Construction of mappings for Hamiltonian systems and their applications:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
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Berlin [u.a.]
Springer
2006
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| Series: | Lecture notes in physics
691 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XIV, 378 S. graph. Darst. 24 cm |
| ISBN: | 9783540309154 3540309152 |
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| 100 | 1 | |a Abdullaev, Sadrilla S. |d 1951- |e Verfasser |0 (DE-588)131424955 |4 aut | |
| 245 | 1 | 0 | |a Construction of mappings for Hamiltonian systems and their applications |c Sadrilla S. Abdullaev |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2006 | |
| 300 | |a XIV, 378 S. |b graph. Darst. |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture notes in physics |v 691 | |
| 650 | 4 | |a Applications (Mathématiques) | |
| 650 | 4 | |a Systèmes hamiltoniens | |
| 650 | 4 | |a Hamiltonian systems | |
| 650 | 4 | |a Mappings (Mathematics) | |
| 650 | 0 | 7 | |a Symplektische Abbildung |0 (DE-588)4332207-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Hamiltonsches System |0 (DE-588)4139943-2 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Hamiltonsches System |0 (DE-588)4139943-2 |D s |
| 689 | 0 | 1 | |a Symplektische Abbildung |0 (DE-588)4332207-4 |D s |
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| 830 | 0 | |a Lecture notes in physics |v 691 |w (DE-604)BV000003166 |9 691 | |
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| adam_text | SADRILLA S. ABDULLAEV CONSTRUCTION OF MAPPINGS FOR HAMILTONIAN SYSTEMS
AND THEIR APPLICATIONS 4Y SPRINGER CONTENTS 1 BASICS OF HAMILTONIAN
MECHANICS 1 1.1 HAMILTON EQUATIONS 1 1.1.1 INVARIANTS BF MOTION 2 1.1.2
HAMILTONIAN EQUATIONS IN EXTENDED PHASE SPACE 3 1.1.3 FORMULATION OF
HAMILTONIAN EQUATIONS WITH ONE COORDINATE AS INDEPENDENT VARIABLE
INSTEAD OF T 4 1.2 THE HAMILTON-JACOBI METHOD 4 1.2.1 CANONICAL CHANGE
OF VARIABLES 4 1.2.2 THE HAMILTON-JACOBI EQUATION 5 1.2.3 THE JACOBI S
THEOREM 6 1.3 ACTION-ANGLE VARIABLES 7 1.3.1 INTEGRABLE HAMILTONIAN
SYSTEMS 7 1.3.2 HAMILTONIAN IN ACTION-ANGLE VARIABLES 8 1.3.3 SYSTEMS
WITH ONE DEGREE OF FREEDOM 9 1.3.4 MANY-DIMENSIONAL SYSTEMS WITH
SEPARABLE VARIABLES . 9 1.4 PARTICLES IN A WAVE FIELD 12 1.4.1 TRAPPED
MOTION OF PARTICLES 13 1.4.2 UNTRAPPED MOTION OF PARTICLES: H LUQ 15 ;
1.4.3 MOTION ON SEPARATRIX 16 ; 1.5 ON SYMPLECTIC NUMERICAL INTEGRATION
OF HAMILTONIAN SYSTEMS 171 1.6 BIBLIOGRAPHIC NOTES / 18 2 PERTURBATION
THEORY FOR NEARLY INTEGRABLE SYSTEMS 21 2.1 PERTURBATION METHODS IN
INFINITE TIME INTERVALS 21 2.1.1 A FUNDAMENTAL PROBLEM OF DYNAMICS : 22
2.1.2 THE MAIN IDEA OF AVERAGING PROCEDURE 22 2.1.3 DETERMINATION OF THE
