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Global Aspects of Classical Integrable Systems

This book gives a complete global geometric description of the motion of the two di­ mensional hannonic oscillator, the Kepler problem, the Euler top, the spherical pendulum and the Lagrange top. These classical integrable Hamiltonian systems one sees treated in almost every physics book on classica...

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Bibliographic Details
Main Authors: Cushman, Richard H., Bates, Larry M. (Author)
Format: eBook
Language:English
Published: Basel Birkhäuser Basel 1997, 1997
Edition:1st ed. 1997
Subjects:
Mathematical Physics
Theoretical, Mathematical And Computational Physics
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 4. Reduction of the left S1 action
  • 5. The Poisson structure
  • 6. The Euler-Poisson equations
  • 7. The energy momemtum mapping
  • 8. The Hamiltonian Hopf bifurcation
  • 9. Exercises
  • Appendix A. Fundamental concepts
  • 1. Symplectic linear algebra
  • 2. Symplectic manifolds
  • 3. Hamilton’s equations
  • 4. Poisson algebras and manifolds
  • 5. Exercises
  • Appendix B. Systems with symmetry
  • 1. Smooth group actions
  • 2. Orbit spaces
  • 2.1 Orbit space of a proper action
  • 2.2 Orbit space of a free action
  • 2.3 Orbit space of a locally free action
  • 3. Momentum mappings
  • 3.1 General properties
  • 3.2 Normal form
  • 4. Reduction: the regular case
  • 5. Reduction: the singular case
  • 6. Exercises
  • Appendix C. Ehresmann connections
  • 1. Basic properties
  • 2. The Ehresmann theorems
  • 3. Exercises
  • Appendix D. Action angle coordinates
  • 1. Local action angle coordinates
  • 2. Monodromy
  • 3. Exercises
  • Appendix E. Basic Morse theory
  • 1. Preliminaries
  • I. The harmonic oscillator
  • 1. Hamilton’s equations and Sl symmetry
  • 2. S1 energy momentum mapping
  • 3. U(2) momentum mapping
  • 4. The Hopf fibration
  • 5. Invariant theory and reduction
  • 6. Exercises
  • II. Geodesics on S3
  • 1. The geodesic and Delaunay vector fields
  • 2. The SO(4) momentum mapping
  • 3. The Kepler problem
  • 4. Exercises
  • III The Euler top
  • 1. Facts about SO(3)
  • 2. Left invariant geodesics
  • 3. Symmetry and reduction
  • 4. Qualitative behavior of the reduced system
  • 5. Analysis of the energy momentum map
  • 6. Integration of the Euler-Arnol’d equations
  • 7. The rotation number
  • 8. A twisting phenomenon
  • 9. Exercises
  • IV. The spherical pendulum
  • 1. Liouville integrability
  • 2. Reduction of the Sl symmetry
  • 3. The energy momentum mapping
  • 4. Rotation number and first return time
  • 5. Monodromy
  • 6. Exercises
  • V. The Lagrange top
  • 1. The basic model
  • 2. Liouville integrability
  • 3. Reduction of the right Sl action
  • 2. The Morse lemma
  • 3. The Morse isotopy lemma
  • 4. Exercises
  • Notes
  • References
  • Acknowledgements

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