Symplectic twist maps: global variational techniques
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Singapore [u.a.]
World Scientific
2001
|
| Schriftenreihe: | Advanced series in nonlinear dynamics
18 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references (p. [293]-301) and index |
| Beschreibung: | XVIII, 301 S. graph. Darst. |
| ISBN: | 9810205899 |
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| 100 | 1 | |a Golé, Christophe |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Symplectic twist maps |b global variational techniques |c Christophe Golé |
| 264 | 1 | |a Singapore [u.a.] |b World Scientific |c 2001 | |
| 300 | |a XVIII, 301 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Advanced series in nonlinear dynamics |v 18 | |
| 500 | |a Includes bibliographical references (p. [293]-301) and index | ||
| 650 | 7 | |a Análise global |2 larpcal | |
| 650 | 4 | |a Applications déviantes (Mathématiques) | |
| 650 | 4 | |a Twist mappings (Mathematics) | |
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Datensatz im Suchindex
| _version_ | 1804133179286618112 |
|---|---|
| adam_text | A D V A N C E D S E R I E S I I N O N L I N E A R D Y N A M I C S V O L
U M E 1 8 SYMPLECTIC TWIST MAPS GLOBAL VARIATIONAL TECHNIQUES CHRISTOPHE
GOLE SMITH COLLEGE, NORTHAMPTON, MASSACHUSETTS, USA WORLD SCIENTIFIC
SINGAPORE »NEW JERSEY LONDON* HONG KONG CONTENT INTRODUCTION 1 1 FALL
FROM PARADISE 1 2 BILLIARDS AND BROKEN GEODESIES 4 3 AN ANCESTOR OF
SYMPLECTIC TOPOLOGY 8 TWIST MAPS OF THE ANNULUS 11 4 MONOTONE TWIST MAPS
OF THE ANNULUS 11 A DEFINITIONS 11 B COMMENTS ON THE DEFINITION 12 C
TWIST MAPS OF THE CYLINDER 16 5 GENERATING FUNCTIONS AND VARIATIONAL
SETTING 17 A GENERATING FUNCTIONS 17 B VARIATIONAL PRINCIPLE 18 C
PERIODIC ORBITS 20 D ROTATION NUMBERS 21 6 EXAMPLES . 21 A STANDARD MAP
21 B ELLIPTIC FIXED POINTS OF AREA PRESERVING MAPS 23 C THE
FRENKEL-KONTOROVA MODEL 24 D BILLIARD MAPS 25 7 THE POINCARE-BIRKHOFF
THEOREM 27 THE AUBRY-MATHER THEOREM 31 8 INTRODUCTION 31 A MOTIVATION
AND STATEMENT OF THE THEOREM 31 B FROM THE ANNULUS TO THE CYLINDER 33 9
CYCLICALLY ORDERED SEQUENCES AND ORBITS 34 10 MINIMIZING ORBITS 36 11 CO
ORBITS OF ALL ROTATION NUMBERS 40 A EXISTENCE OF CO PERIODIC ORBITS 40 *
B EXISTENCE OF CO ORBITS OF IRRATIONAL ROTATION NUMBERS 41 12
AUBRY-MATHER SETS 41 XIV CONTENT 13 APPENDIX: CYCLICALLY ORDERED
SEQUENCES AND CIRCLE MAPS 46 A PROOFS OF LEMMAS 9.1 AND 9.2 47 B
DYNAMICS OF CIRCLE HOMEOMORPHISMS 49 3 GHOST CIRCLES 53 14 GRADIENT FLOW
OF THE ACTION 54 A DEFINITION OF THE FLOW 54 B ORDER PROPERTIES OF THE
FLOW 55 15 THE GRADIENT FLOW AND THE AUBRY-MATHER THEOREM 57 16 GHOST
CIRCLES 59 17 CONSTRUCTION OF GHOST CIRCLES 63 A GHOST CIRCLES THROUGH
ANY AUBRY-MATHER SETS 63 B SMOOTH, RATIONAL GHOST CIRCLES 64 18
CONSTRUCTION OF DISJOINT GHOST CIRCLES 67 19 PROOF OF LEMMA 18.5 70 20