GENERATING FUNCTION 23 2.1.4 VON ZEIPEL S METHOD .-.--.. 25 2.2 LIE
TRANSFORM METHODS .- 26 ,,_ 2.3 TIME-DEPENDENT PERTURBATION SERIES 28
2.3.1 CAUCHY PROBLEM 29 2.3.2 GENERATION FUNCTIONS 29 2.3.3 REMARKS 31
2.3.4 TIME-DEPENDENT LIE TRANSFORM METHOD . . . . 33 X CONTENTS 2.3.5
EXAMPLE DUFFING OSCILLATOR 33 2.4 METHOD OF SUCCESSIVE TRANSFORMATIONS
36 2.5 BIBLIOGRAPHIC NOTES 37 3 MAPPINGS FOR PERTURBED SYSTEMS . ~.; 39
3.1 POINCARE MAPPINGS 39 3.2 METHOD OF A PRIORI ASSUMPTION 41 3.3 METHOD
OF DELTA FUNCTIONS 44 3.4 THE STANDARD MAP 47 3.5 EXACT MAPPINGS OF
HAMILTONIAN SYSTEMS 49 3.6 DIFFICULTIES IN CONSTRUCTING MAPPINGS 50 4
METHOD OF CANONICAL TRANSFORMATION FOR CONSTRUCTING MAPPINGS , 53 4.1
CANONICAL TRANSFORMATION AND MAPPING 54 4.1.1 NONSYMMETRIC FORMS OF MAPS
55 4.1.2 SYMMETRIC MAP . . . . 56 4.1.3 THE GENERATING FUNCTION OF
MAPPINGS 57 4.2 ACCURACY OF MAPS 58 4.2.1 PARTICLES IN A
SINGLE-FREQUENCY WAVE FIELD 59 4.2.2 ACCURACY^ OF THE SYMMETRIC MAP 62
4.3 MAPPINGS FOR HAMILTONIAN SYSTEMS WITH A BROAD PERTURBATION SPECTRUM
64 4.3.1 NON-SYMMETRIC FORMS OF MAPS 68 4.3.2 STANDARD HAMILTONIAN AND
CORRESPONDING MAPPINGS . . 69 4.4 MAPPINGS WITH LIE GENERATING FUNCTIONS
71 4.5 POINCARE MAPS AT ARBITRARY SECTIONS OF PHASE-SPACE 73 4.6 METHOD
OF SUCCESSIVE CANONICAL TRANSFORMATIONS 78 4.7 SUMMARY .. 80 5 MAPPINGS
NEAR SEPARATRIX.X.THEORY 83 5.1 SEPARATRIX AND MAPPINGS . 83 5.1.1 THE
CONVENTIONAL SEPARATRIX (WHISKER) MAP 85 5.1.2 SHORTCOMINGS OF
CONVENTIONAL SEPARATRIX MAPPINGS ... 87 5.2 THE HAMILTON-JACOBI METHOD
TO CONSTRUCT MAPS NEAR A SEPARATRIX 88 5.2.1 MAPPING ALONG SINGLE
SADDLE-SADDLE CONNECTION 90 5.2.2 CALCULATION OF THE GENERATING FUNCTION
93 5.2.3 SYMMETRIC MAPPINGS 94 5.2.4 NONSYMMETRIC MAPPINGS 94 5.2.5
PROPERTIES OF THE MELNIKOV TYPE INTEGRALS K N (H) AND L N {H) 96 5.3
SIMPLIFICATION OF MAPPINGS 97 5.3.1 PRIMARY RESONANT APPROXIMATION 98
5.3.2 SIMPLIFIED FORM OF MAPPINGS 98 CONTENTS XI 5.3.3 SEPARATRIX
MAPPING APPROXIMATION 99 5.4 MAPPING AT ARBITRARY SECTIONS OF PHASE
SPACE 100 5.4.1 MAPPING TO A SECTION E C 101 5.4.2 MAPPING TO SECTION E$
=R) 103 5.5 CONCLUSION 77 104 MAPPINGS NEAR SEPARATRIX. EXAMPLES 105 6.1
MOTION IN A PERTURBED DOUBLE-WELL POTENTIAL 105 6.2 MAPPING FOR THE
PERIODICALLY DRIVEN PENDULUM 114 6.2.1 BEHAVIOR OF INTEGRALS K N (H) AND
L N (H) (5.31) 116 6.2.2 MAPPINGJTO SECTIONS E S 118 6.2.3 MAPPING TO
SECTIONS E C 122 6.3 MAPPING FOR THE 6.3.1 MAPPING 127 6.3.2 THE
SYMMETRIC MAPPING 127 6.3.3 A NONSYMMETRIC MAPPING 130 6.3.4 COMPARISON
WITH A NUMERICAL INTEGRATION 131 6.4 THE KEPLER MAP 132 6.5 COMMENTS ON
SEPARATRIX MAP METHODS 136 6.6 BIBLIOGRAPHIC NOTES : 137 THE KAM THEORY
CHAOS NONTWIST AND NONSMOOTH MAPS 139 7.1 CONSERVATION OF CONDITIONALLY
PERIODIC MOTIONS. THE KAM THEORY 139 7.1.1 INVARIANT TORI FOR MAPPING -.
