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...
Clasificación: | Libro Electrónico |
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Autores principales: | , |
Autor Corporativo: | |
Formato: | Electrónico eBook |
Idioma: | Inglés |
Publicado: |
Basel :
Springer Basel : Imprint: Birkhäuser,
2015.
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Edición: | 2nd ed. 2015. |
Temas: | |
Acceso en línea: | Texto Completo |
Tabla de Contenidos:
- 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.