Integrable Hamiltonian systems: geometry, topology, classification
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English Russian |
| Published: |
Boca Raton u.a.
Chapman & Hall/CRC
2004
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XV, 730 S. Ill., graph. Darst. |
| ISBN: | 0415298059 |
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| 240 | 1 | 0 | |a Integriruemyj gamiltonovyj sistemy |
| 245 | 1 | 0 | |a Integrable Hamiltonian systems |b geometry, topology, classification |c A. V. Bolsinov and A. T. Fomenko |
| 264 | 1 | |a Boca Raton u.a. |b Chapman & Hall/CRC |c 2004 | |
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| 650 | 7 | |a Topologia |2 larpcal | |
| 650 | 4 | |a Geodesic flows | |
| 650 | 4 | |a Geodesics (Mathematics) | |
| 650 | 4 | |a Hamiltonian systems | |
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| adam_text | MEGRABLE HAMLTOMAN SYSTEMS GEOMETRY, TOPOLOGY, CLASSIFICATION A. V.
BOLSINOV AND A. T. FOMENKO CHAPMAN & HALL/CRC A CRC PRESS COMPANY N *
BOCA RATON LONDON NEW YORK WASHINGTON, D.C. CONTENTS PREFACE XIII
CHAPTER 1. BASIC NOTIONS 1 SECTION 1.1. LINEAR SYMPLECTIC GEOMETRY 1
SECTION 1.2. SYMPLECTIC AND POISSON MANIFOLDS 4 1.2.1. COTANGENT BUNDLES
5 1.2.2. THE COMPLEX SPACE C * AND ITS COMPLEX SUBMANIFOLDS. KAHLER
MANIFOLDS 6 1.2.3. ORBITS OF COADJOINT REPRESENTATION 6 SECTION 1.3. THE
DARBOUX THEOREM 11 SECTION 1.4. LIOUVILLE INTEGRABLE HAMILTONIAN
SYSTEMS. THE LIOUVILLE THEOREM 14 SECTION 1.5. NON-RESONANT AND RESONANT
SYSTEMS 21 SECTION 1.6. ROTATION NUMBER 22 SECTION 1.7. THE MOMENTUM
MAPPING OF AN INTEGRABLE SYSTEM AND ITS-BIFURCATION DIAGRAM 25 SECTION
1.8. NON-DEGENERATE CRITICAL POINTS OF THE MOMENTUM MAPPING 27 1.8.1.
THE CASE OF TWO DEGREES OF FREEDOM 27 1.8.2. BOTT INTEGRALS FROM THE
VIEWPOINT OF THE FOUR-DIMENSIONAL SYMPLECTIC MANIFOLD 30 1.8.3.
NON-DEGENERATE SINGULARITIES IN THE CASE OF MANY DEGREES OF FREEDOM 37
1.8.4. TYPES OF NON-DEGENERATE SINGULARITIES IN THE MULTIDIMENSIONAL
CASE 40 SECTION 1.9. MAIN TYPES OF EQUIVALENCE OF DYNAMICAL SYSTEMS 46
CHAPTER 2. THE TOPOLOGY OF FOLIATIONS ON TWO-DIMENSIONAL SURFACES
GENERATED BY MORSE FUNCTIONS 49 SECTION 2.1. SIMPLE MORSE FUNCTIONS 49
SECTION 2.2. REEB GRAPH OF A MORSE FUNCTION 51 SECTION 2.3. NOTION OF AN
ATOM 52 SECTION 2.4. SIMPLE ATOMS 54 2.4.1. THE CASE OF MINIMUM AND
MAXIMUM. THE ATOM A 54 2.4.2. THE CASE OF AN ORIENTABLE SADDLE. THE ATOM
B 55 2.4.3. THE CASE OF A NON-ORIENTABLE SADDLE. THE ATOM B 56 2.4.4.
