Body with Internal Rotors
The Hamiltonian Structure with Feedback Controls
Feedback Stabilization of a Rigid Body with a Single Rotor
Phase Shifts
The Kaluza-Klein Description of Charged Particles
Optimal Control and Yang-Mills Particles
Discrete reduction
Fixed Point Sets and Discrete Reduction
Cotangent Bundles
Examples
Sub-Block Diagonalization with Discrete Symmetry
Discrete Reduction of Dual Pairs
Mechanical Integrators
Definitions and Examples
Limitations on Mechanical Integrators
Symplectic Integrators and Generating Functions
Symmetric Symplectic Algorithms Conserve J
Energy-Momentum Algorithms
The Lie-Poisson Hamilton-Jacobi Equation
Example: The Free Rigid Body
Variational Considerations
Hamiltonian Bifurcation
Some Introductory Examples
The Role of Symmetry
The One to One Resonance and Dual Pairs
Bifurcations in the Double Spherical Pendulum
Continuous Symmetry Groups and Solution Space Singularities
The Poincare-Melnikov Method
The Role of Dissipation
References
Index