PROOF OF THEOREM 18.1 73 A RATIONAL CJS 74 B IRRATIONAL C^ S 74 21*
REMARKS AND APPLICATIONS 78 A* REMARKS 78 B APPROXIMATE ACTION-ANGLE
VARIABLES FOR AN ARBITRARY TWIST MAP 79 C* PARTITION FOR TRANSPORT 80 D*
AN EXTENSION OF AUBRY S FUNDAMENTAL LEMMA 81 22 PROOFS OF MONOTONICITY
AND OF THE STURMIAN LEMMA 82 A PROOF OF STRICT MONOTONOCITY - 82 B PROOF
OF THE STURMIAN LEMMA 83 4 SYMPLECTIC TWIST MAPS 87 23 SYMPLECTIC TWIST
MAPS OF T X IR N 88 A DEFINITION 88 B COMMENTS ON THE DEFINITION 89 C
THE VARIATIONAL SETTING 89 24 EXAMPLES 91 A THE GENERALIZED STANDARD MAP
91 B HAMILTONIAN SYSTEMS 92 C ELLIPTIC FIXED POINTS 93 CONTENT XV 25
MORE ON GENERATING FUNCTIONS 95 A HOMEOMORPHISM BETWEEN TWIST MAPS AND
GENERATING FUNCTIONS 95 B LOCAL VS. GLOBAL TWIST 96 C DIFFERENTIAL OF
THE MAP VS. GENERATING FUNCTION 98 26 SYMPLECTIC TWIST MAPS ON GENERAL
COTANGENT BUNDLES OF COMPACT MANIFOLDS 99 A DEFINITION 99 B MAPS VS.
FUNCTIONS, REVISITED 100 C EXAMPLES 101 PERIODIC ORBITS FOR SYMPLECTIC
TWIST MAPS OF T N X ]R N 103 27 PRESENTATION OF THE RESULTS 103 A
PERIODIC ORBITS AND ROTATION VECTORS 103 B THEOREMS OF EXISTENCE OF
PERIODIC ORBITS 104 C COMMENTS ON THE ASYMPTOTIC CONDITIONS 105 D
HISTORY 106 28 FINITE DIMENSIONAL VARIATIONAL SETTING 107 29 SECOND
VARIATION AND NONDEGENERATE PERIODIC ORBITS 110 30 THE COERCIVE CASE 112
31 ASYMPTOTICALLY LINEAR SYSTEMS 114 32 GHOST TORI 116 33 HYPERBOLICITY
VS. ACTION MINIMIZERS 118 INVARIANT MANIFOLDS 123 34 THE THEORY OF
KOLMOGOROV-AMOLD-MOSER 123 35 PROPERTIES OF INVARIANT TORI 127 A
RECURRENT INVARIANT TORIC GRAPHS ARE LAGRANGIAN 127 B ORBITS ON
LAGRANGIAN INVARIANT TORI ARE MINIMIZERS 128 C BIRKHOFF S GRAPH THEOREM
129 D* AUBRY-MATHER THEOREM VIA TRIMMING 132 E* GENERALIZATIONS OF
BIRKHOFF S GRAPH THEOREM TO HIGHER DIMENSIONS 133 36 (UN)STABLE
MANIFOLDS AND HETEROCLINIC ORBITS 135 A (UN)STABLE MANIFOLDS 135 B
VARIATIONAL APPROACH TO HETEROCLINIC ORBITS 136 C SPLITTING OF
SEPARATRICES AND PQINCARE-MELNIKOV FUNCTION 138 37* INSTABILITY,
TRANSPORT AND DIFFUSION 141 A* SOME QUESTIONS ABOUT STABILITY 141 XVI
CONTENT B* ANSWER TO QUESTION 1: SHADOWING OF AUBRY-MATHER SETS 141 C*
PARTIAL ANSWER TO QUESTION 2: UNBOUNDED ORBITS 142 D* PARTIAL ANSWER TO
QUESTION 3: CONVERSE KAM THEORY 1 4 3 7 HAMILTONIAN SYSTEMS VS. TWIST
MAPS 145 38 CASE STUDY: THE GEODESIC FLOW 146 A A FEW FACTS ABOUT
RIEMANNIAN GEOMETRY 146 B THE GEODESIC FLOW AS A TWIST MAP 1 49 C THE
METHOD OF BROKEN GEODESIES 1 51 D THE STANDARD MAP ON COTANGENT BUNDLES
OF HYPERBOLIC MANIFOLDS 152 39 DECOMPOSITION OF HAMILTONIAN MAPS INTO
TWIST MAPS 154 A LEGENDRE CONDITION VS. TWIST CONDITION 1 54 B
LAGRANGIAN FORMULATION OF THEOREM 39.1 157 C GLOBAL TWIST: THE CASE OF
THE TORUS 1 59 D DECOMPOSITION OF HAMILTONIAN MAPS INTO TWIST MAPS , 162
40 RETURN MAPS IN HAMILTONIAN SYSTEMS 164 41 SUSPENSION OF SYMPLECTIC
