Global Aspects of Classical Integrable Systems

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.- 3.1 The Kepler vector field.- 3.2 The so(4) momentum map.- 3.3 Kepler's equation.- 3.4 Regularization of the Kepler vector field.- 4. Exercises.- III The Euler top.- 1. Facts about SO(3).- 1.1 The standard model.- 1.2 The exponential map.- 1.3 The solid ball model.- 1.4 The sphere bundle model.- 2. Left invariant geodesics.- 2.1 Euler-Arnol'd equations on SO(3).- 2.2 Euler-Arnol'd equations on T1S2 × R3.- 3. Symmetry and reduction.- 3.1 Construction of the reduced phase space.- 3.2 Geometry of the reduction map.- 3.3 Euler's equations.- 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.- 7.1 An analytic formula.- 7.2 Poinsot's construction.- 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.- 3.1 Reduction to the Euler-Poisson equations.- 3.2 The magnetic spherical pendulum.- 4. Reduction of the left S1 action.- 5. The Poisson structure.- 6. The Euler-Poisson equations.- 6.1 The Poisson structure.- 6.2 The energy momentum mapping.- 6.3 Motion of the tip of the figure axis.- 7. The energy momemtum mapping.- 7.1 Topology of ???1(h,a,b) and H?1(h).- 7.2 The discriminant locus.- 7.3 The period lattice and monodromy.- 8. The Hamiltonian Hopf bifurcation.- 8.1 The linear case.- 8.2 The nonlinear case.- 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.- 2. The Morse lemma.- 3. The Morse isotopy lemma.- 4. Exercises.- Notes.- References.- Acknowledgements.