140 7.1.2 DESTRUCTION OF RESONANT ORBITS: NONLINEAR RESONANCE . 142
7.1.3 CHAOTIC LAYER NEAR A SEPARATRIX 144 7.1.4 LYAPUNOV EXPONENTS 146
7.2 APPLICABILITY OF KAM THEORY 148 7.2.1 ON THE SMALLNESS OF
PERTURBATIONS 148 : 7.2.2 ON THE SMOOTHNESS OFJPERTURBATIONS 150; 7.3
NON-TWIST MAPS !-, 150 7.3.1 DYNAMICS OF SYSTEMS WITH A NON-MONOTONIC
FREQUENCY 151 7.3.2 BEHAVIOR NEAR LOCAL MAXIMA OR MINIMA 152 7.3.3
BEHAVIOR NEAR A BENDING POINT 155 7.3.4 NON-TWIST STANDARD MAPS 155
7.3.5 FIXED POINTS AND TRANSIT TO CHAOS 159 7.4 NON-SMOOTH MAPPINGS 160
7.4.1 INTERMITTENCE IN NONTWIST SYSTEMS -. . 162 7.4.2 SIMPLIFIED
NON-SMOOTH MAPPINGS 168 7.4.3 A NONTWIST MAP AND INTERMITTENCY 169 7.5
SUPPRESSION OF CHAOS IN SMOOTH HAMILTONIAN SYSTEMS 173 7.6 BIBLIOGRAPHIC
NOTES 174 XII CONTENTS 8 RESCALING INVARIANCE OF HAMILTONIAN SYSTEMS
NEAR SADDLE POINTS 175 8.1 RESCALING INVARIANCE NEAR SADDLE POINTS AND
SEPARATRIX MAPS 176 8.1.1 STRUCTURE OF PHASE SPACE NEAR SADDLE POINTS
176 8.1.2 UNIVERSAL RESCALING INVARIANCE 177 8.1.3 PROOF OF THE
RESCALING INVARIANCE OF HAMILTONIAN EQUATIONS 179 8.1.4 SEPARATRIX
MAPPING APPROACH 181 8.1.5 RESCALING INVARIANCE IN PARAMETER SPACE 182
8.2 RESCALING INVARIANCE DUE TO THE SYMMETRY OF HAMILTONIANS . . 183
8.2.1 SEPARATRIX MAPPING ANALYSIS 184 8.3 2D-PERIODIC VORTICAL FLOW 187
* 8.3.1 MODEL 187 8.3.2 RESCALING INVARIANCE PROPERTY 189 8.3.3
SEPARATRIX MAPS OF THE SYSTEM 191 8.3.4 ON THE VALIDITY CONDITIONS OF
THE RESCALING INVARIANCE PROPERTY 193 8.4 SUMMARY 195 9 CHAOTIC
TRANSPORT IN STOCHASTIC LAYERS 197 9.1 STATISTICAL DESCRIPTION OF
CHAOTIC DYNAMICAL SYSTEMS 197 9.1.1 ERGODICITY AND MIXING 197 9.1.2
KINETIC DESCRIPTION 199 9.1.3 ANOMALOUS DIFFUSION 201 9.2 NON-GAUSSIAN
STATISTICS IN STOCHASTIC LAYERS 202 9.2.1 MEAN RESIDENCE TIME 202 9.2.2
STATISTICS OF POINCARE RECURRENCES 203 9.3 CHAOTIC TRANSPORT IN
STOCHASTIC LAYERS. THREE-WAVE FIELD MODEL 206 9.3.1 ADVECTION 206 .
9.3.2 ANOMALOUS DIFFUSION 206 9.3.3 PROBABILITY DENSITY FUNCTION 208 9.4
CHAOTIC TRANSPORT IN 2D-PERIODIC VORTICAL FLOW 210 9.4.1 VARIATION OF
DIFFUSION REGIMES 211 9.4.2 SUPERDIFFUSIVE REGIME. LEVY FLIGHTS 213
9.4.3 FIXED POINTS OF FLIGHT ISLANDS 215 9.5 CONCLUSIONS 217 10 MAGNETIC
FIELD LINES IN FUSION PLASMAS .-. .... 219 10. 1 MAGNETIC FIELD LINES AS
HAMILTONIAN SYSTEM 219 10.1.1 EQUILIBRIUM MAGNETIC FIELD 220 10.1.2
HAMILTONIAN FIELD LINE EQUATIONS 220 10.1.3 HAMILTONIAN FORMULATION OF
FIELD LINE EQUATIONS IN A TOROIDAL SYSTEM 221 10.1.4 THE STANDARD
MAGNETIC FIELD * 224 CONTENTS XIII 10.1.5 EQUILIBRIUM MAGNETIC FIELD
WITH THE SHAFRANOV SHIFT . 225 10.2 HAMILTONIAN EQUATIONS IN THE
PRESENCE OF MAGNETIC PERTURBATIONS 229 10.2.1 CYLINDRICAL MODEL OF
PLASMAS 230 10.2.2 MAGNETIC PERTURBATIONS IN TOROIDAL PLASMAS 232 10.2.3
ASYMPTOTICS OF THE TRANSFORMATION MATRIX ELEMENTS S MM (IP) 233 10.2.4
ASYMPTOTIC BEHAVIOR OF H MN (IP) 236 10.3 MAPPING OF FIELD LINES 237
10.4 MAPPINGS AS MODELS FOR MAGNETIC FIELD LINES 240 C 10.4.1 THE
STANDARD MAP AND ITS GENERALIZATIONS 240 10.4.2 THE WOBIG-MENDONGA MAP
241 10.4.3 THE TOKAMAP 242 R 10.5 CONTINUOUS HAMILTONIAN SYSTEM AND
TOKAMAP 244 10.5.1 THE SYMMETRIC TOKAMAP 246 10.5.2 COMPARISON OF THE
TOKAMAP AND THE SYMMETRIC TOKAMAP 247 10.5.3 THE REVTOKAMAP AND THE
