solved problems in lagrangian and hamiltonian mechanics978-90-481-2393-3/1 · the solved problems...

Solved Problems in Lagrangianand Hamiltonian Mechanics

Grenoble Sciences

Grenoble Sciences pursues a triple aim:� to publish works responding to a clearly defined project, with no curriculum orvogue constraints,� to guarantee the selected titles’ scientific and pedagogical qualities,� to propose books at an affordable price to the widest scope of readers.

Each project is selected with the help of anonymous referees, followed by anaverage one-year interaction between the authors and a Readership Committeewhose members’ names figure in the front pages of the book. Grenoble Sciencesthen signs a co-publishing agreement with the most adequate publisher.

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Scientific Director of Grenoble Sciences: Jean BORNAREL,Professor at the Joseph Fourier University, Grenoble, France

Grenoble Sciences is supported by the French Ministry of HigherEducation and Research and the “Region Rhone-Alpes”.

The Solved Problems in Lagrangian and Hamiltonian MechanicsReading Committee included the following members:

� Robert ARVIEU, Professor at the Joseph Fourier University, Grenoble, France� Jacques MEYER, Professor at the Nuclear Physics Institute, Claude BernardUniversity, Lyon, France

with the contribution of:� Myriam REFFAY and Bertrand RUPH

The translation of “Problemes corriges de Mecanique et resumes de cours. DeLagrange a Hamilton”, published by Grenoble Sciences in partnership with EDPSciences, was performed by:� Bernard SILVESTRE-BRAC and Anthony John COLE, Senior researcher at the

CNRS, Grenoble, France

Front cover illustration: composed by Alice GIRAUD

Claude Gignoux · Bernard Silvestre-Brac

Solved Problemsin Lagrangianand Hamiltonian Mechanics

123

Dr. Claude GignouxUniversite Joseph FourierLab. Physique Subatomique et

CosmologieCNRS-IN2P353 avenue des Martyrs38026 Grenoble [email protected]

Dr. Bernard Silvestre-BracUniversite Joseph FourierLab. Physique Subatomique et

CosmologieCNRS-IN2P353 avenue des Martyrs38026 Grenoble [email protected]

ISBN 978-90-481-2392-6 e-ISBN 978-90-481-2393-3DOI 10.1007/978-90-481-2393-3Springer Dordrecht Heidelberg London New York

Library of Congress Control Number: 2009926952

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Foreword

Mechanics is an old science, but it acquired its great reputation at theend of the 17th century, due to Newton’s works. A century later, Eulerand, above all, Lagrange renewed it and led it towards a formulation notonly aesthetically elegant but also capable of applications to other fields ofphysics. Fifty years later, Hamilton and Jacobi gave their names to veryimportant further contributions. Lastly, at the end of the xix

th century,Poincaré took a new step with the introduction of geometry in the analysisof physical problems. During the xx

th century, physicists produced newdevelopments from the works of their famous predecessors.

This book is addressed to readers already familiar with the Newtonianapproach for mechanics. Several training textbooks – some of them excellent– rely on this approach. On the other hand, it seems that exercises based onLagrangian and Hamiltonian formulations are rather scarce in the literature;we hope that the present work may help to fill this gap.

In a previous book published in French by EDP, Grenoble-Sciences col-lection, under the name La mécanique: de la formulation lagrangienne auchaos hamiltonien, we proposed, for undergraduate students, a comprehen-sible synthesis of all the modern facets of mechanics and their relationshipsto other domains of physics. This textbook contains a great number of ex-ercises and problems, many of them original, dealing with the theories ofLagrange, Hamilton and Poincaré. We gave only the results or brief hintsfor solving these problems. Some problems can be considered as difficult,or even disconcerting, and readers encouraged us to provide the solution ofthose exercises which illustrate all the topics presented in the book. Thisis the aim of the present work. We retained from the foregoing book mostof the problems presented here, very often trying to make them clearer,sometimes trying to find interesting extensions. We also proposed new onesbetter suited to our pedagogic goal. In the same spirit, others have beenwithdrawn because we judged them less instructive for physics, even if themathematical points they dealt with were logical consequences of featurestreated in the textbook.

