Global Aspects of Classical Integrable Systems / / by Richard H. Cushman, Larry M. Bates

• Record Nr.: UNINA9910768439303321
• Autore: Cushman Richard H
• Pubbl/distr/stampa: Basel : , : Springer Basel : , : Imprint : Birkhäuser, , 2015
• ISBN: 3-0348-0918-2
• Edizione: [2nd ed. 2015.]
• Descrizione fisica: 1 online resource (493 p.)
• Disciplina: 530; 530.1
• Soggetti: Mathematical physics; Theoretical, Mathematical and Computational Physics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Includes index.
• Nota di contenuto: 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.
• Sommario/riassunto: 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.