THE CLASSIFICATION OF SIMPLE ATOMS 57 VLLL CONTENTS SECTION 8.4. THREE
GENERAL PRINCIPLES FOR CONSTRUCTING INVARIANTS 268 8.4.1. FIRST GENERAL
PRINCIPLE 268 8.4.2. SECOND GENERAL PRINCIPLE 269 8.4.3. THIRD GENERAL
PRINCIPLE 269 SECTION 8.5. ADMISSIBLE SUPERFLUOUS I-FRAMES AND A
REALIZATION THEOREM 269 8.5.1. REALIZATION OF A FRAME ON AN ATOM . . 269
8.5.2. REALIZATION OF A FRAME ON AN EDGE OF A MOLECULE 273 8.5.3.
REALIZATION OF A FRAME ON THE WHOLE MOLECULE 276 SECTION 8.6.
CONSTRUCTION OF ORBITAL INVARIANTS IN THE TOPOLOGICAL CASE. A ^-MOLECULE
278 8.6.1. THE IZ-INVARIANT AND THE INDEX OF A SYSTEM ON AN EDGE 278
8.6.2. BJNVARIANT (ON THE RADICALS OF A MOLECULE) 280 8.6.3.
^I-INVARIANT 282 8.6.4. AZ[&] -INVARIANT 283 8.6.5. FINAL DEFINITION OF
A T-MOLECULE FOR AN INTEGRABLE SYSTEM 285 SECTION 8.7. THEOREM ON THE
TOPOLOGICAL ORBITAL CLASSIFICATION OF INTEGRABLE SYSTEMS WITH TWO
DEGREES OF FREEDOM 286 SECTION 8.8. A PARTICULAR CASE: SIMPLE INTEGRABLE
SYSTEMS 291 SECTION 8.9. SMOOTH ORBITAL CLASSIFICATION 292 CHAPTER 9.
LIOUVILLE CLASSIFICATION OF INTEGRABLE SYSTEMS WITH TWO DEGREES OF
FREEDOM IN FOUR-DIMENSIONAL NEIGHBORHOODS OF SINGULAR POINTS 299 SECTION
9.1. /-TYPE OF A FOUR-DIMENSIONAL SINGULARITY 299 SECTION 9.2. THE LOOP
MOLECULE OF A FOUR-DIMENSIONAL SINGULARITY 304 SECTION 9.3.
CENTER-CENTER CASE 306 SECTION 9.4. CENTER-SADDLE CASE 308 SECTION 9.5.
SADDLE-SADDLE CASE 313 9.5.1. THE STRUCTURE OF A SINGULAR LEAF 313
9.5.2. CZ-TYPE OF A SINGULARITY 318 . 9.5.3. THE LIST OF SADDLE-SADDLE
SINGULARITIES OF SMALL COMPLEXITY 322 SECTION 9.6. ALMOST DIRECT PRODUCT
REPRESENTATION OF A FOUR-DIMENSIONAL SINGULARITY .. . 339 SECTION 9.7.
PROOF OF THE CLASSIFICATION THEOREMS 348 9.7:1. PROOF OF THEOREM 9.3 348
9.7.2. PROOF OF THEOREM 9.4 (REALIZATION THEOREM) 349 SECTION 9.8.
FOCUS-FOCUS CASE S 350 9.8.1. THE STRUCTURE OF A SINGULAR LEAF OF
FOCUS-FOCUS TYPE 350 9.8.2. CLASSIFICATION OF FOCUS-FOCUS SINGULARITIES
353 9.8.3. MODEL EXAMPLE OF A FOCUS-FOCUS SINGULARITY AND THE
REALIZATION THEOREM 356 9.8.4. THE LOOP MOLECULE AND MONODROMY GROUP OF
A FOCUS-FOCUS SINGULARITY 358 SECTION 9.9. ALMOST DIRECT PRODUCT
REPRESENTATION FOR MULTIDIMENSIONAL NON-DEGENERATE SINGULARITIES OF
LIOUVILLE FOLIATIONS 363 CHAPTER 10. METHODS OF CALCULATION OF
TOPOLOGICAL INVARIANTS OF INTEGRABLE HAMILTONIAN SYSTEMS SECTION 10.1.
