Solved problems in Lagrangian and Hamiltonian mechanics:
Gespeichert in:
| Hauptverfasser: | , |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Dordrecht [u.a.]
Springer
2009
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVIII, 464 S. Ill., graph. Darst. |
| ISBN: | 9789048123926 |
Internformat
MARC
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| 240 | 1 | 0 | |a Problèmes corrigés de Mécanique et résumés de cours. De Lagrange à Hamilton. |
| 245 | 1 | 0 | |a Solved problems in Lagrangian and Hamiltonian mechanics |c Claude Gignoux ; Bernard Silvestre-Brac |
| 264 | 1 | |a Dordrecht [u.a.] |b Springer |c 2009 | |
| 300 | |a XVIII, 464 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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Datensatz im Suchindex
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| adam_text | Titel: Solved problems in Lagrangian and Hamiltonian mechanics
Autor: Gignoux, Claude
Jahr: 2009
Contents
Foreword ......................................................... v
Contents ......................................................... xi
Synoptic Tables of the Problems .............................. 1
Chapter 1. The Lagrangian Formulation .................... 9
Summary ........................................................ 9
1.1. Generalized Coordinates ................................. 9
1.2. Lagrange s Equations ................................... 10
1.3. Generalized Forces ...................................... 12
1.4. Lagrange Multipliers .................................... 13
Problem Statements ........................................... 14
1.1. The Wheel Jack ........................................ 14
1.2. TheSling ............................................... 15
1.3. Rope Slipping on a Table ............................... 16
1.4. Reaction Force for a Bead on a Hoop ................... 16
1.5. Huygens Pendulum ..................................... 17
1.6. Cylinder Rolling on a Moving Tray ..................... 18
1.7. Motion of a Badly Balanced Cylinder ................... 18
1.8. Free Axle on a Inclined Piane ........................... 19
1.9. The Turn Indicator ..................................... 21
1.10. An Experiment to Measure the Rotational Velocity
of the Earth ............................................ 22
1.11. Generalized Inertial Forces .............................. 23
Problem Solutions ............................................. 24
1.1. The Wheel Jack ........................................ 24
1.2. TheSling ............................................... 26
1.3. Rope Slipping on a Table ............................... 27
1.4. Reaction Force for a Bead on a Hoop ................... 28
1.5. The Huygens Pendulum ................................ 31
1.6. Cylinder Rolling on a Moving Tray ..................... 33
1.7. Motion of a Badly Balanced Cylinder ................... 35
1.8. Free Axle on a Inclined Piane ........................... 39
1.9. The Turn Indicator ..................................... 43
1.10. An Experiment to Measure the Rotational Velocity
of the Earth ............................................ 46
1.11. Generalized Inertial Forces .............................. 48
xii Contents
Chapter 2. Lagrangian Systems .............................. 51
Summary ....................................................... 51
2.1. Generalized Potential ................................... 51
2.2. Lagrangian System ..................................... 52
2.3. Constants of the Motion ................................ 53
2.4. Two-body System with Central Force ................... 55
2.5. Small Oscillations ....................................... 56
Problem Statements ........................................... 57
2.1. Disc on a Movable Inclined Piane ....................... 57
2.2. Painlevé s Integrai ...................................... 58
2.3. Application of Noether s Theorem ...................... 58
2.4. Foucault s Pendulum ................................... 59
2.5. Three-particle System .................................. 61
2.6. Vibration of a Linear Triatomic Molecule:
The Soft Mode ........................................ 63
2.7. Elastic Transversai Waves in a Solid .................... 64
2.8. Lagrangian in a Rotating Frame ........................ 65
2.9. Particle Drift in a Constant Electromagnetic Field ...... 66
2.10. The Penning Trap ...................................... 67
2.11. Equinox Precession ..................................... 68
2.12. Flexion Vibration of a Biade ............................ 71
2.13. Solitary Waves .......................................... 73
2.14. Vibrational Modes of an Atomic Chain ................. 75
Problem Solutions ............................................. 76
2.1. Disc on a Movable Inclined Piane ....................... 76
