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Lectures on mechanics /
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 s...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge [England] ; New York :
Cambridge University Press,
1992.
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| Colección: | London Mathematical Society lecture note series ;
174. |
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Cover; Series Page; Title; Copyright; Contents; Preface; Chapter 1 Introduction; 1.1 The Classical Water Molecule and the Ozone Molecule; 1.2 Hamiltonian Formulation; 1.3 Geometry, Symmetry, and Reduction; 1.4 Stability; 1.5 Geometric Phases; 1.6 The Rotation Group and the Poincare Sphere; Chapter 2 A Crash Course in Geometric Mechanics; 2.1 Symplectic and Poisson Manifolds; 2.2 The Flow of a Hamiltonian Vector Field; 2.3 Cotangent Bundles; 2.4 Lagrangian Mechanics; 2.5 Lie-Poisson Structures; 2.6 The Rigid Body; 2.7 Momentum Maps; 2.8 Reduction; 2.9 Singularities and Symmetry
- 2.10 A Particle in a Magnetic FieldChapter 3 Cotangent Bundle Reduction; 3.1 Mechanical G-systems; 3.2 The Classical Water Molecule; 3.3 The Mechanical Connection; 3.4 The Geometry and Dynamics of Cotangent Bundle Reduction; 3.5 Examples; 3.6 Lagrangian Reduction; 3. 7 Coupling to a Lie group; Chapter 4 Relative Equilibria; 4.1 Relative Equilibria on Symplectic Manifolds; 4.2 Cotangent Relative Equilibria; 4.3 Examples; 4.4 The Rigid Body; Chapter 5 The Energy-Momentum Method; 5.1 The General Technique; 5.2 Example: The Rigid Body; 5.3 Block Diagonalization
- 5.4 The Normal Form for the Symplectic Structure5.5 Stability of Relative Equilibria for the Double Spherical Pendulum; Chapter 6 Geometric Phases; 6.1 A Simple Example; 6.2 Reconstruction; 6.3 Cotangent Bundle Phases
- a Special Case; 6.4 Cotangent Bundles
- General Case; 6.5 Rigid Body Phases; 6.6 Moving Systems; 6.7 The Bead on the Rotating Hoop; Chapter 7 Stabilization and Control; 7.1 The Rigid Body with Internal Rotors; 7.2 The Hamiltonian Structure with Feedback Controls; 7.3 Feedback Stabilization of a Rigid Body with a Single Rotor; 7.4 Phase Shifts
- 7.5 The Kaluza-Klein Description of Charged Particles7.6 Optimal Control and Yang-Mills Particles; Chapter 8 Discrete reduction; 8.1 Fixed Point Sets and Discrete Reduction; 8.2 Cotangent Bundles; 8.3 Examples; 8.4 Sub-Block Diagonalization with Discrete Symmetry; 8.5 Discrete Reduction of Dual Pairs; Chapter 9 Mechanical Integrators; 9.1 Definitions and Examples; 9.2 Limitations on Mechanical Integrators; 9.3 Symplectic Integrators and Generating Functions; 9.4 Symmetric Symplectic Algorithms Conserve J; 9.5 Energy-Momentum Algorithms; 9.6 The Lie-Poisson Hamilton-Jacobi Equation
- 9.7 Example: The Free Rigid Body9.8 Variational Considerations; Chapter 10 Hamiltonian Bifurcation; 10.1 Some Introductory Examples; 10.2 The Role of Symmetry; 10.3 The One to One Resonance and Dual Pairs; 10.4 Bifurcations in the Double Spherical Pendulum; 10.5 Continuous Symmetry Groups and Solution Space Singularities; 10.6 The Poincare-Melnikov Method; 10.7 The Role of Dissipation; References; Index