Of course, the present work is a natural complement of the course-book;nevertheless, we have tried to make it self-contained and, with this in mind,

vi Foreword

we add in each chapter a succinct, although clear and complete, summary ofeach topic. These summaries are adequate to tackle and solve the problemspresented afterwards; every concept or notion necessary for obtaining thesolution is presented and developed therein. Our aim is to make the readerfamiliar with the Lagrangian and Hamiltonian approaches, which may bedifficult to grasp, to demonstrate the power of this formalism and help todevelop skills for managing the techniques essential for this kind of study.The problems are selected with this purpose and they illustrate very oftenpractical physical situations and sometimes aspects of everyday life.

This book is built around eight chapters entitled:1. The Lagrangian formulation

2. Lagrangian systems

3. Hamilton’s principle (also called the least action principle)

4. The Hamiltonian formalism

5. The Hamilton-Jacobi formalism

6. Integrable systems

7. Quasi-integrable systems

8. From order to chaosIn each chapter, the reader will find:

• A clear, succinct and rather deep summary of all the notions that mustbe understood, the important points that must be memorized and thenotations and symbols used in the problems.

• The statements of the problems which are presented consecutively. Thesestatements are sufficiently detailed so that, with the help of the lessonsummaries, it is unnecessary for the reader to search for other sourcesof information. Whenever a figure turns out to be essential for a goodunderstanding of the text, it appears in the statement. The progressivedifficulty of the problems is symbolized with an increasing number ofstars (from 1 to 3) added to the title.

• Detailed answers to the problems which are grouped together at the endof the chapter. A number of additional figures are inserted in the cor-responding text in order to exhibit essential points or to avoid lengthycircumlocutions.At the beginning of the book, a synoptic table gathers, chapter by chap-

ter, the set of all the proposed problems, giving, for each of them, its ref-erence (number, title, page number), its difficulty (1 to 3 stars), as well asthe important features or the peculiar aspects treated in this problem.

Foreword vii

Two purposes are pursued• To concretize, through simple and often academic examples, notions that

seem apparently very abstract. The methods to find the solution are notnecessarily the most elegant or the quickest, but it is important to check,via these simple examples, one’s understanding of these new tools formechanics. Sometimes the same problem, or the same physical situation,is studied once more in a subsequent chapter with new tools in order toemphasize some novel feature of the method.

• To emphasize the power of these new tools in physics applied to fields asmiscellaneous as traditional mechanics, optics, electromagnetism, wavesin general, and quantum mechanics. Concerning these fascinating and upto date subjects in physics, we will focus only on the mechanical aspect.However, the reader could satisfy his curiosity, with help of keywords,by looking for further information firstly in a good and complete ency-clopedia, then on the web using a engine such as Google (or, for moreexotic subjects, Yahoo). He will find exhaustive lectures as well as recentarticles.

We strongly recommend that the reader carries out some applications anddraws the figures proposed in the detailed solutions. For the drawing orplotting of curves, the authors have used the freewares xfig or xmgrace thatcan be downloaded from the web. The differential equations have beensolved with the help of the Mathematica software package.

Acknowledgment

We are very grateful to the reading committee for all remarks and sug-gestions which were very helpful in achieving a greater degree of consistencyand an improved pedagogical quality in this work. We also very much ap-preciated contributions from our students whose enthusiasm and variousquestions allowed us to make the text more understandable.

Finally, we wish to thank members of the editorial board, Nicole Sauvaland Jean Bornarel, for their competent advice, Sylvie Bordage, Julie Ridardand Thierry Morturier for their role in improving our figures and KonstantinProtasov for his helpful work concerning the layout of the manuscript.

