Solved Problems in Lagrangian and Hamiltonian Mechanics

by Gignoux, Claude; Silvestre-brac, Bernard

ISBN13: 9789048123926

ISBN10: 9048123925

Format: Hardcover

Pub. Date: 2009-05-01

Publisher(s): Springer Verlag

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Summary

The aim of this work is to bridge the gap between the well-known Newtonian mechanics and the studies on chaos, ordinarily reserved to experts. Several topics are treated: Lagrangian, Hamiltonian and Jacobi formalisms, studies of integrable and quasi-integrable systems. The chapter devoted to chaos also enables a simple presentation of the KAM theorem. All the important notions are recalled in summaries of the lectures. They are illustrated by many original problems, stemming from real-life situations, the solutions of which are worked out in great detail for the benefit of the reader.This book will be of interest to undergraduate students as well as others whose work involves mechanics, physics and engineering in general.

Foreword	p. v
Contents	p. xi
Synoptic Tables of the Problems	p. 1
The Lagrangian Formulation	p. 9
Summary	p. 9
Generalized Coordinates	p. 9
Lagrange's Equations	p. 10
Generalized Forces	p. 12
Lagrange Multipliers	p. 13
Problem Statements	p. 14
The Wheel Jack	p. 14
The Sling	p. 15
Rope Slipping on a Table	p. 16
Reaction Force for a Bead on a Hoop	p. 16
Huygens Pendulum	p. 17
Cylinder Rolling on a Moving Tray	p. 18
Motion of a Badly Balanced Cylinder	p. 18
Free Axle on a Inclined Plane	p. 19
The Turn Indicator	p. 21
An Experiment to Measure the Rotational Velocity of the Earth	p. 22
Generalized Inertial Forces	p. 23
Problem Solutions	p. 24
The Wheel Jack	p. 24
The Sling	p. 26
Rope Slipping on a Table	p. 27
Reaction Force for a Bead on a Hoop	p. 28
The Huygens Pendulum	p. 31
Cylinder Rolling on a Moving Tray	p. 33
Motion of a Badly Balanced Cylinder	p. 35
Free Axle on a Inclined Plane	p. 39
The Turn Indicator	p. 43
An Experiment to Measure the Rotational Velocity of the Earth	p. 46
Generalized Inertial Forces	p. 48
Lagrangian Systems	p. 51
Summary	p. 51
Generalized Potential	p. 51
Lagrangian System	p. 52
Constants of the Motion	p. 53
Two-body System with Central Force	p. 55
Small Oscillations	p. 56
Problem Statements	p. 57
Disc on a Movable Inclined Plane	p. 57
Painlevé's Integral	p. 58
Application of Noether's Theorem	p. 58
Foucault's Pendulum	p. 59
Three-particle System	p. 61
Vibration of a Linear Triatomic Molecule: The "Soft" Mode	p. 63
Elastic Transversal Waves in a Solid	p. 64
Lagrangian in a Rotating Frame	p. 65
Particle Drift in a Constant Electromagnetic Field	p. 66
The Penning Trap	p. 67
Equinox Precession	p. 68
Flexion Vibration of a Blade	p. 71
Solitary Waves	p. 73
Vibrational Modes of an Atomic Chain	p. 75
Problem Solutions	p. 76
Disc on a Movable Inclined Plane	p. 76
Painlevé's Integral	p. 77
Application of Noether's Theorem	p. 78
Foucault's Pendulum	p. 79
Three-particle System	p. 82
Vibration of a Linear Triatomic Molecule: The "Soft" Mode	p. 86
Elastic Transversal Waves in a Solid	p. 88
Lagrangian in a Rotating Frame	p. 89
Particle Drift in a Constant Electromagnetic Field	p. 91
The Penning Trap	p. 94
Equinox Precession	p. 97
Flexion Vibration of a Blade	p. 102
Solitary Waves	p. 105
Vibrational Modes of an Atomic Chain	p. 107
Hamilton's Principle	p. 111
Summary	p. 111
Statement of the Principle	p. 111
Action Functional	p. 112
Action and Field Theory	p. 112
Some Well Known Actions	p. 113
The Calculus of Variations	p. 114
Problem Statements	p. 116
The Lorentz Force	p. 116
Relativistic Particle in a Central Force Field	p. 117
Principle of Least Action?	p. 118
Minimum or Maximum Action?	p. 119
Is There Only One Solution Which Makes the Action Stationary?	p. 120
The Principle of Maupertuis	p. 121
Fermat's Principle	p. 122
The Skier Strategy	p. 122
Free Motion on an Ellipsoid	p. 123
Minimum Area for a Fixed Volume	p. 124
The Form of Soap Films	p. 125
Laplace's Law for Surface Tension	p. 127
Chain of Pendulums	p. 128
Wave Equation for a Flexible Blade	p. 128
Precession of Mercury's Orbit	p. 128
