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Global Aspects of Classical Integrable Systems

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...

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Bibliographic Details
Main Authors: Cushman, Richard H., Bates, Larry M. (Author)
Format: eBook
Language:English
Published: Basel Birkhäuser 2015, 2015
Edition:2nd ed. 2015
Subjects:
Mathematical Physics
Theoretical, Mathematical And Computational Physics
Online Access:
Collection: Springer eBooks 2005- - Collection details see MPG.ReNa
Table of Contents:
  • Action angle coordinates
  • Monodromy
  • Morse theory.-References
  • Acknowledgments.-Index
  • 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
  • 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
  • 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
  • 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
  • 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

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