Global Aspects of Classical Integrable Systems:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Basel
Springer Basel
2015
|
| Ausgabe: | 2. ed. |
| Schlagworte: | |
| Online-Zugang: | TUM01 UBT01 Volltext Inhaltsverzeichnis Abstract |
| Beschreibung: | 1 Online-Ressource (XVIII, 477 p. 86 illus., 69 illus. in color) |
| ISBN: | 9783034809184 |
| DOI: | 10.1007/978-3-0348-0918-4 |
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Datensatz im Suchindex
| _version_ | 1804174858546839552 |
|---|---|
| adam_text | GLOBAL ASPECTS OF CLASSICAL INTEGRABLE SYSTEMS
/ CUSHMAN, RICHARD H.
: 2015
TABLE OF CONTENTS / INHALTSVERZEICHNIS
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
DIESES SCHRIFTSTUECK WURDE MASCHINELL ERZEUGT.
GLOBAL ASPECTS OF CLASSICAL INTEGRABLE SYSTEMS
/ CUSHMAN, RICHARD H.
: 2015
ABSTRACT / INHALTSTEXT
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
DIESES SCHRIFTSTUECK WURDE MASCHINELL ERZEUGT.
|
| any_adam_object | 1 |
| author | Cushman, Richard H. 1942- Bates, Larry M. 1957- |
| author_GND | (DE-588)11485999X (DE-588)173019013 |
| author_facet | Cushman, Richard H. 1942- Bates, Larry M. 1957- |
| author_role | aut aut |
| author_sort | Cushman, Richard H. 1942- |
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| dewey-full | 530.1 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.1 |
| dewey-search | 530.1 |
| dewey-sort | 3530.1 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
| doi_str_mv | 10.1007/978-3-0348-0918-4 |
| edition | 2. ed. |
| format | Electronic eBook |
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| indexdate | 2024-07-10T07:07:04Z |
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| publisher | Springer Basel |
| record_format | marc |
| spelling | Cushman, Richard H. 1942- Verfasser (DE-588)11485999X aut Global Aspects of Classical Integrable Systems Richard H. Cushman, Larry M. Bates 2. ed. Basel Springer Basel 2015 1 Online-Ressource (XVIII, 477 p. 86 illus., 69 illus. in color) txt rdacontent c rdamedia cr rdacarrier Physics Theoretical, Mathematical and Computational Physics Programmierung (DE-588)4076370-5 gnd rswk-swf Globale Analysis (DE-588)4021285-3 gnd rswk-swf Programm (DE-588)4047394-6 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Integrables System (DE-588)4114032-1 gnd rswk-swf Integrables System (DE-588)4114032-1 s Globale Analysis (DE-588)4021285-3 s DE-604 Mathematik (DE-588)4037944-9 s Programm (DE-588)4047394-6 s Programmierung (DE-588)4076370-5 s Bates, Larry M. 1957- Verfasser (DE-588)173019013 aut Erscheint auch als Druckausgabe 978-3-0348-0917-7 https://doi.org/10.1007/978-3-0348-0918-4 Verlag Volltext Springer Fremddatenuebernahme application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028101393&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Springer Fremddatenuebernahme application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028101393&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Abstract |
| spellingShingle | Cushman, Richard H. 1942- Bates, Larry M. 1957- Global Aspects of Classical Integrable Systems Physics Theoretical, Mathematical and Computational Physics Programmierung (DE-588)4076370-5 gnd Globale Analysis (DE-588)4021285-3 gnd Programm (DE-588)4047394-6 gnd Mathematik (DE-588)4037944-9 gnd Integrables System (DE-588)4114032-1 gnd |
| subject_GND | (DE-588)4076370-5 (DE-588)4021285-3 (DE-588)4047394-6 (DE-588)4037944-9 (DE-588)4114032-1 |
| title | Global Aspects of Classical Integrable Systems |
| title_auth | Global Aspects of Classical Integrable Systems |
| title_exact_search | Global Aspects of Classical Integrable Systems |
| title_full | Global Aspects of Classical Integrable Systems Richard H. Cushman, Larry M. Bates |
| title_fullStr | Global Aspects of Classical Integrable Systems Richard H. Cushman, Larry M. Bates |
| title_full_unstemmed | Global Aspects of Classical Integrable Systems Richard H. Cushman, Larry M. Bates |
| title_short | Global Aspects of Classical Integrable Systems |
| title_sort | global aspects of classical integrable systems |
| topic | Physics Theoretical, Mathematical and Computational Physics Programmierung (DE-588)4076370-5 gnd Globale Analysis (DE-588)4021285-3 gnd Programm (DE-588)4047394-6 gnd Mathematik (DE-588)4037944-9 gnd Integrables System (DE-588)4114032-1 gnd |
| topic_facet | Physics Theoretical, Mathematical and Computational Physics Programmierung Globale Analysis Programm Mathematik Integrables System |
| url | https://doi.org/10.1007/978-3-0348-0918-4 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028101393&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028101393&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT cushmanrichardh globalaspectsofclassicalintegrablesystems AT bateslarrym globalaspectsofclassicalintegrablesystems |