Problem Solutions	p. 131
The Lorentz Force	p. 131
Relativistic Particle in a Central Force Field	p. 132
Principle of Least Action?	p. 135
Minimum or Maximum Action?	p. 137
Is There Only One Solution Which Makes the Action Stationary?	p. 138
The Principle of Maupertuis	p. 141
Fermat's Principle	p. 144
The Skier Strategy	p. 146
Free Motion on an Ellipsoid	p. 150
Minimum Area for a Fixed Volume	p. 152
The Form of Soap Films	p. 154
Laplace's Law for Surface Tension	p. 158
Chain of Pendulums	p. 160
Wave Equation for a Flexible Blade	p. 161
Precession of Mercury's Orbit	p. 162
Hamiltonian Formalism	p. 165
Summary	p. 165
Generalized Momentum	p. 165
Hamilton's Function	p. 165
Hamilton's Equations	p. 166
Liouville's Theorem	p. 167
Autonomous One-dimensional Systems	p. 168
Periodic One-dimensional Hamiltonian Systems	p. 169
Problem Statements	p. 171
Electric Charges Trapped in Conductors	p. 171
Symmetry of the Trajectory	p. 171
Hamiltonian in a Rotating Frame	p. 172
Identical Hamiltonian Flows	p. 173
The Runge-Lenz Vector	p. 173
Quicker and More Ecologic than a Plane	p. 174
Hamiltonian of a Charged Particle	p. 176
The First Integral Invariant	p. 177
What About Non-Autonomous Systems?	p. 178
The Reverse Pendulum	p. 178
The Paul Trap	p. 180
Optical Hamilton's Equations	p. 181
Application to Billiard Balls	p. 183
Parabolic Double Well	p. 184
Stability of Circular Trajectories in a Central Potential	p. 185
The Bead on the Hoop	p. 186
Trajectories in a Central Force Field	p. 188
Problem Solutions	p. 188
Electric Charges Trapped in Conductors	p. 188
Symmetry of the Trajectory	p. 190
Hamiltonian in a Rotating Frame	p. 192
Identical Hamiltonian Flows	p. 194
The Runge-Lenz Vector	p. 195
Quicker and More Ecologic than a Plane	p. 198
Hamiltonian of a Charged Particle	p. 200
The First Integral Invariant	p. 204
What About Non-Autonomous Systems?	p. 206
The Reverse Pendulum	p. 207
The Paul Trap	p. 211
Optical Hamilton's Equations	p. 214
Application to Billiard Balls	p. 216
Parabolic Double Well	p. 219
Stability of Circular Trajectories in a Central Potential	p. 222
The Bead on the Hoop	p. 224
Stability of Circular Trajectories in a Central Potential	p. 228
Hamilton-Jacobi Formalism	p. 233
Summary	p. 233
The Action Function	p. 233
Reduced Action	p. 234
Maupertuis' Principle	p. 235
Jacobi's Theorem	p. 236
Separation of Variables	p. 236
Huygens' Construction	p. 238
Problem Statements	p. 239
How to Manipulate the Action and the Reduced Action	p. 239
Action for a One-dimensional Harmonic Oscillator	p. 241
Motion on a Surface and Geodesic	p. 241
Wave Surface for Free Fall	p. 242
Peculiar Wave Fronts	p. 243
Electrostatic Lens	p. 243
Maupertuis' Principle with an Electromagnetic Field	p. 245
Separable Hamiltonian, Separable Action	p. 246
Stark Effect	p. 247
Orbits of Earth's Satellites	p. 248
Phase and Group Velocities	p. 251
Problem Solutions	p. 252
How to Manipulate the Action and the Reduced Action	p. 252
Action for a One-Dimensional Harmonic Oscillator	p. 258
Motion on a Surface and Geodesic	p. 260
Wave Surface for Free Fall	p. 261
Peculiar Wave Fronts	p. 264
Electrostatic Lens	p. 265
Maupertuis' Principle with an Electromagnetic Field	p. 268
Separable Hamiltonian, Separable Action	p. 270
Stark Effect	p. 271
Orbits of Earth's Satellites	p. 275
Phase and Group Velocities	p. 279
Integrable Systems	p. 281
Summary	p. 281
Basic Notions	p. 281
Some Definitions	p. 281
Good Coordinates: The Angle-Action Variables	p. 283
Complements	p. 286
Building the Angle Variables	p. 286
Flow/Poisson Bracket/Involution	p. 287
Criterion to Obtain a Canonical Transformation	p. 288
Problem Statements	p. 289
Expression of the Period for a One-Dimensional Motion	p. 289
One-dimensional Particle in a Box	p. 290
Ball Bouncing on the Ground	p. 290
Particle in a Constant Magnetic Field	p. 291
Actions for the Kepler Problem	p. 292
The Sommerfeld Atom	p. 293
Energy as a Function of Actions	p. 294
Invariance of the Circulation Under a Continuous Deformation	p. 296
Ball Bouncing on a Moving Tray	p. 297
Harmonic Oscillator with a Variable Frequency	p. 298