GENERAL SCHEME FOR TOPOLOGICAL ANALYSIS OF THE LIOUVILLE FOLIATION
10.1.1. MOMENTUM MAPPING 10.1.2. CONSTRUCTION OF THE BIFURCATION DIAGRAM
10.1.3. VERIFICATION OF THE NON-DEGENERACY CONDITION 10.1.4. DESCRIPTION
OF THE ATOMS OF THE SYSTEM 10.1.5. CONSTRUCTION OF THE MOLECULE OF THE
SYSTEM ON A GIVEN ENERGY LEVEL 10.1.6. COMPUTATION OF MARKS CONTENTS IX
SECTION 10.2. METHODS FOR COMPUTING MARKS 377 SECTION 10.3. THE LOOP
MOLECULE METHOD 378 SECTION 10.4. LIST OF TYPICAL LOOP MOLECULES 382
10.4.1. LOOP MOLECULES OF REGULAR POINTS OF THE BIFURCATION DIAGRAM 382
10.4.2. LOOP MOLECULES OF NON-DEGENERATE SINGULARITIES .384 SECTION
10.5. THE STRUCTURE OF THE LIOUVILLE FOLIATION FOR TYPICAL DEGENERATE
SINGULARITIES 386 SECTION 10.6. TYPICAL LOOP MOLECULES CORRESPONDING TO
DEGENERATE ONE-DIMENSIONAL ORBITS 389 SECTION 10.7. COMPUTATION OF R-
AND -MARKS BY MEANS OF ROTATION FUNCTIONS 395 SECTION 10.8. COMPUTATION
OF THE N-MARK BY MEANS OF ROTATION FUNCTIONS 398 SECTION 10.9.
RELATIONSHIP BETWEEN THE MARKS OF THE MOLECULE AND THE TOPOLOGY OF Q 3
402 CHAPTER 11. INTEGRABLE GEODESIC FLOWS ON TWO-DIMENSIONAL SURFACES 40
9 SECTION 11.1. STATEMENT OF THE PROBLEM 409 SECTION 11.2. TOPOLOGICAL
OBSTRUCTIONS TO INTEGRABILITY OF GEODESIC FLOWS ON TWO-DIMENSIONAL
SURFACES 412 SECTION 11.3. TWO EXAMPLES OF INTEGRABLE GEODESIC FLOWS 416
11.3.1. SURFACES OF REVOLUTION 416 11.3.2. LIOUVILLE METRICS 418 SECTION
11.4. RIEMANNIAN METRICS WHOSE GEODESIC FLOWS ARE INTEGRABLE BY MEANS OF
LINEAR OR QUADRATIC INTEGRALS. LOCAL THEORY 420 11.4.1. SOME GENERAL
PROPERTIES OF POLYNOMIAL INTEGRALS OF GEODESIC FLOWS. LOCAL THEORY 420
11.4.2. RIEMANNIAN METRICS WHOSE GEODESIC FLOWS ADMIT A LINEAR INTEGRAL.
LOCAL THEORY 423 11.4.3. RIEMANNIAN METRICS WHOSE GEODESIC FLOWS ADMIT A
QUADRATIC INTEGRAL. LOCAL THEORY 424 SECTION 11.5. LINEARLY AND
QUADRATICALLY INTEGRABLE GEODESIC FLOWS ON CLOSED SURFACES 434 11.5.1.