TWIST MAPS BY HAMILTONIAN FLOWS 165 A SUSPENSION WITH FIBER CONVEXITY 1
66 B SUSPENSION WITHOUT CONVEXITY 170 8 PERIODIC ORBITS FOR HAMILTONIAN
SYSTEMS 173 42 PERIODIC ORBITS IN THE COTANGENT OF THE N-TORUS 174 A
OPTICAL HAMILTONIANS 1 74 B ASYMPTOTICALLY QUADRATIC HAMILTONIANS 175 C
REMARKS ABOUT THE ABOVE RESULTS 176 43 PERIODIC ORBITS IN GENERAL
COTANGENT SPACES 177 A THE DISCRETE VARIATIONAL SETTING 178 B THE
ISOLATING BLOCK 179 C END OF PROOF OF THEOREM 43.1 182 D PERIODIC ORBITS
OF DIFFERENT HOMOTOPY CLASSES 184 44 LINKING OF SPHERES 186 9
GENERALIZATIONS OF THE AUBRY-MATHER THEOREM 191 45* THEORY FOR FUNCTIONS
ON LATTICES AND PDE S 192 A* FUNCTIONS ON LATTICES 192 B* PDE S 193 C*
LAMINATIONS BY MINIMAL SURFACES 195 CONTENT XVII 46* MONOTONE RECURRENCE
RELATIONST 196 47* ANTI-LNTEGRABLE LIMIT 197 48* MATHER S THEORY OF
MINIMAL MEASURES 199 A* LAGRANGIAN MINIMIZERS 200 B* ERGODIC THEORY 201
C* MINIMAL MEASURES 204 D* EXAMPLES AND COUNTEREXAMPLES 209 49* THE CASE
OF HYPERBOLIC MANIFOLDS 212 A* HYPERBOLIC COUNTEREXAMPLE 212 B* ALL
ROTATION DIRECTIONS IN HYPERBOLIC MANIFOLDS 213 50* CONCLUDING REMARKS
216 10 GENERATING PHASES AND SYMPLECTIC TOPOLOGY 217 51 CHAPERON S
METHOD AND THE THEOREM OF CONLEY-ZEHNDER 218 A A NEW ACTION FUNCTION 21
8 B INTERPRETATION OF W AS ACTION OF BROKEN GEODESIC 221 C THE
CONLEY-ZEHNDER THEOREM 22 2 52 GENERATING PHASES AND SYMPLECTIC GEOMETRY
224 A GENERATING PHASES AND LAGRANGIAN MANIFOLDS 224 B SYMPLECTIC
PROPERTIES OF GENERATING PHASES 226 C THE ACTION FUNCTION GENERATES THE
GRAPH OF F 227 D SYMPLECTIC REDUCTION 229 E FURTHER APPLICATIONS OF
GENERATING PHASES 231 A1 OVERVIEW OF SYMPLECTIC GEOMETRY 233 53
SYMPLECTIC VECTOR SPACES 233 54 SUBSPACES OF A SYMPLECTIC VECTOR SPACE
236 55 SYMPLECTIC LINEAR MAPS 238 56 SYMPLECTIC MANIFOLDS 241 57
COTANGENT BUNDLES 243 A SOME DEFINITIONS 243 B COTANGENT BUNDLES AS
SYMPLECTIC MANIFOLD 244 C NOTABLE LAGRANGIAN SUBMANIFOLDS OF COTANGENT
BUNDLES 246 58 HAMILTONIAN SYSTEMS 247 A LAGRANGIAN SYSTEMS VERSUS
HAMILTONIAN SYSTEMS 247 B HAMILTONIAN SYSTEMS ON SYMPLECTIC MANIFOLDS
251 C INVARIANTS OF THE HAMILTONIAN FLOW 252 XVIII CONTENT A2 SOME
TOPOLOGICAL TOOLS 259 59* HANDS ON INTRODUCTION TO HOMOLOGY THEORY 260
A* FINITE CELL COMPLEXES 260 B* CELLULAR HOMOLOGY 26 2 C* COHOMOLOGY 266
60* MORSE THEORY 267 61 CONTROLLING THE TOPOLOGY OF INVARIANT SETS 273
62 TOPOLOGICAL PROOFS 276 A PROOF OF THE CUPLENGTH ESTIMATE IN THEOREM
61.2 276 B* THE BETTI NUMBER ESTIMATE OF THEOREM 61.2 AND CONLEY S
THEORY 278 C FLOER S LEMMA 280 D PROOF OF PROPOSITION 61.4 282 E*
FLOER S THEOREM OF GLOBAL CONTINUATION OF HYPERBOLIC INVARIANT SETS 283
63 GENERATING PHASES QUADRATIC AT INFINITY 284 GENERATING PHASES ON
PRODUCT SPACES 284 GENERATING PHASES ON VECTOR BUNDLES 285 64* COVERING
SPACES, LIFTS AND FUNDAMENTAL GROUP 288 BIBLIOGRAPHY 293 INDEX 303
