Global Aspects of Classical Integrable Systems

• Κύριοι συγγραφείς: Cushman, Richard H. (Συγγραφέας), Bates, Larry M. (Συγγραφέας)
• Συγγραφή απο Οργανισμό/Αρχή: SpringerLink (Online service)
• Μορφή: Ηλεκτρονική πηγή Ηλ. βιβλίο
• Γλώσσα: English
• Έκδοση: Basel : Springer Basel : Imprint: Birkhäuser, 2015.
• Έκδοση: 2nd ed. 2015.
• Θέματα: Physics.; Theoretical, Mathematical and Computational Physics.

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...

Πίνακας περιεχομένων:

• 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.