Global Aspects of Classical Integrable Systems

Definition of monodromy.-  5.2 Monodromy of the bundle of period lattices.-6. Exercises.-  V. The Lagrange top.-1.The basic model.- 2. Liouville integrability.- 3. Reduction of the right S1 action.- 3.1 Reduction to the Euler-Poisson equations.-3.2 The magnetic spherical pendulum.-  4. Reduction of the left S1 action.-4.1 Induced action on P a.- 4.2 The orbit space (J a )  1 (b)/S1.-  4.3 Some differential  spaces.- 4.4 Poisson structure on C.(P a ).- 5. The Euler-Poisson equations.- 5.1 The twice reduced system.- 5.2 The energy momentum mapping.- 5.3 Motion of the tip of the figure axis.-6. The energy momentum mapping.-  6.1 Topology of EM   1 (h, a, b) and H  1 (h).-  6.2 The discriminant locus.- 6.3 The period lattice.- 6.4 Monodromy.- 7. The Hamiltonian  Hopf bifurcation.- 7.1 The linear case.- 7.2 The nonlinear case.- 8. Exercises.-  Part II. Theory.-VI. Fundamental concepts.-1. Symplectic linear algebra.- 2. Symplectic manifolds.- 3. Hamilton's equations.- 4. Poisson algebras and manifolds.-  5. Exercises.-  VII. Systems with symmetry.-1. Smooth group actions.- 2. Orbit spaces.- 2.1 Orbit space of a proper action.-2.2 Orbit space of a proper free action.- 3. Differential  spaces.- 3.1 Differential  structure.- 3.2 An orbit space as a differential  space.- 3.3 Subcartesian spaces.- 3.4 Stratification of an orbit space by orbit types.- 3.5 Minimality of S.- 4. Vector fields on a differential  space.- 4.1 Definition of a vector field.- 4.2 Vector field on a stratified differential  space.- 4.3 Vector fields on an orbit space.- 5. Momentum mappings.-5.1 General properties.- 5.2 Normal form.- 6. Regular reduction.- 6.1 The standard approach.- 6.2 An alternative approach.- 7. Singular reduction.- 7.1 Singular  reduced space and dynamics.- 7.2 Stratification of the singular reduced space.- 8. Exercises.- VIII. Ehresmann connections.-1. Basic properties.- 2. The Ehresmann theorems.- 3. Exercises.- IX. Action angle coordinates.- 1. Liouville integrable systems.- 2. Local action angle coordinates.- 3. Exercises.- X.Monodromy.-1. The period lattice bundle.-2. The geometric mondromy theorem.- 2.1 The singular fiber.- 2.2 Nearby singular fibers.- 2.3 Monodromy.- 3. The hyperbolic billiard.- 3.1 The basic model.- 3.2 Reduction of the S1 symmetry.- 3.3 Partial reconstruction.- 3.4 Full reconstruction.- 3.5 The first return time and rotation angle.-3.6 The action functions.- 4. Exercises.- XI. Morse theory.-1. Preliminaries.- 2. The Morse lemma.- 3. The Morse isotopy lemma.- 4. Exercises.- Notes.-Forward and Introduction.- Harmonic oscillator.- Geodesics on S3.- Euler top.- Spherical pendulum.- Lagrange top.- Fundamental concepts.- Systems with symmetry.- Ehresmann connections.- Action angle coordinates.- Monodromy.- Morse theory.-References.- Acknowledgments.-Index.