Solved Problems in Lagrangian and Hamiltonian Mechanics

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Springer Science & Business Media, Jul 14, 2009 - Science - 464 pages
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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.

 

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Contents

414 Parabolic Double Well
219
415 Stability of Circular Trajectories in a Central Potential
222
416 The Bead on the Hoop Statement p 186
224
417 Stability of Circular Trajectories in a Central Potential
228
HamiltonJacobi Formalism
232
52 Reduced Action
234
53 Maupertuis Principle
235
54 Jacobis Theorem
236

16 Cylinder Rolling on a Moving Tray
18
18 Free Axle on a Inclined Plane
19
19 The Turn Indicator
21
110 An Experiment to Measure the Rotational Velocity of the Earth
22
111 Generalized Inertial Forces
23
Problem Solutions
24
12 The Sling Statement and Figure p 15
26
13 Rope Slipping on a Table
27
14 Reaction Force for a Bead on a Hoop
28
15 The Huygens Pendulum
31
16 Cylinder Rolling on a Moving Tray
33
17 Motion of a Badly Balanced Cylinder
35
18 Free Axle on a Inclined Plane
39
19 The Turn Indicator Statement p 21
43
110 An Experiment to Measure the Rotational Velocity of the Earth
46
111 Generalized Inertial Forces
48
Lagrangian Systems
51
22 Lagrangian System
52
23 Constants of the Motion
53
24 Twobody System with Central Force
55
25 Small Oscillations
56
Problem Statements
57
22 Painlevés Integral Solution p 77
58
24 Foucaults Pendulum
59
25 Threeparticle System
61
The Soft Mode
63
27 Elastic Transversal Waves in a Solid F Waves Solution p 88
64
28 Lagrangian in a Rotating Frame
65
29 Particle Drift in a Constant Electromagnetic Field
66
210 The Penning Trap
67
211 Equinox Precession Solution p 97
68
212 Flexion Vibration of a Blade
71
213 Solitary Waves
73
214 Vibrational Modes of an Atomic Chain Solution and Figure p 107
75
Problem Solutions
76
22 Painlevés Integral Statement p 58
77
24 Foucaults Pendulum
79
25 Threeparticle System Statement p 61
82
The Soft Mode
86
27 Elastic Transversal Waves in a Solid
88
28 Lagrangian in a Rotating Frame
89
29 Particle Drift in a Constant Electromagnetic Field
91
210 The Penning Trap Statement p 67
94
211 Equinox Precession
97
212 Flexion Vibration of a Blade
102
213 Solitary Waves statement p 73
105
214 Vibrational Modes of an Atomic Chain
107
Hamiltons Principle
111
32 Action Functional
112
34 Some Well Known Actions
113
Problem Statements
116
32 Relativistic Particle in a Central Force Field Solution p 132
117
33 Principle of Least Action?
118
34 Minimum or Maximum Action?
119
35 Is There Only One Solution Which Makes the Action Stationary?
120
36 The Principle of Maupertuis
121
37 Fermats Principle
122
39 Free Motion on an Ellipsoid
123
310 Minimum Area for a Fixed Volume
124
311 The Form of Soap Films
125
312 Laplaces Law for Surface Tension
127
313 Chain of Pendulums Solution p 160
128
Problem Solutions
131
32 Relativistic Particle in a Central Force Field Statement p 117
132
33 Principle of Least Action?
135
34 Minimum or Maximum Action?
137
35 Is There Only One Solution Which Makes the Action Stationary?
138
36 The Principle of Maupertuis
141
37 Fermats Principle Statement p 122
144
38 The Skier Strategy
146
39 Free Motion on an Ellipsoid
150
310 Minimum Area for a Fixed Volume
152
311 The Form of Soap Films
154
312 Laplaces Law for Surface Tension
158
313 Chain of Pendulums statement p 128
160
314 Wave Equation for a Flexible Blade
161
315 Precession of Mercurys Orbit
162
Hamiltonian Formalism
165
42 Hamiltons Function
166
43 Hamiltons Equations
167
45 Autonomous Onedimensional Systems
168
46 Periodic Onedimensional Hamiltonian Systems
169
Problem Statements
171
43 Hamiltonian in a Rotating Frame
172
44 Identical Hamiltonian Flows
173
46 Quicker and More Ecologic than a Plane Solution p 198
174
47 Hamiltonian of a Charged Particle
176
48 The First Integral Invariant
177
49 What About NonAutonomous Systems?
178
411 The Paul Trap
180
412 Optical Hamiltons Equations
181
413 Application to Billiard Balls
183
414 Parabolic Double Well
184
415 Stability of Circular Trajectories in a Central Potential
185
416 The Bead on the Hoop
186
417 Trajectories in a Central Force Field
188
43 Hamiltonian in a Rotating Frame
192
44 Identical Hamiltonian Flows
194
45 The RungeLenz Vector
195
46 Quicker and More Ecologic than a Plane Statement and Figure p 174
198
47 Hamiltonian of a Charged Particle
200
48 The First Integral Invariant
204
49 What About NonAutonomous Systems?
206
410 The Reverse Pendulum
207
411 The Paul Trap
211
412 Optical Hamiltons Equations
214
413 Application to Billiard Balls