2.2. Painlevé s Integrai ...................................... 77
2.3. Application of Noether s Theorem ...................... 78
2.4. Foucault s Pendulum ................................... 79
2.5. Three-particle System .................................. 82
2.6. Vibration of a Linear Triatomic Molecule:
The Soft Mode ........................................ 86
2.7. Elastic Transversai Waves in a Solid .................... 88
2.8. Lagrangian in a Rotating Frame ........................ 89
2.9. Particle Drift in a Constant Electromagnetic Field ...... 91
2.10. The Penning Trap ...................................... 94
2.11. Equinox Precession ..................................... 97
2.12. Flexion Vibration of a Biade ........................... 102
2.13. Solitary Waves ........................................ 105
2.14. Vibrational Modes of an Atomic Chain ................ 107
Chapter 3. Hamilton s Principle ............................ Ili
Summary ...................................................... Ili
3.1. Statement of the Principle ............................. Ili
3.2. Action Functional ..................................... 112
Contents xiii
3.3. Action and Field Theory ............................... 112
3.4. Some Well Known Actions ............................. 113
3.5. The Calculus of Variations ............................. 114
Problem Statements .......................................... 116
3.1. The Lorentz Force ..................................... 116
3.2. Relativistic Particle in a Central Force Field ........... 117
3.3. Principle of Least Action? ............................. 118
3.4. Minimum or Maximum Action? ....................... 119
3.5. Is There Only One Solution
Which Makes the Action Stationary? .................. 120
3.6. The Principle of Maupertuis ........................... 121
3.7. Fermat s Principle ..................................... 122
3.8. The Skier Strategy .................................... 122
3.9. Free Motion on an Ellipsoid ........................... 123
3.10. Minimum Area for a Fixed Volume .................... 124
3.11. The Form of Soap Films ............................... 125
3.12. Laplace s Law for Surface Tension ..................... 127
3.13. Chain of Pendulums ................................... 128
3.14. Wave Equation for a Flexible Biade .................... 128
3.15. Precession of Mercury s Orbit .......................... 128
Problem Solutions ............................................ 131
3.1. The Lorentz Force ..................................... 131
3.2. Relativistic Particle in a Central Force Field ........... 132
3.3. Principle of Least Action? ............................. 135
3.4. Minimum or Maximum Action? ....................... 137
3.5. Is There Only One Solution
Which Makes the Action Stationary? .................. 138
3.6. The Principle of Maupertuis ........................... 141
3.7. Fermat s Principle ..................................... 144
3.8. The Skier Strategy .................................... 146
3.9. Free Motion on an Ellipsoid ........................... 150
3.10. Minimum Area for a Fixed Volume .................... 152
3.11. The Form of Soap Films ............................... 154
3.12. Laplace s Law for Surface Tension ..................... 158
3.13. Chain of Pendulums ................................... 160
3.14. Wave Equation for a Flexible Biade .................... 161
3.15. Precession of Mercury s Orbit .......................... 162
Chapter 4. Hamiltonian Formalism ......................... 165
Summary ...................................................... 165
4.1. Generalized Momentum ................................ 165
4.2. Hamilton s Function ................................... 166
4.3. Hamilton s Equations .................................. 167
4.4. Liouville s Theorem .................................... 167
xiv Contents
4.5. Autonomous One-dimensional Systems ................. 168
4.6. Periodic One-dimensional Hamiltonian Systems ........ 169
Problem Statements .......................................... 171
4.1. Electric Charges Trapped in Conductors ............... 171
4.2. Symmetry of the Trajectory ........................... 171
4.3. Hamiltonian in a Rotating Frame ...................... 172
4.4. Identical Hamiltonian Flows ........................... 173
4.5. The Runge-Lenz Vector ................................ 173
4.6. Quicker and More Ecologie than a Piane ............... 174
4.7. Hamiltonian of a Charged Particle ..................... 176
4.8. The First Integrai Invariant ............................ 177
4.9. What About Non-Autonomous Systems? .............. 178
4.10. The Reverse Pendulum ................................ 178
4.11. The Paul Trap ......................................... 180
4.12. Optical Hamilton s Equations .......................... 181
4.13. Application to Billiard Balls ........................... 183