A work such as ours is never perfect neither in the form nor in theessence of the matter. If, despite the care brought by the authors in theconstruction of this work, you find mistakes, shortcomings or incoherences,we will be grateful to you for mentioning them via one of the followingelectronic addresses:

[email protected]@lpsc.in2p3.fr

ContentsForeword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Synoptic Tables of the Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 1. The Lagrangian Formulation . . . . . . . . . . . . . . . . . . . . 9Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1. Generalized Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2. Lagrange’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3. Generalized Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4. Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Problem Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.1. The Wheel Jack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2. The Sling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3. Rope Slipping on a Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4. Reaction Force for a Bead on a Hoop . . . . . . . . . . . . . . . . . . . 161.5. Huygens Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.6. Cylinder Rolling on a Moving Tray . . . . . . . . . . . . . . . . . . . . . 181.7. Motion of a Badly Balanced Cylinder . . . . . . . . . . . . . . . . . . . 181.8. Free Axle on a Inclined Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.9. The Turn Indicator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.10. An Experiment to Measure the Rotational Velocity

of the Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.11. Generalized Inertial Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Problem Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.1. The Wheel Jack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.2. The Sling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.3. Rope Slipping on a Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.4. Reaction Force for a Bead on a Hoop . . . . . . . . . . . . . . . . . . . 281.5. The Huygens Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.6. Cylinder Rolling on a Moving Tray . . . . . . . . . . . . . . . . . . . . . 331.7. Motion of a Badly Balanced Cylinder . . . . . . . . . . . . . . . . . . . 351.8. Free Axle on a Inclined Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 391.9. The Turn Indicator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431.10. An Experiment to Measure the Rotational Velocity

of the Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461.11. Generalized Inertial Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

xii Contents

Chapter 2. Lagrangian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.1. Generalized Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.2. Lagrangian System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.3. Constants of the Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.4. Two-body System with Central Force . . . . . . . . . . . . . . . . . . . 552.5. Small Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Problem Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.1. Disc on a Movable Inclined Plane . . . . . . . . . . . . . . . . . . . . . . . 572.2. Painlevé’s Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.3. Application of Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . 582.4. Foucault’s Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.5. Three-particle System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.6. Vibration of a Linear Triatomic Molecule:

The “Soft” Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.7. Elastic Transversal Waves in a Solid . . . . . . . . . . . . . . . . . . . . 642.8. Lagrangian in a Rotating Frame . . . . . . . . . . . . . . . . . . . . . . . . 652.9. Particle Drift in a Constant Electromagnetic Field . . . . . . 662.10. The Penning Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.11. Equinox Precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.12. Flexion Vibration of a Blade . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712.13. Solitary Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.14. Vibrational Modes of an Atomic Chain . . . . . . . . . . . . . . . . . 75

Problem Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762.1. Disc on a Movable Inclined Plane . . . . . . . . . . . . . . . . . . . . . . . 762.2. Painlevé’s Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772.3. Application of Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . 782.4. Foucault’s Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792.5. Three-particle System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822.6. Vibration of a Linear Triatomic Molecule:

The “Soft” Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862.7. Elastic Transversal Waves in a Solid . . . . . . . . . . . . . . . . . . . . 882.8. Lagrangian in a Rotating Frame . . . . . . . . . . . . . . . . . . . . . . . . 892.9. Particle Drift in a Constant Electromagnetic Field . . . . . . 912.10. The Penning Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942.11. Equinox Precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972.12. Flexion Vibration of a Blade . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022.13. Solitary Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052.14. Vibrational Modes of an Atomic Chain . . . . . . . . . . . . . . . . 107

Chapter 3. Hamilton’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

3.1. Statement of the Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113.2. Action Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Contents xiii

3.3. Action and Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123.4. Some Well Known Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133.5. The Calculus of Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Problem Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1163.1. The Lorentz Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1163.2. Relativistic Particle in a Central Force Field . . . . . . . . . . . 1173.3. Principle of Least Action? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183.4. Minimum or Maximum Action? . . . . . . . . . . . . . . . . . . . . . . . 1193.5. Is There Only One Solution