THE TORUS .. 434 11-.5.2. THE KLEIN BOTTLE 447 11.5.3. THE SPHERE 457
11.5.4. THE PROJECTIVE PLANE 472 CHAPTER 12. LIOUVILLE CLASSIFICATION OF
INTEGRABLE GEODESIC FLOWS ON TWO-DIMENSIONAL SURFACES : 477 SECTION
12.1. THE TORUS 477 SECTION 12.2. THE KLEIN BOTTLE 491 12.2.1.
QUADRATICALLY INTEGRABLE GEODESIC FLOW ON THE KLEIN BOTTLE 491 12.2.2.
LINEARLY INTEGRABLE GEODESIC FLOWS ON THE KLEIN BOTTLE 496 12.2.3.
QUASI-LINEARLY INTEGRABLE GEODESIC FLOWS ON THE KLEIN BOTTLE 497 12.2.4.
QUASI-QUADRATICALLY INTEGRABLE GEODESIC FLOWS ON THE KLEIN BOTTLE 498
SECTION 12.3. THE SPHERE 500 12.3.1. QUADRATICALLY INTEGRABLE GEODESIC
FLOWS ON THE SPHERE 500 12.3.2. LINEARLY INTEGRABLE GEODESIC FLOWS ON
THE SPHERE 510 SECTION 12.4. THE PROJECTIVE PLANE 515 12.4.1.
QUADRATICALLY INTEGRABLE GEODESIC FLOWS ON THE PROJECTIVE PLANE 515
12.4.2. LINEARLY INTEGRABLE GEODESIC FLOWS ON THE PROJECTIVE PLANE 518
CONTENTS CHAPTER 13. ORBITAL CLASSIFICATION OF INTEGRABLE GEODESIC FLOWS
ON TWO-DIMENSIONAL SURFACES SECTION 13.1. CASE OF THE TORUS 13.1.1.
FLOWS WITH SIMPLE BIFURCATIONS (ATOMS) 13.1.2. FLOWS WITH COMPLICATED
BIFURCATIONS (ATOMS) SECTION 13.2. CASE OF THE SPHERE SECTION 13.3.
EXAMPLES OF INTEGRABLE GEODESIC FLOWS ON THE SPHERE 13.3.1. THE TRIAXIAL
ELLIPSOID 13.3.2. THE STANDARD SPHERE 13.3.3. THE POISSON SPHERE SECTION
13.4. NON-TRIVIALITY OF ORBITAL EQUIVALENCE CLASSES AND METRICS WITH
CLOSED GEODESIES CHAPTER 14. THE TOPOLOGY OF LIOUVILLE FOLIATIONS IN
CLASSICAL INTEGRABLE CASES IN RIGID BODY DYNAMICS SECTION 14.1.
INTEGRABLE CASES IN RIGID BODY DYNAMICS SECTION 14.2. TOPOLOGICAL TYPE
OF ISOENERGY 3-SURFACES 14.2.1. THE TOPOLOGY OF THE ISOENERGY SURFACE
AND THE BIFURCATION DIAGRAM 14.2.2. EULER CASE 14.2.3. LAGRANGE CASE
14.2.4. KOVALEVSKAYA CASE 14.2.5. ZHUKOVSKIL CASE 14.2.6.
GORYACHEV-CHAPLYGIN-SRETENSKII CASE 14.2.7. CLEBSCH CASE 14.2.8. STEKLOV
CASE SECTION 14.3. LIOUVILLE CLASSIFICATION OF SYSTEMS IN THE EULER CASE
SECTION 14.4. LIOUVILLE CLASSIFICATION OF SYSTEMS IN THE LAGRANGE CASE
SECTION 14.5. LIOUVILLE CLASSIFICATION OF SYSTEMS IN THE KOVALEVSKAYA
CASE SECTION 14.6. LIOUVILLE CLASSIFICATION OF SYSTEMS IN THE
GORYACHEV-CHAPLYGIN-SRETENSKII CASE SECTION 14.7. LIOUVILLE
CLASSIFICATION OF SYSTEMS IN THE ZHUKOVSKIL CASE SECTION 14.8. ROUGH
LIOUVILLE CLASSIFICATION OF SYSTEMS IN THE CLEBSCH CASE SECTION 14.9.