4.14. Parabolic Doublé Well ................................. 184
4.15. Stability of Circular Trajectories in a Central Potential 185
4.16. The Bead on the Hoop ................................ 186
4.17. Trajectories in a Central Force Field ................... 188
Problem Solutions ............................................ 188
4.1. Electric Charges Trapped in Conductors ............... 188
4.2. Symmetry of the Trajectory ........................... 190
4.3. Hamiltonian in a Rotating Frame ...................... 192
4.4. Identical Hamiltonian Flows ........................... 194
4.5. The Runge-Lenz Vector ................................ 195
4.6. Quicker and More Ecologie than a Piane ............... 198
4.7. Hamiltonian of a Charged Particle ..................... 200
4.8. The First Integrai Invariant ............................ 204
4.9. What About Non-Autonomous Systems? .............. 206
4.10. The Reverse Pendulum ................................ 207
4.11. The Paul Trap ......................................... 211
4.12. Optical Hamilton s Equations .......................... 214
4.13. Application to Billiard Balls ........................... 216
4.14. Parabolic Doublé Well ................................. 219
4.15. Stability of Circular Trajectories in a Central Potential 222
4.16. The Bead on the Hoop ................................ 224
4.17. Stability of Circular Trajectories in a Central Potential 228
Chapter 5. Hamilton-Jacobi Formalism .................... 233
Summary ...................................................... 233
5.1. The Action Function ................................... 233
5.2. Reduced Action ....................................... 234
5.3. Maupertuis Principle .................................. 235
Contents xv
5.4. Jacobi s Theorem ...................................... 236
5.5. Separation of Variables ................................ 236
5.6. Huygens Construction ................................. 238
Problem Statements .......................................... 239
5.1. How to Manipulate the Action and the Reduced Action 239
5.2. Action for a One-dimensional Harmonic Oscillator ..... 241
5.3. Motion on a Surface and Geodesie ..................... 241
5.4. Wave Surface for Free Fall ............................. 242
5.5. Peculiar Wave Fronts .................................. 243
5.6. Electrostatic Lens ..................................... 243
5.7. Maupertuis Principle with an Electromagnetic Field ... 245
5.8. Separable Hamiltonian, Separable Action .............. 246
5.9. StarkEffect ........................................... 247
5.10. Orbits of Earth s Satellites ............................. 248
5.11. Phase and Group Velocities ............................ 251
Problem Solutions ............................................ 252
5.1. How to Manipulate the Action and the Reduced Action 252
5.2. Action for a One-Dimensional Harmonic Oscillator ..... 258
5.3. Motion on a Surface and Geodesie ..................... 260
5.4. Wave Surface for Free Fall ............................. 261
5.5. Peculiar Wave Fronts .................................. 264
5.6. Electrostatic Lens ..................................... 265
5.7. Maupertuis Principle with an Electromagnetic Field ... 268
5.8. Separable Hamiltonian, Separable Action .............. 270
5.9. StarkEffect ........................................... 271
5.10. Orbits of Earth s Satellites ............................. 275
5.11. Phase and Group Velocities ............................ 279
Chapter 6. Integrable Systems .............................. 281
Summary ...................................................... 281
6.1. Basic Notions .......................................... 281
6.1.1. Some Definitions ................................ 281
6.1.2. Good Coordinatesi The Angle-Action Variables . 283
6.2. Complements .......................................... 286
6.2.1. Building the Angle Variables .................... 286
6.2.2. Flow/Poisson Bracket/Involution ................ 287
6.2.3. Criterion to Obtain a Canonica! Transformation . 288
Problem Statements .......................................... 289
6.1. Expression of the Period for a One-Dimensional Motion 289
6.2. One-dimensional Particle in a Box ..................... 290
6.3. Ball Bouncing on the Ground .......................... 290
6.4. Particle in a Constant Magnetic Field .................. 291
6.5. Actions for the Kepler Problem ........................ 292
6.6. The Sommerfeld Atom ................................. 293
6.7. Energy as a Function of Actions ....................... 294
i Contents
6.8. Invariance of the Circulation Under a Continuous
Deformation ........................................... 296
6.9. Ball Bouncing on a Moving Tray ....................... 297
6.10. Harmonic Oscillator with a Variable Frequency ........ 298