SYMMETRIC REVTOKAMAP .... 251 10.6 OTHER MAPPING MODELS OF FIELD LINES
252 10.6.1 ANALYTICAL MODELS 252 10.6.2 NUMERICAL MAPPING MODELS 253
10.7 CONCLUSIONS 254 11 MAPPING OF FIELD LINES IN ERGODIC DIVERTOR
TOKAMAKS .... 255 11.1 ERGODIC DIVERTOR CONCEPT 255 11.1.1 MAPPINGS TO
STUDY ERGODIC DIVERTORS 256 11.2 MAGNETIC STRUCTURE OF THE DED 257
11.2.1 SET OF DIVERTOR COILS AND MAGNETIC PERTURBATIONS 257 11.3
SPECTRUM OF MAGNETIC PERTURBATIONS 259 11.4 FORMATION OF THE ERGODIC
ZONE 263 : 11.5 STATISTICAL PROPERTIES OF FIELD LINES 264 V 11.5.1
GLOBAL AND LOCAL DIFFUSION COEFFICIENTS 265 ILR5.2 QUASILINEAR DIFFUSION
COEFFICIENTS 266 11.5.3 NUMERICAL CALCULATION OF FIELD LINE DIFFUSION
COEFFICIENTS 266 11.6 ERGODIC DIVERTOR AS A CHAOTIC SCATTERING SYSTEM
268 11.6.1 BASIN BOUNDARY STRUCTURE, AT THE PLASMA EDGE 269 11.6.2
MAGNETIC FOOTPRINTS 270 11.7 CONCLUSION .-273 12 MAPPINGS OF MAGNETIC
FIELD LINES IN POLOIDAL DIVERTOR TOKAMAKS 275 12.1 FIELD LINES IN
EQUILIBRIUM PLASMAS NEAR THE SEPARATRIX 277 12.1.1 MAGNETIC
PERTURBATIONS 280 12.1.2 SEPARATRIX MAP 280 XIV CONTENTS 12.1.3 MAPPINGS
TO THE DIVERTOR PLATES 284 12.2 TWO-WIRE MODEL OF THE PLASMA 284 12.2.1
MAGNETIC FIELD PERTURBATIONS 288 12.2.2 THE STRUCTURE OF THE STOCHASTIC
LAYER 292 12.2.3 STRUCTURE OF MAGNETIC FOOTPRINTS 296 12.3 CONCLUSION
298 13 MISCELLANEOUS 299 13.1 RAY DYNAMICS IN WAVEGUIDE MEDIA 299 13.1.1
RAYS AS A HAMILTONIAN SYSTEM 299 13.1.2 MAPPING MODELS OF RAY
PROPAGATION IN WAVEGUIDE MEDIA 302 13.1.3 RAY DYNAMICS IN THE WAVEGUIDE
MODEL 305 13.1.4 OTHER MAPPING MODELS OF RAYS 308 13.2 MAPPING METHODS
IN ACCELERATOR PHYSICS 308 13.3 MAPPINGS IN DYNAMICAL ASTRONOMY 313 A
THE SECOND ORDER GENERATING FUNCTION 317 B ASYMPTOTIC ESTIMATIONS OF THE
INTEGRAL K(H) AND L(H) NEAR SEPARATRIX 321 B.I GENERAL STRUCTURE OF
INTEGRALS 321 B.I.I UNPERTURBED ORBITS NEAR THE SEPARATRIX 322 B.I.2
PERTURBATION HAMILTONIAN IN NORMAL COORDINATES , R) NEAR THE SADDLE
POINTS 324 B.I.3 INTEGRALS OVER THE POWERS OF ORBITS (T,T),R](T,H) NEAR
THE SEPARATRIX 325 B.1.4 OSCILLATORY PARTS OF R(H) 327 B.2
PERIODICALLY-DRIVEN PENDULUM 330 B.3 THE INTEGRAL K(H) IN THE PROBLEM OF
DRIVEN MORSE OSCILLATOR ,332 C PROOF OF RESCALING INVARIANCE OF THE
EQUATIONS OF MOTION . 335 C.I THE CASE OF LINEAR APPROXIMATION 335 C.2
THE CASE OF NONLINEAR APPROXIMATION 338 D RELATION BETWEEN *D AND 6 341
E ASYMPTOTIC ESTIMATION OF THE INTEGRAL S MM (10.49) 345 F SAMPLE
PROGRAM FOR IMPLEMENTING A MAPPING PROCEDURE . 349 REFERENCES 359 INDEX
377
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| adam_txt |
SADRILLA S. ABDULLAEV CONSTRUCTION OF MAPPINGS FOR HAMILTONIAN SYSTEMS
AND THEIR APPLICATIONS 4Y SPRINGER CONTENTS 1 BASICS OF HAMILTONIAN
MECHANICS 1 1.1 HAMILTON EQUATIONS 1 1.1.1 INVARIANTS'BF MOTION 2 1.1.2
HAMILTONIAN EQUATIONS IN EXTENDED PHASE SPACE 3 1.1.3 FORMULATION OF
HAMILTONIAN EQUATIONS WITH ONE COORDINATE AS INDEPENDENT VARIABLE
INSTEAD OF T 4 1.2 THE HAMILTON-JACOBI METHOD 4 1.2.1 CANONICAL CHANGE
OF VARIABLES 4 1.2.2 THE HAMILTON-JACOBI EQUATION 5 1.2.3 THE JACOBI'S
THEOREM 6 1.3 ACTION-ANGLE VARIABLES 7 1.3.1 INTEGRABLE HAMILTONIAN
SYSTEMS 7 1.3.2 HAMILTONIAN IN ACTION-ANGLE VARIABLES 8 1.3.3 SYSTEMS
WITH ONE DEGREE OF FREEDOM 9 1.3.4 MANY-DIMENSIONAL SYSTEMS WITH
SEPARABLE VARIABLES . 9 1.4 PARTICLES IN A WAVE FIELD 12 1.4.1 TRAPPED