216
56 Huygens Construction
238
Problem Statements
239
52 Action for a Onedimensional Harmonic Oscillator Solution p 258
241
54 Wave Surface for Free Fall
242
55 Peculiar Wave Fronts
243
with an Electromagnetic Field
245
Separable Action Solution p 270
246
59 Stark Effect
247
510 Orbits of Earths Satellites
248
511 Phase and Group Velocities
251
Problem Solutions
252
Harmonic Oscillator statement p 241
258
53 Motion on a Surface and Geodesic
260
54 Wave Surface for Free Fall
261
55 Peculiar Wave Fronts statement p 243
264
57 Maupertuis Principle with an Electromagnetic Field
268
58 Separable Hamiltonian Separable Action statement p 246
270
59 Stark Effect Statement p 247
271
510 Orbits of Earths Satellites
275
511 Phase and Group Velocities
279
Integrable Systems
281
The AngleAction Variables
283
62 Complements
286
622 FlowPoisson BracketInvolution
287
623 Criterion to Obtain a Canonical Transformation
288
Problem Statements
289
62 OneDimensional Particle in a Box
290
64 Particle in a Constant Magnetic Field
291
65 Actions for the Kepler Problem
292
66 The Sommerfeld Atom
293
67 Energy as a Function of Actions
294
68 Invariance of the Circulation Under a Continuous Deformation
296
69 Ball Bouncing on a Moving Tray
297
610 Harmonic Oscillator with
298
612 Invariance of the Poisson Bracket Under a Canonical Transformation
299
614 OneDimensional Free Fall
300
615 OneDimensional Free Fall Again
301
617 From the Harmonic Oscillator to Coulombs Problem Solution p 333
302
618 Generators for Fundamental Transformations Solution p 336
303
Problem Solutions
305
62 OneDimensional Particle in a Box
306
63 Ball Bouncing on the Ground
308
64 Particle in a Constant Magnetic Field
310
65 Actions for the Kepler Problem
314
66 The Sommerfeld Atom statement p 293
316
67 Energy as a Function of Actions
318
68 Invariance of the Circulation Under a Continuous Deformation
322
69 Ball Bouncing on a Moving Tray
324
611 Choice of the Momentum
325
612 Invariance of the Poisson Bracket Under a Canonical Transformation
326
613 Canonicity for a Contact Transformation Statement p 299
327
614 Onedimensional Free Fall
329
615 Onedimensional Free Fall Again
330
616 Scale Dilation as a Function of Time
332
617 From the Harmonic Oscillator to Coulombs Problem Statement p 301
333
618 Generators for Fundamental Transformations Statement p 303
336
QuasiIntegrable Systems
341
72 Perturbation Theory
342
74 Adiabatic Invariants
345
Problem Statements
347
73 First Canonical Correction for the Pendulum Solution and Figure p 363
348
74 Beyond the First Order Correction
349
75 Adiabatic Invariant in an Elevator
350
76 Adiabatic Invariant and Adiabatic Relaxation Solution and Figure p 372
351
77 Charge in a Slowly Varying Magnetic
352
78 Illuminations Concerning the Aurora Borealis Solution and Figure p 379
354
Hannays Phase
356
Problem Solutions
358
72 Noncanonical Versus Canonical Perturbative Expansion Statement p 347
361
73 First Canonical Correction for the Pendulum Statement p 348
363
74 Beyond the First Order Correction
367
75 Adiabatic Invariant in an Elevator Statement p 350
370
76 Adiabatic Invariant and Adiabatic Relaxation Statement and Figure p 351
372
77 Charge in a Slowly Varying Magnetic Field Statement p 352
375
78 Illuminations Concerning the Aurora Borealis Statement p 354
379
Hannays Phase
382
From Order to Chaos
385
82 The Model of the Kicked Rotor
386
83 Poincarés Sections
388
85 Poincarés Sections for the Kicked Rotor
390
86 How to Recognize Fixed Points
393
87 SeparatricesHomocline PointsChaos
394
88 Complements
395
Problem Statements
396
the Standard Mapping Solution p 418
398
A Slight Difference Solution p 423
399
A Curiosity of the Standard Mapping Solution p 425
401
88 Anosovs Mapping or Arnolds Cat
403
89 Fermis Accelerator
405
810 Damped Pendulum and Standard Mapping Solution and Figure p 443
407
811 Stability of Periodic Orbits on a Billiard Table Solution p 447
409
Jupiters Greeks and Trojans solution p 450
412
Problem Solutions
415
82 Continuous Fractions or How to Play with Irrational Numbers
417
83 Properties of the Phase Space of the Standard Mapping statement p 398
418
1 for the Standard Mapping statement p 398
419
A Slight Difference Statement p 399
423
A Curiosity of the Standard Mapping statement p 401
425
87 Demonstration of a Kicked Rotor?
427
88 Anosovs Mapping or Arnolds Cat
432
89 Fermis Accelerator
438
810 Damped Pendulum and Standard Mapping Statement p 407
443
811 Stability of Periodic Orbits on a Billiard Table Statement and Figure p 409
447
Jupiters Greeks and Trojans Statement and Figure p 412
450
Bibliography
456
Index
461
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