6.11. Choice of the Momentum .............................. 298
6.12. Invariance of the Poisson Bracket
Under a Canonical Transformation ..................... 299
6.13. Canonicity for a Contact Transformation .............. 299
6.14. One-Dimensional Free Fall ............................. 300
6.15. One-Dimensional Free Fall Again ...................... 301
6.16. Scale Dilation as a Function of Time ................... 301
6.17. From the Harmonic Oscillator to Coulomb s Problem .. 302
6.18. Generators for Fundamental Transformations .......... 303
Problem Solutions ............................................ 305
6.1. Expression of the Period for a One-Dimensional Motion 305
6.2. One-Dimensional Particle in a Box ..................... 306
6.3. Ball Bouncing on the Ground .......................... 308
6.4. Particle in a Constant Magnetic Field .................. 310
6.5. Actions for the Kepler Problem ........................ 314
6.6. The Sommerfeld Atom ................................. 316
6.7. Energy as a Function of Actions ....................... 318
6.8. Invariance of the Circulation Under a Continuous
Deformation ........................................... 322
6.9. Ball Bouncing on a Moving Tray ....................... 324
6.10. Harmonic Oscillator with a Variable Frequency ........ 324
6.11. Choice of the Momentum .............................. 325
6.12. Invariance of the Poisson Bracket
Under a Canonical Transformation ..................... 326
6.13. Canonicity for a Contact Transformation .............. 327
6.14. One-dimensional Free Fall ............................. 329
6.15. One-dimensional Free Fall Again ....................... 330
6.16. Scale Dilation as a Function of Time ................... 332
6.17. From the Harmonic Oscillator to Coulomb s Problem .. 333
6.18. Generators for Fundamental Transformations .......... 336
Chapter 7. Quasi-Integrable Systems ...................... 341
Summary ...................................................... 341
7.1. Introduction ........................................... 341
7.2. Perturbation Theory ................................... 342
7.3. Canonical Perturbation Theory ........................ 342
7.4. Adiabatic Invariants................................... 345
Problem Statements .......................................... 347
7.1. Limits of the Perturbative Expansion .................. 347
7.2. Non-canonical Versus Canonical Perturbative Expansion 347
Contents xvii
7.3. First Canonical Correction for the Pendulum .......... 348
7.4. Beyond the First Order Correction ..................... 349
7.5. Adiabatic Invariant in an Elevator ..................... 350
7.6. Adiabatic Invariant and Adiabatic Relaxation .......... 351
7.7. Charge in a Slowly Varying Magnetic Field ............ 352
7.8. Illuminations Concerning the Aurora Borealis .......... 354
7.9. Bead on a Rigid Wire: Hannay s Phase ................ 356
Problem Solutions ............................................ 358
7.1. Limits of the Perturbative Expansion .................. 358
7.2. Non-canonical Versus Canonical Perturbative Expansion 361
7.3. First Canonical Correction for the Pendulum .......... 363
7.4. Beyond the First Order Correction ..................... 367
7.5. Adiabatic Invariant in an Elevator ..................... 370
7.6. Adiabatic Invariant and Adiabatic Relaxation .......... 372
7.7. Charge in a Slowly Varying Magnetic Field ............ 375
7.8. Illuminations Concerning the Aurora Borealis .......... 379
7.9. Bead on a Rigid Wire: Hannay s Phase ................ 382
Chapter 8. From Order to Chaos ........................... 385
Summary ...................................................... 385
8.1. Introduction ........................................... 385
8.2. The Model of the Kicked Rotor ........................ 386
8.3. Poincaré s Sections .................................... 388
8.4. The Rotor for a Nuli Perturbation ..................... 388
8.5. Poincaré s Sections for the Kicked Rotor ............... 390
8.6. How to Recognize Fixed Points ........................ 393
8.7. Separatrices/Homocline Points/Chaos ................. 394
8.8. Complements .......................................... 395
Problem Statements .......................................... 396
8.1. Disappearance of Resonant Tori ....................... 396
8.2. Continuous Fractions or How to Play with Irrational
Numbers .............................................. 396
8.3. Properties of the Phase Space of the Standard Mapping 398
8.4. Bifurcation of the Periodic Trajectory 1:1
for the Standard Mapping ............................. 398
8.5. Chaos-Ergodicity : A Slight Difference ................ 399
8.6. Acceleration Modes: A Curiosity of the Standard
Mapping ............................................... 401
8.7. Demonstration of a Kicked Rotor? ..................... 401
