Global Aspects of Classical Integrable Systems (eBook)

Foreword.- Introduction.- The mathematical pendulum.- Exercises.- Part I. Examples.- I. The harmonic oscillator.- 1. Hamilton’s equations and S1 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.- 4 Regularization of the Kepler vector field.- 4.1 Moser’s regularization.-  4.1 Ligon-Schaaf regularization.-  5. 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) ⇥ R3.-  2.2 Euler-Arnol’d equations on T1 S2 ⇥ R3.- 3. Symmetry and reduction.- 3.1 SO(3) symmetry.- 3.2 Construction of the reduced phase space.- 3.3 Geometry of the reduction map.- 3.4 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 S1 symmetry.-2.1 The orbit space T S2 /S1.-  2.2 The singular reduced space.- 2.3 Differential  structure on Pj ).- 2.4 Poisson brackets on C• (Pj ).-  2.5 Dynamics  on the reduced space Pj.-  3. The energy momentum mapping.- 3.1 Critical points of EM.-  3.2 Critical values of EM.-  3.3 Level sets of the reduced Hamiltonian  H j |Pj.-  3.4 Level sets of the energy momentum mapping EM.- 4. First return time and rotation number.-  4.1 Definition of first return time and rotation number.-4.2 Analytic properties of the rotation number.-4.3 Analytic properties of first return time.-5. Monodromy.-  5.1 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.