Which Makes the Action Stationary? . . . . . . . . . . . . . . . . . . 1203.6. The Principle of Maupertuis . . . . . . . . . . . . . . . . . . . . . . . . . . . 1213.7. Fermat’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223.8. The Skier Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223.9. Free Motion on an Ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . 1233.10. Minimum Area for a Fixed Volume . . . . . . . . . . . . . . . . . . . . 1243.11. The Form of Soap Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1253.12. Laplace’s Law for Surface Tension . . . . . . . . . . . . . . . . . . . . . 1273.13. Chain of Pendulums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283.14. Wave Equation for a Flexible Blade . . . . . . . . . . . . . . . . . . . . 1283.15. Precession of Mercury’s Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Problem Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313.1. The Lorentz Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313.2. Relativistic Particle in a Central Force Field . . . . . . . . . . . 1323.3. Principle of Least Action? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1353.4. Minimum or Maximum Action? . . . . . . . . . . . . . . . . . . . . . . . 1373.5. Is There Only One Solution

Which Makes the Action Stationary? . . . . . . . . . . . . . . . . . . 1383.6. The Principle of Maupertuis . . . . . . . . . . . . . . . . . . . . . . . . . . . 1413.7. Fermat’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1443.8. The Skier Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1463.9. Free Motion on an Ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . 1503.10. Minimum Area for a Fixed Volume . . . . . . . . . . . . . . . . . . . . 1523.11. The Form of Soap Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1543.12. Laplace’s Law for Surface Tension . . . . . . . . . . . . . . . . . . . . . 1583.13. Chain of Pendulums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1603.14. Wave Equation for a Flexible Blade . . . . . . . . . . . . . . . . . . . . 1613.15. Precession of Mercury’s Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Chapter 4. Hamiltonian Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 165Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

4.1. Generalized Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1654.2. Hamilton’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1664.3. Hamilton’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1674.4. Liouville’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

xiv Contents

4.5. Autonomous One-dimensional Systems . . . . . . . . . . . . . . . . . 1684.6. Periodic One-dimensional Hamiltonian Systems . . . . . . . . 169

Problem Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1714.1. Electric Charges Trapped in Conductors . . . . . . . . . . . . . . . 1714.2. Symmetry of the Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . 1714.3. Hamiltonian in a Rotating Frame . . . . . . . . . . . . . . . . . . . . . . 1724.4. Identical Hamiltonian Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 1734.5. The Runge-Lenz Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1734.6. Quicker and More Ecologic than a Plane . . . . . . . . . . . . . . . 1744.7. Hamiltonian of a Charged Particle . . . . . . . . . . . . . . . . . . . . . 1764.8. The First Integral Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1774.9. What About Non-Autonomous Systems? . . . . . . . . . . . . . . 1784.10. The Reverse Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1784.11. The Paul Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1804.12. Optical Hamilton’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 1814.13. Application to Billiard Balls . . . . . . . . . . . . . . . . . . . . . . . . . . . 1834.14. Parabolic Double Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1844.15. Stability of Circular Trajectories in a Central Potential 1854.16. The Bead on the Hoop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1864.17. Trajectories in a Central Force Field . . . . . . . . . . . . . . . . . . . 188

Problem Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1884.1. Electric Charges Trapped in Conductors . . . . . . . . . . . . . . . 1884.2. Symmetry of the Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . 1904.3. Hamiltonian in a Rotating Frame . . . . . . . . . . . . . . . . . . . . . . 1924.4. Identical Hamiltonian Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 1944.5. The Runge-Lenz Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1954.6. Quicker and More Ecologic than a Plane . . . . . . . . . . . . . . . 1984.7. Hamiltonian of a Charged Particle . . . . . . . . . . . . . . . . . . . . . 2004.8. The First Integral Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2044.9. What About Non-Autonomous Systems? . . . . . . . . . . . . . . 2064.10. The Reverse Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2074.11. The Paul Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2114.12. Optical Hamilton’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 2144.13. Application to Billiard Balls . . . . . . . . . . . . . . . . . . . . . . . . . . . 2164.14. Parabolic Double Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2194.15. Stability of Circular Trajectories in a Central Potential 2224.16. The Bead on the Hoop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2244.17. Stability of Circular Trajectories in a Central Potential 228

Chapter 5. Hamilton-Jacobi Formalism . . . . . . . . . . . . . . . . . . . . 233Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

5.1. The Action Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2335.2. Reduced Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2345.3. Maupertuis’ Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