ROUGH LIOUVILLE CLASSIFICATION OF SYSTEMS IN THE STEKLOV CASE SECTION
14.10. ROUGH LIOUVILLE CLASSIFICATION OF INTEGRABLE FOUR-DIMENSIONAL
RIGID BODY SYSTEMS SECTION 14.11. THE COMPLETE LIST OF MOLECULES
APPEARING IN INTEGRABLE CASES OF RIGID BODY DYNAMICS CHAPTER 15.
MAUPERTUIS PRINCIPLE AND GEODESIC EQUIVALENCE SECTION 15.1. GENERAL
MAUPERTUIS PRINCIPLE SECTION 15.2. MAUPERTUIS PRINCIPLE IN RIGID BODY
DYNAMICS SECTION 15.3. CLASSICALICASES OF INTEGRABILITY IN RIGID BODY
DYNAMICS AND RELATED INTEGRABLE GEODESIC FLOWS ON THE SPHERE 15.3.1.
EULER CASE AND THE POISSON SPHERE 15.3.2. LAGRANGE CASE AND METRICS OF
REVOLUTION 15.3.3. CLEBSCH CASE AND GEODESIC FLOW ON THE ELLIPSOID
15.3.4. GORYACHEV-CHAPLYGIN CASE AND THE CORRESPONDING INTEGRABLE
GEODESIC FLOW ON THE SPHERE 15.3.5. KOVALEVSKAYA CASE AND THE
CORRESPONDING INTEGRABLE GEODESIC FLOW ON THE SPHERE SECTION 15.4.
CONJECTURE ON GEODESIC FLOWS WITH INTEGRALS OF HIGH DEGREE SECTION 15.5.
DINI THEOREM AND THE GEODESIC EQUIVALENCE OF RIEMANNIAN METRICS SECTION
15.6. GENERALIZED DINI-MAUPERTUIS PRINCIPLE 631 644 647 647 652 656 657
658 658 660 661 663 669 677 CONTENTS XI SECTION 15.7. SECTION 15.8.
CHAPTER 16. SECTION 16.1. SECTION 16.2. SECTION 16.3. SECTION 16.4.
SECTION 16.5. SECTION 16.6. SECTION 16.7. REFERENCES SUBJECT INDEX
ORBITAL EQUIVALENCE OF THE NEUMANN PROBLEM AND THE JACOBI PROBLEM
EXPLICIT FORMS OF SOME REMARKABLE HAMILTONIANS AND THEIR INTEGRALS IN
SEPARATING VARIABLES EULER CASE IN RIGID BODY DYNAMICS AND JACOBI
PROBLEM ABOUT GEODESIES ON THE ELLIPSOID. ORBITAL ISOMORPHISM
INTRODUCTION JACOBI PROBLEM AND EULER CASE LIOUVILLE FOLIATIONS ROTATION
FUNCTIONS THE MAIN THEOREM SMOOTH INVARIANTS TOPOLOGICAL NON-CONJUGACY
OF THE JACOBI PROBLEM AND THE EULER CASE 679 681 687 687 688 690 692 697
698 701 705 725
|
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| author | Bolsinov, Aleksej V. Fomenko, Anatolij Timofeevič 1945- |
| author_GND | (DE-588)119092689 |
| author_facet | Bolsinov, Aleksej V. Fomenko, Anatolij Timofeevič 1945- |
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| author_sort | Bolsinov, Aleksej V. |
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| dewey-full | 515/.39 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.39 |
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| id | DE-604.BV017732782 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:21:17Z |
| institution | BVB |
| isbn | 0415298059 |
| language | English Russian |
| lccn | 2003067457 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-010657912 |
| oclc_num | 53840099 |
| open_access_boolean | |
| owner | DE-824 DE-703 DE-384 DE-83 DE-11 |
| owner_facet | DE-824 DE-703 DE-384 DE-83 DE-11 |
| physical | XV, 730 S. Ill., graph. Darst. |
| publishDate | 2004 |
| publishDateSearch | 2004 |
| publishDateSort | 2004 |
| publisher | Chapman & Hall/CRC |
| record_format | marc |
