Topological methods in hydrodynamics:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
1998
|
| Series: | Applied mathematical sciences
125 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 374 S. Ill., graph. Darst. |
| ISBN: | 038794947X 9780387949475 |
Staff View
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| 100 | 1 | |a Arnolʹd, V. I. |d 1937-2010 |e Verfasser |0 (DE-588)119540878 |4 aut | |
| 245 | 1 | 0 | |a Topological methods in hydrodynamics |c Vladimir I. Arnold ; Boris A. Khesin |
| 264 | 1 | |a New York [u.a.] |b Springer |c 1998 | |
| 300 | |a XV, 374 S. |b Ill., graph. Darst. | ||
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| 650 | 4 | |a Topologie | |
| 650 | 7 | |a Topologie |2 ram | |
| 650 | 7 | |a Topologische dynamica |2 gtt | |
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Record in the Search Index
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| adam_text | APPLIED MATHEMATICAL SCIENCES
VOLUME 125
EDITORS
J.E. MARSDEN L. SIROVICH
ADVISORS
J.K. HALE T. KAMBE
J. KELLER K. KIRCHGASSNER
B.J. MATKOWSKY C.S. PESKIN
SPRINGER
NEW YORK
BERLIN
HEIDELBERG
BARCELONA
BUDAPEST
HONG KONG
LONDON
MILAN
PARIS
SANTA CLARA
SINGAPORE
TOKYO
CONTENTS
PREFACE V
ACKNOWLEDGMENTS IX
I. GROUP AND HAMILTONIAN STRUCTURES OF FLUID DYNAMICS 1
§1. SYMMETRY GROUPS FOR A RIGID BODY AND AN IDEAL FLUID 1
§2. LIE GROUPS, LIE ALGEBRAS, AND ADJOINT REPRESENTATION 3
§3. COADJOINT REPRESENTATION OF A LIE GROUP 10
3.A. DEFINITION OF THE COADJOINT REPRESENTATION 10
3.B. DUAL OF THE SPACE OF PLANE DIVERGENCE-FREE VECTOR FIELDS 11
3.C. THE LIE ALGEBRA OF DIVERGENCE-FREE VECTOR FIELDS AND ITS
DUAL IN ARBITRARY DIMENSION 13
§4. LEFT-INVARIANT METRICS AND A RIGID BODY FOR AN ARBITRARY GROUP 14
§5. APPLICATIONS TO HYDRODYNAMICS 19
§6. HAMILTONIAN STRUCTURE FOR THE EULER EQUATIONS 25
§7. IDEAL HYDRODYNAMICS ON RIEMANNIAN MANIFOLDS 31
7.A. THE EULER HYDRODYNAMIC EQUATION ON MANIFOLDS 31
7.B. DUAL SPACE TO THE LIE ALGEBRA OF DIVERGENCE-FREE FIELDS 32
L.C INERTIA OPERATOR OF AN N-DIMENSIONAL FLUID 36
§8. PROOFS OF THEOREMS ABOUT THE LIE ALGEBRA OF DIVERGENCE-FREE
FIELDS AND ITS DUAL 39
§9. CONSERVATION LAWS IN HIGHER-DIMENSIONAL HYDRODYNAMICS 42
§10. THE GROUP SETTING OF IDEAL MAGNETOHYDRODYNAMICS 49
10.A. EQUATIONS OF MAGNETOHYDRODYNAMICS AND THE
KIRCHHOFF EQUATIONS 49
10.B. MAGNETIC EXTENSION OF ANY LIE GROUP 50
10.C. HAMILTONIAN FORMULATION OF THE KIRCHHOFF AND
MAGNETOHYDRODYNAMICS EQUATIONS 53
§11. FINITE-DIMENSIONAL APPROXIMATIONS OF THE EULER EQUATION 56
11 .A. APPROXIMATIONS BY VORTEX SYSTEMS IN THE PLANE 56
11 .B. NONINTEGRABILITY OF FOUR OR MORE POINT VORTICES 58
LL.C. HAMILTONIAN VORTEX APPROXIMATIONS IN THREE
DIMENSIONS 59
11 .D. FINITE-DIMENSIONAL APPROXIMATIONS OF DIFFEOMORPHISM
GROUPS 60
XII CONTENTS