Choice of the Momentum	p. 298
Invariance of the Poisson Bracket Under a Canonical Transformation	p. 299
Canonicity for a Contact Transformation	p. 299
One-Dimensional Free Fall	p. 300
One-Dimensional Free Fall Again	p. 301
Scale Dilation as a Function of Time	p. 301
From the Harmonic Oscillator to Coulomb's Problem	p. 302
Generators for Fundamental Transformations	p. 303
Problem Solutions	p. 305
Expression of the Period for a One-Dimensional Motion	p. 305
One-Dimensional Particle in a Box	p. 306
Ball Bouncing on the Ground	p. 308
Particle in a Constant Magnetic Field	p. 310
Actions for the Kepler Problem	p. 314
The Sommerfeld Atom	p. 316
Energy as a Function of Actions	p. 318
Invariance of the Circulation Under a Continuous Deformation	p. 322
Ball Bouncing on a Moving Tray	p. 324
Harmonic Oscillator with a Variable Frequency	p. 324
Choice of the Momentum	p. 325
Invariance of the Poisson Bracket Under a Canonical Transformation	p. 326
Canonicity for a Contact Transformation	p. 327
One-dimensional Free Fall	p. 329
One-dimensional Free Fall Again	p. 330
Scale Dilation as a Function of Time	p. 332
From the Harmonic Oscillator to Coulomb's Problem	p. 333
Generators for Fundamental Transformations	p. 336
Quasi-Integrable Systems	p. 341
Summary	p. 341
Introduction	p. 341
Perturbation Theory	p. 342
Canonical Perturbation Theory	p. 342
Adiabatic Invariants	p. 345
Problem Statements	p. 347
Limits of the Perturbative Expansion	p. 347
Non-canonical Versus Canonical Perturbative Expansion	p. 347
First Canonical Correction for the Pendulum	p. 348
Beyond the First Order Correction	p. 349
Adiabatic Invariant in an Elevator	p. 350
Adiabatic Invariant and Adiabatic Relaxation	p. 351
Charge in a Slowly Varying Magnetic Field	p. 352
Illuminations Concerning the Aurora Borealis	p. 354
Bead on a Rigid Wire: Hannay's Phase	p. 356
Problem Solutions	p. 358
Limits of the Perturbative Expansion	p. 358
Non-canonical Versus Canonical Perturbative Expansion	p. 361
First Canonical Correction for the Pendulum	p. 363
Beyond the First Order Correction	p. 367
Adiabatic Invariant in an Elevator	p. 370
Adiabatic Invariant and Adiabatic Relaxation	p. 372
Charge in a Slowly Varying Magnetic Field	p. 375
Illuminations Concerning the Aurora Borealis	p. 379
Bead on a Rigid Wire: Hannay's Phase	p. 382
From Order to Chaos	p. 385
Summary	p. 385
Introduction	p. 385
The Model of the Kicked Rotor	p. 386
Poincaré's Sections	p. 388
The Rotor for a Null Perturbation	p. 388
Poincaré's Sections for the Kicked Rotor	p. 390
How to Recognize Fixed Points	p. 393
Separatrices/Homocline Points/Chaos	p. 394
Complements	p. 395
Problem Statements	p. 396
Disappearance of Resonant Tori	p. 396
Continuous Fractions or How to Play with Irrational Numbers	p. 396
Properties of the Phase Space of the Standard Mapping	p. 398
Bifurcation of the Periodic Trajectory 1:1 for the Standard Mapping	p. 398
Chaos-Ergodicity: A Slight Difference	p. 399
Acceleration Modes: A Curiosity of the Standard Mapping	p. 401
Demonstration of a Kicked Rotor?	p. 401
Anosov's Mapping (or Arnold's Cat)	p. 403
Fermi's Accelerator	p. 405
Damped Pendulum and Standard Mapping	p. 407
Stability of Periodic Orbits on a Billiard Table	p. 409
Lagrangian Points: Jupiter's Greeks and Trojans	p. 412
Problem Solutions	p. 415
Disappearance of Resonant Tori	p. 415
Continuous Fractions or How to Play with Irrational Numbers	p. 417
Properties of the Phase Space of the Standard Mapping	p. 418
Bifurcation of the Periodic Trajectory 1:1 for the Standard Mapping	p. 419
Chaos-Ergodicity: A Slight Difference	p. 423
Acceleration Modes: A Curiosity of the Standard Mapping	p. 425
Demonstration of a Kicked Rotor?	p. 427
Anosov's Mapping (or Arnold's Cat)	p. 432
Fermi's Accelerator	p. 438
Damped Pendulum and Standard Mapping	p. 443
Stability of Periodic Orbits on a Billiard Table	p. 447
Lagrangian Points: Jupiter's Greeks and Trojans	p. 450
Bibliography	p. 457
Index	p. 461
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Solved Problems in Lagrangian and Hamiltonian Mechanics

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