| spelling | Bolsinov, Aleksej V. Verfasser aut Integriruemyj gamiltonovyj sistemy Integrable Hamiltonian systems geometry, topology, classification A. V. Bolsinov and A. T. Fomenko Boca Raton u.a. Chapman & Hall/CRC 2004 XV, 730 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Flots géodésiques Geometria larpcal Géodésiques (Mathématiques) Sistemas dinâmicos larpcal Sistemas hamiltonianos larpcal Systèmes hamiltoniens Topologia larpcal Geodesic flows Geodesics (Mathematics) Hamiltonian systems Integrables System (DE-588)4114032-1 gnd rswk-swf Hamiltonsches System (DE-588)4139943-2 gnd rswk-swf Topologie (DE-588)4060425-1 gnd rswk-swf Geometrie (DE-588)4020236-7 gnd rswk-swf Hamiltonsches System (DE-588)4139943-2 s Topologie (DE-588)4060425-1 s DE-604 Geometrie (DE-588)4020236-7 s Integrables System (DE-588)4114032-1 s Fomenko, Anatolij Timofeevič 1945- Verfasser (DE-588)119092689 aut GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010657912&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bolsinov, Aleksej V. Fomenko, Anatolij Timofeevič 1945- Integrable Hamiltonian systems geometry, topology, classification Flots géodésiques Geometria larpcal Géodésiques (Mathématiques) Sistemas dinâmicos larpcal Sistemas hamiltonianos larpcal Systèmes hamiltoniens Topologia larpcal Geodesic flows Geodesics (Mathematics) Hamiltonian systems Integrables System (DE-588)4114032-1 gnd Hamiltonsches System (DE-588)4139943-2 gnd Topologie (DE-588)4060425-1 gnd Geometrie (DE-588)4020236-7 gnd |
| subject_GND | (DE-588)4114032-1 (DE-588)4139943-2 (DE-588)4060425-1 (DE-588)4020236-7 |
| title | Integrable Hamiltonian systems geometry, topology, classification |
| title_alt | Integriruemyj gamiltonovyj sistemy |
| title_auth | Integrable Hamiltonian systems geometry, topology, classification |
| title_exact_search | Integrable Hamiltonian systems geometry, topology, classification |
| title_full | Integrable Hamiltonian systems geometry, topology, classification A. V. Bolsinov and A. T. Fomenko |
| title_fullStr | Integrable Hamiltonian systems geometry, topology, classification A. V. Bolsinov and A. T. Fomenko |
| title_full_unstemmed | Integrable Hamiltonian systems geometry, topology, classification A. V. Bolsinov and A. T. Fomenko |
| title_short | Integrable Hamiltonian systems |
| title_sort | integrable hamiltonian systems geometry topology classification |
| title_sub | geometry, topology, classification |
| topic | Flots géodésiques Geometria larpcal Géodésiques (Mathématiques) Sistemas dinâmicos larpcal Sistemas hamiltonianos larpcal Systèmes hamiltoniens Topologia larpcal Geodesic flows Geodesics (Mathematics) Hamiltonian systems Integrables System (DE-588)4114032-1 gnd Hamiltonsches System (DE-588)4139943-2 gnd Topologie (DE-588)4060425-1 gnd Geometrie (DE-588)4020236-7 gnd |
| topic_facet | Flots géodésiques Geometria Géodésiques (Mathématiques) Sistemas dinâmicos Sistemas hamiltonianos Systèmes hamiltoniens Topologia Geodesic flows Geodesics (Mathematics) Hamiltonian systems Integrables System Hamiltonsches System Topologie Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010657912&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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