§12. THE NAVIER-STOKES EQUATION FROM THE GROUP VIEWPOINT
II. TOPOLOGY OF STEADY FLUID FLOWS
§1. CLASSIFICATION OF THREE-DIMENSIONAL STEADY FLOWS
1 .A. STATIONARY EULER SOLUTIONS AND BERNOULLI FUNCTIONS
1 .B. STRUCTURAL THEOREMS
§2. VARIATIONAL PRINCIPLES FOR STEADY SOLUTIONS AND APPLICATIONS TO
TWO-DIMENSIONAL FLOWS
2.A. MINIMIZATION OF THE ENERGY
2.B. THE DIRICHLET PROBLEM AND STEADY FLOWS
2.C. RELATION OF TWO VARIATIONAL PRINCIPLES
2.D. SEMIGROUP VARIATIONAL PRINCIPLE FOR TWO-DIMENSIONAL
STEADY FLOWS
§3. STABILITY OF STATIONARY POINTS ON LIE ALGEBRAS
§4. STABILITY OF PLANAR FLUID FLOWS
4.A. STABILITY CRITERIA FOR STEADY FLOWS
4.B. WANDERING SOLUTIONS OF THE EULER EQUATION
§5. LINEAR AND EXPONENTIAL STRETCHING OF PARTICLES AND RAPIDLY
OSCILLATING PERTURBATIONS
5.A. THE LINEARIZED AND SHORTENED EULER EQUATIONS
5.B. THE ACTION-ANGLE VARIABLES
5.C. SPECTRUM OF THE SHORTENED EQUATION
5.D. THE SQUIRE THEOREM FOR SHEAR FLOWS
5.E. STEADY FLOWS WITH EXPONENTIAL STRETCHING OF PARTICLES
5.F. ANALYSIS OF THE LINEARIZED EULER EQUATION
5.G. INCONCLUSIVENESS OF THE STABILITY TEST FOR SPACE STEADY
FLOWS
§6. FEATURES OF HIGHER-DIMENSIONAL STEADY FLOWS
6.A. GENERALIZED BELTRAMI FLOWS
6.B. STRUCTURE OF FOUR-DIMENSIONAL STEADY FLOWS
6.C. TOPOLOGY OF THE VORTICITY FUNCTION
6.D. NONEXISTENCE OF SMOOTH STEADY FLOWS AND SHARPNESS OF
THE RESTRICTIONS
III. TOPOLOGICAL PROPERTIES OF MAGNETIC AND VORTICITY FIELDS
§1. MINIMAL ENERGY AND HELICITY OF A FROZEN-IN FIELD
1 .A. VARIATIONAL PROBLEM FOR MAGNETIC ENERGY
1 .B. EXTREMAL FIELDS AND THEIR TOPOLOGY
1 .C. HELICITY BOUNDS THE ENERGY
1 .D. HELICITY OF FIELDS ON MANIFOLDS
§2. TOPOLOGICAL OBSTRUCTIONS TO ENERGY RELAXATION
2.A. MODEL EXAMPLE: TWO LINKED FLUX TUBES
2.B. ENERGY LOWER BOUND FOR NONTRIVIAL LINKING
§3. SAKHAROV-ZELDOVICH MINIMIZATION PROBLEM
CONTENTS XIII
§4. ASYMPTOTIC LINKING NUMBER 139
4. A. ASYMPTOTIC LINKING NUMBER OF A PAIR OF TRAJECTORIES 140
4.B. DIGRESSION ON THE GAUSS FORMULA 143
4.C. ANOTHER DEFINITION OF THE ASYMPTOTIC LINKING NUMBER 144
4.D. LINKING FORMS ON MANIFOLDS 147
§5. ASYMPTOTIC CROSSING NUMBER 152
5.A. ENERGY MINORATION FOR GENERIC VECTOR FIELDS 152
5.B. ASYMPTOTIC CROSSING NUMBER OF KNOTS AND LINKS 155
5.C. CONFORMAL MODULUS OF A TORUS 159
§6. ENERGY OF A KNOT 160
6.A. ENERGY OF A CHARGED LOOP 160
6.B. GENERALIZATIONS OF THE KNOT ENERGY 163
§7. GENERALIZED HELICITIES AND LINKING NUMBERS 166
7. A. RELATIVE HELICITY 166
7.B. ERGODIC MEANING OF HIGHER-DIMENSIONAL HELICITY
INTEGRALS 168
7.C. HIGHER-ORDER LINKING INTEGRALS 174
7.D. CALUGAREANU INVARIANT AND SELF-LINKING NUMBER 177
7.E. HOLOMORPHIC LINKING NUMBER 179
§8. ASYMPTOTIC HOLONOMY AND APPLICATIONS 184
8.A. JONES-WITTEN INVARIANTS FOR VECTOR FIELDS 184
8.B. INTERPRETATION OF GODBILLON-VEY-TYPE CHARACTERISTIC
CLASSES 191
IV. DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS 195