MOTION OF PARTICLES 13 1.4.2 UNTRAPPED MOTION OF PARTICLES: H LUQ 15 ;
1.4.3 MOTION ON SEPARATRIX 16 ;' 1.5 ON SYMPLECTIC NUMERICAL INTEGRATION
OF HAMILTONIAN SYSTEMS 171 1.6 BIBLIOGRAPHIC NOTES / 18 2 PERTURBATION
THEORY FOR NEARLY INTEGRABLE SYSTEMS 21 2.1 PERTURBATION METHODS IN
INFINITE TIME INTERVALS 21 2.1.1 A FUNDAMENTAL PROBLEM OF DYNAMICS : 22
2.1.2 THE MAIN IDEA OF AVERAGING PROCEDURE 22 2.1.3 DETERMINATION OF THE
GENERATING FUNCTION 23 2.1.4 VON ZEIPEL'S METHOD .-.--. 25 2.2 LIE
TRANSFORM METHODS .- 26 ,,_ 2.3 TIME-DEPENDENT PERTURBATION SERIES 28
2.3.1 CAUCHY PROBLEM 29 2.3.2 GENERATION FUNCTIONS 29 2.3.3 REMARKS 31
2.3.4 TIME-DEPENDENT LIE TRANSFORM METHOD . . . .' 33 X CONTENTS 2.3.5
EXAMPLE DUFFING OSCILLATOR 33 2.4 METHOD OF SUCCESSIVE TRANSFORMATIONS
36 2.5 BIBLIOGRAPHIC NOTES 37 3 MAPPINGS FOR PERTURBED SYSTEMS . ~.; 39
3.1 POINCARE MAPPINGS 39 3.2 METHOD OF A PRIORI ASSUMPTION 41 3.3 METHOD
OF DELTA FUNCTIONS 44 3.4 THE STANDARD MAP 47 3.5 EXACT MAPPINGS OF
HAMILTONIAN SYSTEMS 49 3.6 DIFFICULTIES IN CONSTRUCTING MAPPINGS 50 4
METHOD OF CANONICAL TRANSFORMATION FOR CONSTRUCTING MAPPINGS , 53 4.1
CANONICAL TRANSFORMATION AND MAPPING 54 4.1.1 NONSYMMETRIC FORMS OF MAPS
55 4.1.2 SYMMETRIC MAP . . . .' 56 4.1.3 THE GENERATING FUNCTION OF
MAPPINGS 57 4.2 ACCURACY OF MAPS 58 4.2.1 PARTICLES IN A
SINGLE-FREQUENCY WAVE FIELD 59 4.2.2 ACCURACY^ OF THE SYMMETRIC MAP 62
4.3 MAPPINGS FOR HAMILTONIAN SYSTEMS WITH A BROAD PERTURBATION SPECTRUM
64 4.3.1 NON-SYMMETRIC FORMS OF MAPS 68 4.3.2 STANDARD HAMILTONIAN AND
CORRESPONDING MAPPINGS . . 69 4.4 MAPPINGS WITH LIE GENERATING FUNCTIONS
71 4.5 POINCARE MAPS AT ARBITRARY SECTIONS OF PHASE-SPACE 73 4.6 METHOD
OF SUCCESSIVE CANONICAL TRANSFORMATIONS 78 4.7 SUMMARY . 80 5 MAPPINGS
NEAR SEPARATRIX.X.THEORY 83 5.1 SEPARATRIX AND MAPPINGS .' 83 5.1.1 THE
CONVENTIONAL SEPARATRIX (WHISKER) MAP 85 5.1.2 SHORTCOMINGS OF
CONVENTIONAL SEPARATRIX MAPPINGS . 87 5.2 THE HAMILTON-JACOBI METHOD
TO CONSTRUCT MAPS NEAR A SEPARATRIX 88 5.2.1 MAPPING ALONG SINGLE
SADDLE-SADDLE CONNECTION 90 5.2.2 CALCULATION OF THE GENERATING FUNCTION
93 5.2.3 SYMMETRIC MAPPINGS 94 5.2.4 NONSYMMETRIC MAPPINGS 94 5.2.5
PROPERTIES OF THE MELNIKOV TYPE INTEGRALS K N (H) AND L N {H) 96 5.3
SIMPLIFICATION OF MAPPINGS 97 5.3.1 "PRIMARY RESONANT" APPROXIMATION 98
5.3.2 SIMPLIFIED FORM OF MAPPINGS 98 CONTENTS XI 5.3.3 SEPARATRIX
MAPPING APPROXIMATION 99 5.4 MAPPING AT ARBITRARY SECTIONS OF PHASE
SPACE 100 5.4.1 MAPPING TO A SECTION E C 101 5.4.2 MAPPING TO SECTION E$
=R) 103 5.5 CONCLUSION 77 104 MAPPINGS NEAR SEPARATRIX. EXAMPLES 105 6.1
MOTION IN A PERTURBED DOUBLE-WELL POTENTIAL 105 6.2 MAPPING FOR THE
PERIODICALLY DRIVEN PENDULUM 114 6.2.1 BEHAVIOR OF INTEGRALS K N (H) AND
L N (H) (5.31) 116 6.2.2 MAPPINGJTO SECTIONS E S 118 6.2.3 MAPPING TO
SECTIONS E C 122 6.3 MAPPING FOR THE 6.3.1 MAPPING 127 6.3.2 THE
SYMMETRIC MAPPING 127 6.3.3 A NONSYMMETRIC MAPPING 130 6.3.4 COMPARISON
WITH A NUMERICAL INTEGRATION 131 6.4 THE KEPLER MAP 132 6.5 COMMENTS ON
SEPARATRIX MAP METHODS 136 6.6 BIBLIOGRAPHIC NOTES : 137 THE KAM THEORY
CHAOS NONTWIST AND NONSMOOTH MAPS 139 7.1 CONSERVATION OF CONDITIONALLY
PERIODIC MOTIONS. THE KAM THEORY 139 7.1.1 INVARIANT TORI FOR MAPPING -.