|
| any_adam_object | 1 |
| author | Golé, Christophe |
| author_facet | Golé, Christophe |
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| author_sort | Golé, Christophe |
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| dewey-raw | 514.74 |
| dewey-search | 514.74 |
| dewey-sort | 3514.74 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV019721172 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T20:04:36Z |
| institution | BVB |
| isbn | 9810205899 |
| language | English |
| lccn | 2002284360 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-013048294 |
| oclc_num | 48960458 |
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| owner | DE-29T DE-11 DE-188 |
| owner_facet | DE-29T DE-11 DE-188 |
| physical | XVIII, 301 S. graph. Darst. |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | World Scientific |
| record_format | marc |
| series | Advanced series in nonlinear dynamics |
| series2 | Advanced series in nonlinear dynamics |
| spelling | Golé, Christophe Verfasser aut Symplectic twist maps global variational techniques Christophe Golé Singapore [u.a.] World Scientific 2001 XVIII, 301 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Advanced series in nonlinear dynamics 18 Includes bibliographical references (p. [293]-301) and index Análise global larpcal Applications déviantes (Mathématiques) Twist mappings (Mathematics) Advanced series in nonlinear dynamics 18 (DE-604)BV004464593 18 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013048294&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Golé, Christophe Symplectic twist maps global variational techniques Advanced series in nonlinear dynamics Análise global larpcal Applications déviantes (Mathématiques) Twist mappings (Mathematics) |
| title | Symplectic twist maps global variational techniques |
| title_auth | Symplectic twist maps global variational techniques |
| title_exact_search | Symplectic twist maps global variational techniques |
| title_full | Symplectic twist maps global variational techniques Christophe Golé |
| title_fullStr | Symplectic twist maps global variational techniques Christophe Golé |
| title_full_unstemmed | Symplectic twist maps global variational techniques Christophe Golé |
| title_short | Symplectic twist maps |
| title_sort | symplectic twist maps global variational techniques |
| title_sub | global variational techniques |
| topic | Análise global larpcal Applications déviantes (Mathématiques) Twist mappings (Mathematics) |
| topic_facet | Análise global Applications déviantes (Mathématiques) Twist mappings (Mathematics) |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013048294&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV004464593 |
| work_keys_str_mv | AT golechristophe symplectictwistmapsglobalvariationaltechniques |