§1. THE LOBACHEVSKY PLANE AND PRELIMINARIES IN DIFFERENTIAL
GEOMETRY 196
1 .A. THE LOBACHEVSKY PLANE OF AFFINE TRANSFORMATIONS 196
L.B. CURVATURE AND PARALLEL TRANSLATION 197
L.C. BEHAVIOR OF GEODESIES ON CURVED MANIFOLDS 201
I.D. RELATION OF THE COVARIANT AND LIE DERIVATIVES 202
§2. SECTIONAL CURVATURES OF LIE GROUPS EQUIPPED WITH A ONE-SIDED
INVARIANT METRIC 204
§3. RIEMANNIAN GEOMETRY OF THE GROUP OF AREA-PRESERVING
DIFFEOMORPHISMS OF THE TWO-TORUS 209
3.A. THE CURVATURE TENSOR FOR THE GROUP OF TORUS
DIFFEOMORPHISMS 209
3.B. CURVATURE CALCULATIONS 212
§4. DIFFEOMORPHISM GROUPS AND UNRELIABLE FORECASTS 214
4.A. CURVATURES OF VARIOUS DIFFEOMORPHISM GROUPS 214
4.B. UNRELIABILITY OF LONG-TERM WEATHER PREDICTIONS 218
§5. EXTERIOR GEOMETRY OF THE GROUP OF VOLUME-PRESERVING
DIFFEOMORPHISMS 219
§6. CONJUGATE POINTS IN DIFFEOMORPHISM GROUPS 223
XIV CONTENTS
§7. GETTING AROUND THE FINITENESS OF THE DIAMETER OF THE GROUP OF
VOLUME-PRESERVING DIFFEOMORPHISMS 225
7.A. INTERPLAY BETWEEN THE INTERNAL AND EXTERNAL GEOMETRY
OF THE DIFFEOMORPHISM GROUP 226
7.B. DIAMETER OF THE DIFFEOMORPHISM GROUPS 227
7.C. COMPARISON OF THE METRICS AND COMPLETION OF THE
GROUP OF DIFFEOMORPHISMS 228
7.D. THE ABSENCE OF THE SHORTEST PATH 230
7.E. DISCRETE FLOWS 234
7.F. OUTLINE OF THE PROOFS 235
7.G. GENERALIZED FLOWS 236
7.H. APPROXIMATION OF FLUID FLOWS BY GENERALIZED ONES 238
7.1. EXISTENCE OF CUT AND CONJUGATE POINTS ON
DIFFEOMORPHISM GROUPS 240
§8. INFINITE DIAMETER OF THE GROUP OF HAMILTONIAN DIFFEOMORPHISMS
AND SYMPLECTO-HYDRODYNAMICS 242
8.A. RIGHT-INVARIANT METRICS ON SYMPLECTOMORPHISMS 243
8.B. CALABI INVARIANT 246
8.C. BI-INVARIANT METRICS AND PSEUDOMETRICS ON THE GROUP
OF HAMILTONIAN DIFFEOMORPHISMS 252
8.D. BI-INVARIANT INDEFINITE METRIC AND ACTION FUNCTIONAL ON
THE GROUP OF VOLUME-PRESERVING DIFFEOMORPHISMS OF A
THREE-FOLD 255
V. KINEMATIC FAST DYNAMO PROBLEMS 259
§1. DYNAMO AND PARTICLE STRETCHING 259
1 .A. FAST AND SLOW KINEMATIC DYNAMOS 259
1 .B. NONDISSIPATIVE DYNAMOS ON ARBITRARY MANIFOLDS 262
* §2. DISCRETE DYNAMOS IN TWO DIMENSIONS 264
2.A. DYNAMO FROM THE CAT MAP ON A TORUS 264
2.B. HORSESHOES AND MULTIPLE FOLDINGS IN DYNAMO
CONSTRUCTIONS 267
2.C. DISSIPATIVE DYNAMOS ON SURFACES 271
2.D. ASYMPTOTIC LEFSCHETZ NUMBER 273
§3. MAIN ANTIDYNAMO THEOREMS 273
3.A. COWLING S AND ZELDOVICH S THEOREMS 273
3.B. ANTIDYNAMO THEOREMS FOR TENSOR DENSITIES 274
3.C. DIGRESSION ON THE FOKKER-PLANCK EQUATION 277
3.D. PROOFS OF THE ANTIDYNAMO THEOREMS 281
3.E. DISCRETE VERSIONS OF ANTIDYNAMO THEOREMS 284
§4. THREE-DIMENSIONAL DYNAMO MODELS 285
4.A. ROPE DYNAMO MECHANISM 285
4.B. NUMERICAL EVIDENCE OF THE DYNAMO EFFECT 286
CONTENTS XV
4.C. A DISSIPATIVE DYNAMO MODEL ON A THREE-DIMENSIONAL
RIEMANNIAN MANIFOLD 288
4.D. GEODESIC FLOWS AND DIFFERENTIAL OPERATIONS ON SURFACES
