Global Aspects of Classical Integrable Systems

by Richard H. Cushman, Larry M. Bates

Overview

This book gives a uniquely complete description of the geometry of the energy momentum mapping of five classical integrable systems: the 2-dimensional harmonic oscillator, the geodesic flow on the 3-sphere, the Euler top, the spherical pendulum and the Lagrange top. It presents for the first time in book form a general theory of symmetry reduction which allows one to reduce the symmetries in the spherical pendulum and the Lagrange top. Also the monodromy obstruction to the existence of global action angle coordinates is calculated for the spherical pendulum and the Lagrange top. The book addresses professional mathematicians and graduate students and can be used as a textbook on advanced classical mechanics or global analysis.

Product Details

ISBN-13:	9783034898171
Publisher:	Birkh�user Basel
Publication date:	10/05/2012
Edition description:	Softcover reprint of the original 1st ed. 1997
Pages:	435
Product dimensions:	6.10(w) x 9.25(h) x 0.04(d)

Table of Contents

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.

Editorial Reviews

"Ideal for someone who needs a thorough global understanding of one of these systems [and] who would like to learn some of the tools and language of modern geometric mechanics. The exercises at the end of each chapter are excellent. The book could serve as a good supplementary text for a graduate course in geometric mechanics."

--SIAM Review

From the Publisher