140 7.1.2 DESTRUCTION OF RESONANT ORBITS: NONLINEAR RESONANCE . 142
7.1.3 CHAOTIC LAYER NEAR A SEPARATRIX 144 7.1.4 LYAPUNOV EXPONENTS 146
7.2 APPLICABILITY OF KAM THEORY 148 7.2.1 ON THE SMALLNESS OF
PERTURBATIONS 148 : 7.2.2 ON THE SMOOTHNESS OFJPERTURBATIONS 150; 7.3
NON-TWIST MAPS !-, 150 7.3.1 DYNAMICS OF SYSTEMS WITH A NON-MONOTONIC
FREQUENCY 151 7.3.2 BEHAVIOR NEAR LOCAL MAXIMA OR MINIMA 152 7.3.3
BEHAVIOR NEAR A BENDING POINT 155 7.3.4 NON-TWIST STANDARD MAPS 155
7.3.5 FIXED POINTS AND TRANSIT TO CHAOS 159 7.4 NON-SMOOTH MAPPINGS 160
7.4.1 INTERMITTENCE IN NONTWIST SYSTEMS -. . 162 7.4.2 SIMPLIFIED
NON-SMOOTH MAPPINGS \ 168 7.4.3 A NONTWIST MAP AND INTERMITTENCY 169 7.5
SUPPRESSION OF CHAOS IN SMOOTH HAMILTONIAN SYSTEMS 173 7.6 BIBLIOGRAPHIC
NOTES 174 XII CONTENTS 8 RESCALING INVARIANCE OF HAMILTONIAN SYSTEMS
NEAR SADDLE POINTS 175 8.1 RESCALING INVARIANCE NEAR SADDLE POINTS AND
SEPARATRIX MAPS 176 8.1.1 STRUCTURE OF PHASE SPACE NEAR SADDLE POINTS
176 8.1.2 UNIVERSAL RESCALING INVARIANCE 177 8.1.3 PROOF OF THE
RESCALING INVARIANCE OF HAMILTONIAN EQUATIONS 179 8.1.4 SEPARATRIX
MAPPING APPROACH 181 8.1.5 RESCALING INVARIANCE IN PARAMETER SPACE 182
8.2 RESCALING INVARIANCE DUE TO THE SYMMETRY OF HAMILTONIANS . . 183
8.2.1 SEPARATRIX MAPPING ANALYSIS 184 8.3 2D-PERIODIC VORTICAL FLOW 187
* 8.3.1 MODEL 187 '8.3.2 RESCALING INVARIANCE PROPERTY 189 8.3.3
SEPARATRIX MAPS OF THE SYSTEM 191 8.3.4 ON THE VALIDITY CONDITIONS OF
THE RESCALING INVARIANCE PROPERTY 193 8.4 SUMMARY 195 9 CHAOTIC
TRANSPORT IN STOCHASTIC LAYERS 197 9.1 STATISTICAL DESCRIPTION OF
CHAOTIC DYNAMICAL SYSTEMS 197 9.1.1 ERGODICITY AND MIXING 197 9.1.2
KINETIC DESCRIPTION 199 9.1.3 ANOMALOUS DIFFUSION 201 9.2 NON-GAUSSIAN
STATISTICS IN STOCHASTIC LAYERS 202 9.2.1 MEAN RESIDENCE TIME 202 9.2.2
STATISTICS OF POINCARE RECURRENCES 203 9.3 CHAOTIC TRANSPORT IN
STOCHASTIC LAYERS. THREE-WAVE FIELD MODEL 206 9.3.1 ADVECTION 206 .