OF CONSTANT NEGATIVE CURVATURE 293
4.E. ENERGY BALANCE AND SINGULARITIES OF THE EULER EQUATION 298
§5. DYNAMO EXPONENTS IN TERMS OF TOPOLOGICAL ENTROPY 299
5.A. TOPOLOGICAL ENTROPY OF DYNAMICAL SYSTEMS 299
5.B. BOUNDS FOR THE EXPONENTS IN NONDISSIPATIVE DYNAMO
MODELS 300
5.C. UPPER BOUNDS FOR DISSIPATIVE L
1
-DYNAMOS 301
VI. DYNAMICAL SYSTEMS WITH HYDRODYNAMICAL BACKGROUND 303
§1. THE KORTEWEG-DE VRIES EQUATION AS AN EULER EQUATION 303
LA. VIRASORO ALGEBRA 303
1 .B. THE TRANSLATION ARGUMENT PRINCIPLE AND INTEGRABILITY OF
THE HIGH-DIMENSIONAL RIGID BODY 307
L.C. INTEGRABILITY OF THE KDV EQUATION 312
1 .D. DIGRESSION ON LIE ALGEBRA COHOMOLOGY AND THE
GELFAND-FUCHS COCYCLE 315
§2. EQUATIONS OF GAS DYNAMICS AND COMPRESSIBLE FLUIDS 318
2. A. BAROTROPIC FLUIDS AND GAS DYNAMICS 318
2.B. OTHER CONSERVATIVE FLUID SYSTEMS 322
2.C. INFINITE CONDUCTIVITY EQUATION 324
§3. KAHLER GEOMETRY AND DYNAMICAL SYSTEMS ON THE SPACE OF
KNOTS 326
3.A. GEOMETRIC STRUCTURES ON THE SET OF EMBEDDED CURVES 326
3.B. FILAMENT, NONLINEAR SCHRODINGER, AND HEISENBERG
CHAIN EQUATIONS 331
3.C. LOOP GROUPS AND THE GENERAL LANDAU-LIFSCHITZ
EQUATION 333
§4. SOBOLEV S EQUATION 335
§5. ELLIPTIC COORDINATES FROM THE HYDRODYNAMICAL VIEWPOINT 340
5.A. CHARGES ON QUADRICS IN THREE DIMENSIONS 340
5.B. CHARGES ON HIGHER-DIMENSIONAL QUADRICS 342
REFERENCES 345
INDEX 369
|
| any_adam_object | 1 |
| author | Arnolʹd, V. I. 1937-2010 Khesin, Boris A. 1964- |
| author_GND | (DE-588)119540878 (DE-588)120262487 |
| author_facet | Arnolʹd, V. I. 1937-2010 Khesin, Boris A. 1964- |
| author_role | aut aut |
| author_sort | Arnolʹd, V. I. 1937-2010 |
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| building | Verbundindex |
| bvnumber | BV011960416 |
| callnumber-first | Q - Science |
| callnumber-label | QA1 |
| callnumber-raw | QA1 |
| callnumber-search | QA1 |
| callnumber-sort | QA 11 |
| callnumber-subject | QA - Mathematics |
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| classification_tum | PHY 220f PHY 014f |
| ctrlnum | (OCoLC)36509746 (DE-599)BVBBV011960416 |
| dewey-full | 532/.5 510 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 532 - Fluid mechanics 510 - Mathematics |
| dewey-raw | 532/.5 510 |
| dewey-search | 532/.5 510 |
| dewey-sort | 3532 15 |
| dewey-tens | 530 - Physics 510 - Mathematics |
| discipline | Physik Mathematik |
| format | Book |
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| id | DE-604.BV011960416 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:19:13Z |
| institution | BVB |
| isbn | 038794947X 9780387949475 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008087711 |
| oclc_num | 36509746 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-703 DE-20 DE-29T DE-91G DE-BY-TUM DE-634 DE-83 DE-188 DE-19 DE-BY-UBM |