8.8. Anosov s Mapping (or Arnold s Cat) ................... 403
8.9. Fermi s Accelerator .................................... 405
8.10. Damped Pendulum and Standard Mapping ............ 407
8.11. Stability of Periodic Orbits on a Billiard Table ......... 409
8.12. Lagrangian Points: Jupiter s Greeks and TYojans ....... 412
xviii Contents
Problem Solutions ............................................ 415
8.1. Disappearance of Resonant Tori ....................... 415
8.2. Continuous Fractions or How to Play with Irrational
Numbers .............................................. 417
8.3. Properties of the Phase Space of the Standard Mapping 418
8.4. Bifurcation of the Periodic Trajectory 1:1
for the Standard Mapping ............................. 419
8.5. Chaos-Ergodicity: A Slight Difference ................. 423
8.6. Acceleration Modes: A Curiosity of the Standard
Mapping ............................................... 425
8.7. Demonstration of a Kicked Rotor? ..................... 427
8.8. Anosov s Mapping (or Arnold s Cat) ................... 432
8.9. Fermi s Accelerator .................................... 438
8.10. Damped Pendulum and Standard Mapping ............ 443
8.11. Stability of Periodic Orbits on a Billiard Table ......... 447
8.12. Lagrangian Points: Jupiter s Greeks and Trojans ....... 450
Bibliography ................................................... 457
Index ........................................................... 461
|
| any_adam_object | 1 |
| author | Gignoux, Claude Silvestre-Brac, Bernard |
| author_facet | Gignoux, Claude Silvestre-Brac, Bernard |
| author_role | aut aut |
| author_sort | Gignoux, Claude |
| author_variant | c g cg b s b bsb |
| building | Verbundindex |
| bvnumber | BV025564276 |
| classification_rvk | UF 1000 |
| classification_tum | PHY 200f |
| ctrlnum | (OCoLC)636348892 (DE-599)BVBBV025564276 |
| discipline | Physik |
| format | Book |
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| genre | (DE-588)4143389-0 Aufgabensammlung gnd-content |
| genre_facet | Aufgabensammlung |
| id | DE-604.BV025564276 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:36:34Z |
| institution | BVB |
| isbn | 9789048123926 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020163364 |
| oclc_num | 636348892 |
| open_access_boolean | |
| owner | DE-11 DE-91G DE-BY-TUM |
| owner_facet | DE-11 DE-91G DE-BY-TUM |
| physical | XVIII, 464 S. Ill., graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Springer |
| record_format | marc |
| spelling | Gignoux, Claude Verfasser aut Problèmes corrigés de Mécanique et résumés de cours. De Lagrange à Hamilton. Solved problems in Lagrangian and Hamiltonian mechanics Claude Gignoux ; Bernard Silvestre-Brac Dordrecht [u.a.] Springer 2009 XVIII, 464 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Hamilton-Formalismus (DE-588)4376155-0 gnd rswk-swf Theoretische Mechanik (DE-588)4185100-6 gnd rswk-swf Lagrange-Formalismus (DE-588)4316154-6 gnd rswk-swf (DE-588)4143389-0 Aufgabensammlung gnd-content Lagrange-Formalismus (DE-588)4316154-6 s DE-604 Hamilton-Formalismus (DE-588)4376155-0 s Theoretische Mechanik (DE-588)4185100-6 s Silvestre-Brac, Bernard Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020163364&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gignoux, Claude Silvestre-Brac, Bernard Solved problems in Lagrangian and Hamiltonian mechanics Hamilton-Formalismus (DE-588)4376155-0 gnd Theoretische Mechanik (DE-588)4185100-6 gnd Lagrange-Formalismus (DE-588)4316154-6 gnd |
| subject_GND | (DE-588)4376155-0 (DE-588)4185100-6 (DE-588)4316154-6 (DE-588)4143389-0 |
| title | Solved problems in Lagrangian and Hamiltonian mechanics |
| title_alt | Problèmes corrigés de Mécanique et résumés de cours. De Lagrange à Hamilton. |
| title_auth | Solved problems in Lagrangian and Hamiltonian mechanics |
| title_exact_search | Solved problems in Lagrangian and Hamiltonian mechanics |
| title_full | Solved problems in Lagrangian and Hamiltonian mechanics Claude Gignoux ; Bernard Silvestre-Brac |
| title_fullStr | Solved problems in Lagrangian and Hamiltonian mechanics Claude Gignoux ; Bernard Silvestre-Brac |
| title_full_unstemmed | Solved problems in Lagrangian and Hamiltonian mechanics Claude Gignoux ; Bernard Silvestre-Brac |
| title_short | Solved problems in Lagrangian and Hamiltonian mechanics |
| title_sort | solved problems in lagrangian and hamiltonian mechanics |
| topic | Hamilton-Formalismus (DE-588)4376155-0 gnd Theoretische Mechanik (DE-588)4185100-6 gnd Lagrange-Formalismus (DE-588)4316154-6 gnd |
| topic_facet | Hamilton-Formalismus Theoretische Mechanik Lagrange-Formalismus Aufgabensammlung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020163364&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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