9.3.2 ANOMALOUS DIFFUSION 206 9.3.3 PROBABILITY DENSITY FUNCTION 208 9.4
CHAOTIC TRANSPORT IN 2D-PERIODIC VORTICAL FLOW 210 9.4.1 VARIATION OF
DIFFUSION REGIMES 211 9.4.2 SUPERDIFFUSIVE REGIME. LEVY FLIGHTS 213
9.4.3 FIXED POINTS OF FLIGHT ISLANDS 215 9.5 CONCLUSIONS 217 10 MAGNETIC
FIELD LINES IN FUSION PLASMAS .-. . 219 10. 1 MAGNETIC FIELD LINES AS
HAMILTONIAN SYSTEM 219 10.1.1 EQUILIBRIUM MAGNETIC FIELD 220 10.1.2
HAMILTONIAN FIELD LINE EQUATIONS 220 10.1.3 HAMILTONIAN FORMULATION OF
FIELD LINE EQUATIONS IN A TOROIDAL SYSTEM 221 10.1.4 THE STANDARD
MAGNETIC FIELD * 224 CONTENTS XIII 10.1.5 EQUILIBRIUM MAGNETIC FIELD
WITH THE SHAFRANOV SHIFT . 225 10.2 HAMILTONIAN EQUATIONS IN THE
PRESENCE OF MAGNETIC PERTURBATIONS 229 10.2.1 CYLINDRICAL MODEL OF
PLASMAS 230 10.2.2 MAGNETIC PERTURBATIONS IN TOROIDAL PLASMAS 232 10.2.3
ASYMPTOTICS OF THE TRANSFORMATION MATRIX ELEMENTS S MM (IP) 233 10.2.4
ASYMPTOTIC BEHAVIOR OF H MN (IP) 236 10.3 MAPPING OF FIELD LINES 237
10.4 MAPPINGS AS MODELS FOR MAGNETIC FIELD LINES 240 C 10.4.1 THE
STANDARD MAP AND ITS GENERALIZATIONS 240 10.4.2 THE WOBIG-MENDONGA MAP
241 10.4.3 THE TOKAMAP 242 R 10.5 CONTINUOUS HAMILTONIAN SYSTEM AND
TOKAMAP 244 10.5.1 THE SYMMETRIC TOKAMAP 246 10.5.2 COMPARISON OF THE
TOKAMAP AND THE SYMMETRIC TOKAMAP 247 10.5.3 THE REVTOKAMAP AND THE
SYMMETRIC REVTOKAMAP . 251 10.6 OTHER MAPPING MODELS OF FIELD LINES
252 10.6.1 ANALYTICAL MODELS 252 10.6.2 NUMERICAL MAPPING MODELS 253'
10.7 CONCLUSIONS 254 11 MAPPING OF FIELD LINES IN ERGODIC DIVERTOR
TOKAMAKS . 255 11.1 ERGODIC DIVERTOR CONCEPT 255 11.1.1 MAPPINGS TO
STUDY ERGODIC DIVERTORS 256 11.2 MAGNETIC STRUCTURE OF THE DED 257
11.2.1 SET OF DIVERTOR COILS AND MAGNETIC PERTURBATIONS 257 11.3
SPECTRUM OF MAGNETIC PERTURBATIONS 259 11.4 FORMATION OF THE ERGODIC
ZONE 263 : 11.5 STATISTICAL PROPERTIES OF FIELD LINES 264 V 11.5.1
GLOBAL AND LOCAL DIFFUSION COEFFICIENTS 265 ILR5.2 QUASILINEAR DIFFUSION
COEFFICIENTS 266 11.5.3 NUMERICAL CALCULATION OF FIELD LINE DIFFUSION
COEFFICIENTS 266 11.6 ERGODIC DIVERTOR AS A CHAOTIC SCATTERING SYSTEM
268 11.6.1 BASIN BOUNDARY STRUCTURE, AT THE PLASMA EDGE 269 11.6.2
MAGNETIC FOOTPRINTS 270 11.7 CONCLUSION .-273 12 MAPPINGS OF MAGNETIC
FIELD LINES IN POLOIDAL DIVERTOR TOKAMAKS 275 12.1 FIELD LINES IN
EQUILIBRIUM PLASMAS NEAR THE SEPARATRIX 277 12.1.1 MAGNETIC
PERTURBATIONS 280 12.1.2 SEPARATRIX MAP 280 XIV CONTENTS 12.1.3 MAPPINGS
TO THE DIVERTOR PLATES 284 12.2 TWO-WIRE MODEL OF THE PLASMA 284 12.2.1
MAGNETIC FIELD PERTURBATIONS 288 12.2.2 THE STRUCTURE OF THE STOCHASTIC
LAYER 292 12.2.3 STRUCTURE OF MAGNETIC FOOTPRINTS 296 12.3 CONCLUSION
298 13 MISCELLANEOUS 299 13.1 RAY DYNAMICS IN WAVEGUIDE MEDIA 299 13.1.1
RAYS AS A HAMILTONIAN SYSTEM 299 13.1.2 MAPPING MODELS OF RAY
PROPAGATION IN WAVEGUIDE MEDIA 302 13.1.3 RAY DYNAMICS IN THE WAVEGUIDE
MODEL 305 13.1.4 OTHER MAPPING MODELS OF RAYS 308 13.2 MAPPING METHODS
IN ACCELERATOR PHYSICS 308 13.3 MAPPINGS IN DYNAMICAL ASTRONOMY 313 A
THE SECOND ORDER GENERATING FUNCTION 317 B ASYMPTOTIC ESTIMATIONS OF THE
INTEGRAL K(H) AND L(H) NEAR SEPARATRIX 321 B.I GENERAL STRUCTURE OF
INTEGRALS 321 B.I.I UNPERTURBED ORBITS NEAR THE SEPARATRIX 322 B.I.2
PERTURBATION HAMILTONIAN IN NORMAL COORDINATES , R) NEAR THE SADDLE
POINTS 324 B.I.3 INTEGRALS OVER THE POWERS OF ORBITS (T,T),R](T,H) NEAR
THE SEPARATRIX 325 B.1.4 OSCILLATORY PARTS OF R(H) 327 B.2
PERIODICALLY-DRIVEN PENDULUM 330 B.3 THE INTEGRAL K(H) IN THE PROBLEM OF
DRIVEN MORSE OSCILLATOR ,332 C PROOF OF RESCALING INVARIANCE OF THE
EQUATIONS OF MOTION . 335 C.I THE CASE OF LINEAR APPROXIMATION 335 C.2
THE CASE OF NONLINEAR APPROXIMATION 338 D RELATION BETWEEN *D AND 6 341
E ASYMPTOTIC ESTIMATION OF THE INTEGRAL S MM (10.49) 345 F SAMPLE
PROGRAM FOR IMPLEMENTING A MAPPING PROCEDURE . 349 REFERENCES 359 INDEX
377 |
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| author | Abdullaev, Sadrilla S. 1951- |
| author_GND | (DE-588)131424955 |
| author_facet | Abdullaev, Sadrilla S. 1951- |
| author_role | aut |
| author_sort | Abdullaev, Sadrilla S. 1951- |