| owner_facet | DE-355 DE-BY-UBR DE-703 DE-20 DE-29T DE-91G DE-BY-TUM DE-634 DE-83 DE-188 DE-19 DE-BY-UBM |
| physical | XV, 374 S. Ill., graph. Darst. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Springer |
| record_format | marc |
| series | Applied mathematical sciences |
| series2 | Applied mathematical sciences |
| spelling | Arnolʹd, V. I. 1937-2010 Verfasser (DE-588)119540878 aut Topological methods in hydrodynamics Vladimir I. Arnold ; Boris A. Khesin New York [u.a.] Springer 1998 XV, 374 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applied mathematical sciences 125 Hydrodynamica gtt Hydrodynamique Hydrodynamique ram Topologie Topologie ram Topologische dynamica gtt Hydrodynamics Topology Hydrodynamik (DE-588)4026302-2 gnd rswk-swf Topologie (DE-588)4060425-1 gnd rswk-swf Topologische Methode (DE-588)4312758-7 gnd rswk-swf Hydrodynamik (DE-588)4026302-2 s Topologie (DE-588)4060425-1 s DE-604 Topologische Methode (DE-588)4312758-7 s Khesin, Boris A. 1964- Verfasser (DE-588)120262487 aut Applied mathematical sciences 125 (DE-604)BV000005274 125 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008087711&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Arnolʹd, V. I. 1937-2010 Khesin, Boris A. 1964- Topological methods in hydrodynamics Applied mathematical sciences Hydrodynamica gtt Hydrodynamique Hydrodynamique ram Topologie Topologie ram Topologische dynamica gtt Hydrodynamics Topology Hydrodynamik (DE-588)4026302-2 gnd Topologie (DE-588)4060425-1 gnd Topologische Methode (DE-588)4312758-7 gnd |
| subject_GND | (DE-588)4026302-2 (DE-588)4060425-1 (DE-588)4312758-7 |
| title | Topological methods in hydrodynamics |
| title_auth | Topological methods in hydrodynamics |
| title_exact_search | Topological methods in hydrodynamics |
| title_full | Topological methods in hydrodynamics Vladimir I. Arnold ; Boris A. Khesin |
| title_fullStr | Topological methods in hydrodynamics Vladimir I. Arnold ; Boris A. Khesin |
| title_full_unstemmed | Topological methods in hydrodynamics Vladimir I. Arnold ; Boris A. Khesin |
| title_short | Topological methods in hydrodynamics |
| title_sort | topological methods in hydrodynamics |
| topic | Hydrodynamica gtt Hydrodynamique Hydrodynamique ram Topologie Topologie ram Topologische dynamica gtt Hydrodynamics Topology Hydrodynamik (DE-588)4026302-2 gnd Topologie (DE-588)4060425-1 gnd Topologische Methode (DE-588)4312758-7 gnd |
| topic_facet | Hydrodynamica Hydrodynamique Topologie Topologische dynamica Hydrodynamics Topology Hydrodynamik Topologische Methode |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008087711&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005274 |
| work_keys_str_mv | AT arnolʹdvi topologicalmethodsinhydrodynamics AT khesinborisa topologicalmethodsinhydrodynamics |