| author_variant | s s a ss ssa |
| building | Verbundindex |
| bvnumber | BV021601823 |
| callnumber-first | Q - Science |
| callnumber-label | QC20 |
| callnumber-raw | QC20.7.H35 |
| callnumber-search | QC20.7.H35 |
| callnumber-sort | QC 220.7 H35 |
| callnumber-subject | QC - Physics |
| classification_rvk | UD 8220 |
| classification_tum | PHY 200f PHY 013f |
| ctrlnum | (OCoLC)64626362 (DE-599)BVBBV021601823 |
| dewey-full | 515.39 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.39 |
| dewey-search | 515.39 |
| dewey-sort | 3515.39 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| discipline_str_mv | Physik Mathematik |
| format | Book |
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| illustrated | Illustrated |
| index_date | 2024-07-02T14:47:57Z |
| indexdate | 2024-07-09T20:39:39Z |
| institution | BVB |
| isbn | 9783540309154 3540309152 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014817155 |
| oclc_num | 64626362 |
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| physical | XIV, 378 S. graph. Darst. 24 cm |
| publishDate | 2006 |
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| publisher | Springer |
| record_format | marc |
| series | Lecture notes in physics |
| series2 | Lecture notes in physics |
| spelling | Abdullaev, Sadrilla S. 1951- Verfasser (DE-588)131424955 aut Construction of mappings for Hamiltonian systems and their applications Sadrilla S. Abdullaev Berlin [u.a.] Springer 2006 XIV, 378 S. graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Lecture notes in physics 691 Applications (Mathématiques) Systèmes hamiltoniens Hamiltonian systems Mappings (Mathematics) Symplektische Abbildung (DE-588)4332207-4 gnd rswk-swf Hamiltonsches System (DE-588)4139943-2 gnd rswk-swf Hamiltonsches System (DE-588)4139943-2 s Symplektische Abbildung (DE-588)4332207-4 s DE-604 Lecture notes in physics 691 (DE-604)BV000003166 691 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2710178&prov=M&dok_var=1&dok_ext=htm Inhaltstext GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014817155&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Abdullaev, Sadrilla S. 1951- Construction of mappings for Hamiltonian systems and their applications Lecture notes in physics Applications (Mathématiques) Systèmes hamiltoniens Hamiltonian systems Mappings (Mathematics) Symplektische Abbildung (DE-588)4332207-4 gnd Hamiltonsches System (DE-588)4139943-2 gnd |
| subject_GND | (DE-588)4332207-4 (DE-588)4139943-2 |
| title | Construction of mappings for Hamiltonian systems and their applications |
| title_auth | Construction of mappings for Hamiltonian systems and their applications |
| title_exact_search | Construction of mappings for Hamiltonian systems and their applications |
| title_exact_search_txtP | Construction of mappings for Hamiltonian systems and their applications |
| title_full | Construction of mappings for Hamiltonian systems and their applications Sadrilla S. Abdullaev |
| title_fullStr | Construction of mappings for Hamiltonian systems and their applications Sadrilla S. Abdullaev |
| title_full_unstemmed | Construction of mappings for Hamiltonian systems and their applications Sadrilla S. Abdullaev |
| title_short | Construction of mappings for Hamiltonian systems and their applications |
| title_sort | construction of mappings for hamiltonian systems and their applications |
| topic | Applications (Mathématiques) Systèmes hamiltoniens Hamiltonian systems Mappings (Mathematics) Symplektische Abbildung (DE-588)4332207-4 gnd Hamiltonsches System (DE-588)4139943-2 gnd |
| topic_facet | Applications (Mathématiques) Systèmes hamiltoniens Hamiltonian systems Mappings (Mathematics) Symplektische Abbildung Hamiltonsches System |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=2710178&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014817155&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000003166 |
| work_keys_str_mv | AT abdullaevsadrillas constructionofmappingsforhamiltoniansystemsandtheirapplications |