Contents xv

5.4. Jacobi’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2365.5. Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2365.6. Huygens’ Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

Problem Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2395.1. How to Manipulate the Action and the Reduced Action 2395.2. Action for a One-dimensional Harmonic Oscillator . . . . . 2415.3. Motion on a Surface and Geodesic . . . . . . . . . . . . . . . . . . . . . 2415.4. Wave Surface for Free Fall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2425.5. Peculiar Wave Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2435.6. Electrostatic Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2435.7. Maupertuis’ Principle with an Electromagnetic Field . . . 2455.8. Separable Hamiltonian, Separable Action . . . . . . . . . . . . . . 2465.9. Stark Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2475.10. Orbits of Earth’s Satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2485.11. Phase and Group Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Problem Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2525.1. How to Manipulate the Action and the Reduced Action 2525.2. Action for a One-Dimensional Harmonic Oscillator . . . . . 2585.3. Motion on a Surface and Geodesic . . . . . . . . . . . . . . . . . . . . . 2605.4. Wave Surface for Free Fall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2615.5. Peculiar Wave Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2645.6. Electrostatic Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2655.7. Maupertuis’ Principle with an Electromagnetic Field . . . 2685.8. Separable Hamiltonian, Separable Action . . . . . . . . . . . . . . 2705.9. Stark Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2715.10. Orbits of Earth’s Satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2755.11. Phase and Group Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

Chapter 6. Integrable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

6.1. Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2816.1.1. Some Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2816.1.2. Good Coordinates: The Angle–Action Variables . 283

6.2. Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2866.2.1. Building the Angle Variables . . . . . . . . . . . . . . . . . . . . 2866.2.2. Flow/Poisson Bracket/Involution . . . . . . . . . . . . . . . . 2876.2.3. Criterion to Obtain a Canonical Transformation . 288

Problem Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2896.1. Expression of the Period for a One-Dimensional Motion 2896.2. One-dimensional Particle in a Box . . . . . . . . . . . . . . . . . . . . . 2906.3. Ball Bouncing on the Ground . . . . . . . . . . . . . . . . . . . . . . . . . . 2906.4. Particle in a Constant Magnetic Field . . . . . . . . . . . . . . . . . . 2916.5. Actions for the Kepler Problem . . . . . . . . . . . . . . . . . . . . . . . . 2926.6. The Sommerfeld Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2936.7. Energy as a Function of Actions . . . . . . . . . . . . . . . . . . . . . . . 294

xvi Contents

6.8. Invariance of the Circulation Under a ContinuousDeformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

6.9. Ball Bouncing on a Moving Tray . . . . . . . . . . . . . . . . . . . . . . . 2976.10. Harmonic Oscillator with a Variable Frequency . . . . . . . . 2986.11. Choice of the Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2986.12. Invariance of the Poisson Bracket

Under a Canonical Transformation . . . . . . . . . . . . . . . . . . . . . 2996.13. Canonicity for a Contact Transformation . . . . . . . . . . . . . . 2996.14. One-Dimensional Free Fall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3006.15. One-Dimensional Free Fall Again . . . . . . . . . . . . . . . . . . . . . . 3016.16. Scale Dilation as a Function of Time . . . . . . . . . . . . . . . . . . . 3016.17. From the Harmonic Oscillator to Coulomb’s Problem . . 3026.18. Generators for Fundamental Transformations . . . . . . . . . . 303

Problem Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3056.1. Expression of the Period for a One-Dimensional Motion 3056.2. One-Dimensional Particle in a Box . . . . . . . . . . . . . . . . . . . . . 3066.3. Ball Bouncing on the Ground . . . . . . . . . . . . . . . . . . . . . . . . . . 3086.4. Particle in a Constant Magnetic Field . . . . . . . . . . . . . . . . . . 3106.5. Actions for the Kepler Problem . . . . . . . . . . . . . . . . . . . . . . . . 3146.6. The Sommerfeld Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3166.7. Energy as a Function of Actions . . . . . . . . . . . . . . . . . . . . . . . 3186.8. Invariance of the Circulation Under a Continuous

Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3226.9. Ball Bouncing on a Moving Tray . . . . . . . . . . . . . . . . . . . . . . . 3246.10. Harmonic Oscillator with a Variable Frequency . . . . . . . . 3246.11. Choice of the Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3256.12. Invariance of the Poisson Bracket

Under a Canonical Transformation . . . . . . . . . . . . . . . . . . . . . 3266.13. Canonicity for a Contact Transformation . . . . . . . . . . . . . . 3276.14. One-dimensional Free Fall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3296.15. One-dimensional Free Fall Again . . . . . . . . . . . . . . . . . . . . . . . 3306.16. Scale Dilation as a Function of Time . . . . . . . . . . . . . . . . . . . 3326.17. From the Harmonic Oscillator to Coulomb’s Problem . . 3336.18. Generators for Fundamental Transformations . . . . . . . . . . 336

Chapter 7. Quasi-Integrable Systems . . . . . . . . . . . . . . . . . . . . . . 341Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3417.2. Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3427.3. Canonical Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . 3427.4. Adiabatic Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

Problem Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3477.1. Limits of the Perturbative Expansion . . . . . . . . . . . . . . . . . . 3477.2. Non-canonical Versus Canonical Perturbative Expansion 347

Contents xvii

7.3. First Canonical Correction for the Pendulum . . . . . . . . . . 3487.4. Beyond the First Order Correction . . . . . . . . . . . . . . . . . . . . . 3497.5. Adiabatic Invariant in an Elevator . . . . . . . . . . . . . . . . . . . . . 3507.6. Adiabatic Invariant and Adiabatic Relaxation . . . . . . . . . . 3517.7. Charge in a Slowly Varying Magnetic Field . . . . . . . . . . . . 3527.8. Illuminations Concerning the Aurora Borealis . . . . . . . . . . 3547.9. Bead on a Rigid Wire: Hannay’s Phase . . . . . . . . . . . . . . . . 356

Problem Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3587.1. Limits of the Perturbative Expansion . . . . . . . . . . . . . . . . . . 3587.2. Non-canonical Versus Canonical Perturbative Expansion 3617.3. First Canonical Correction for the Pendulum . . . . . . . . . . 3637.4. Beyond the First Order Correction . . . . . . . . . . . . . . . . . . . . . 3677.5. Adiabatic Invariant in an Elevator . . . . . . . . . . . . . . . . . . . . . 3707.6. Adiabatic Invariant and Adiabatic Relaxation . . . . . . . . . . 3727.7. Charge in a Slowly Varying Magnetic Field . . . . . . . . . . . . 3757.8. Illuminations Concerning the Aurora Borealis . . . . . . . . . . 3797.9. Bead on a Rigid Wire: Hannay’s Phase . . . . . . . . . . . . . . . . 382

Chapter 8. From Order to Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . 385Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3858.2. The Model of the Kicked Rotor . . . . . . . . . . . . . . . . . . . . . . . . 3868.3. Poincaré’s Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3888.4. The Rotor for a Null Perturbation . . . . . . . . . . . . . . . . . . . . . 3888.5. Poincaré’s Sections for the Kicked Rotor . . . . . . . . . . . . . . . 3908.6. How to Recognize Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . 3938.7. Separatrices/Homocline Points/Chaos . . . . . . . . . . . . . . . . . 3948.8. Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

Problem Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3968.1. Disappearance of Resonant Tori . . . . . . . . . . . . . . . . . . . . . . . 3968.2. Continuous Fractions or How to Play with Irrational

Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3968.3. Properties of the Phase Space of the Standard Mapping 3988.4. Bifurcation of the Periodic Trajectory 1:1

for the Standard Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3988.5. Chaos–Ergodicity : A Slight Difference . . . . . . . . . . . . . . . . 3998.6. Acceleration Modes: A Curiosity of the Standard

Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4018.7. Demonstration of a Kicked Rotor? . . . . . . . . . . . . . . . . . . . . . 4018.8. Anosov’s Mapping (or Arnold’s Cat) . . . . . . . . . . . . . . . . . . . 4038.9. Fermi’s Accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4058.10. Damped Pendulum and Standard Mapping . . . . . . . . . . . . 4078.11. Stability of Periodic Orbits on a Billiard Table . . . . . . . . . 4098.12. Lagrangian Points: Jupiter’s Greeks and Trojans . . . . . . . 412

xviii Contents

Problem Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4158.1. Disappearance of Resonant Tori . . . . . . . . . . . . . . . . . . . . . . . 4158.2. Continuous Fractions or How to Play with Irrational

Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4178.3. Properties of the Phase Space of the Standard Mapping 4188.4. Bifurcation of the Periodic Trajectory 1:1

for the Standard Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4198.5. Chaos–Ergodicity: A Slight Difference . . . . . . . . . . . . . . . . . 4238.6. Acceleration Modes: A Curiosity of the Standard

Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4258.7. Demonstration of a Kicked Rotor? . . . . . . . . . . . . . . . . . . . . . 4278.8. Anosov’s Mapping (or Arnold’s Cat) . . . . . . . . . . . . . . . . . . . 4328.9. Fermi’s Accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4388.10. Damped Pendulum and Standard Mapping . . . . . . . . . . . . 4438.11. Stability of Periodic Orbits on a Billiard Table . . . . . . . . . 4478.12. Lagrangian Points: Jupiter’s Greeks and Trojans . . . . . . . 450

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

Synoptic Tablesof the Problems

Chapter 1. Lagrangian Formulation

No Title Level P. Features

1.1 The wheel jack � 14 Lagrangian mechanics.D’Alembert’s principle

1.2 The sling � 15 Lagrange equationsfor a very simple system

1.3 The rope slipping on atable

� 16 Lagrange equationsin the presence of friction

1.4 Reaction force for abead on a hoop

�� 16 Reaction force calculated byadding one generalized coordinate

1.5 Huygens pendulum � � � 17 Work of contact forces

1.6 Cylinder rollingon a movable tray

�� 18 Lagrange equations with twodifferent coordinates

1.7 Motion of a badlybalanced cylinder

� � � 18 Koenig’s theorem.Holonomic constraint

1.8 Free axle on a inclinedplane

� � � 19 Constrained Lagrange equations.Lagrange multipliers

1.9 The turning indicator � � � 21 The gyroscope studied with theLagrangian formalism

1.10 An experiment tomeasure the rotationalvelocity of the Earth

� � � 22 A well controlled gyroscope asan alternative experiment toFoucault’s pendulum

1.11 Generalized inertialforces

� � � 23 Lagrange equations in a nonGalilean frame

2 Synoptic Tables of the Problems

Chapter 2. Lagrangian Systems


2.1 Disc on a movableinclined plane

� 57 Example of a time-dependentLagrangian

2.2 Painlevé’s integral �� 58 Search for a first integral for atime-dependent Lagrangian

2.3 Application ofNoether’s theorem

� 58 Very simple application ofNoether’s theorem

2.4 Foucault’s pendulum �� 59 A famous experiment explainedwith Lagrangian formalism

2.5 Three-particle system �� 61 Appearance of symmetries by achange of variables

2.6 Vibration of a lin-ear triatomic molecule:the “soft” mode

�� 63 Eigenmodes for an oscillatingsystem

2.7 Elastic transversalwaves in a solid

�� 64 Passing from a discrete model to acontinuous model for the study ofa solid bar

2.8 Lagrangian in arotating frame

�� 65 Modification of the Lagrangian inthe passage to a rotating frame

2.9 Particle drift in a con-stant electromagneticfield

�� 66 Drift motion for a particle in anelectromagnetic field exhibited inthe Lagrangian formalism

2.10 The Penning trap � � � 67 An astute electromagnetic systemto trap a particle

2.11 Equinox precession � � � 68 Lagrangian formalism applied toan astronomical phenomenon

2.12 Flexion vibration of ablade

� � � 71 Passing from a discrete model toa continuous model for the studyof the vibration of an embeddedblade

2.13 Solitary waves �� 73 Non-linear equation for a wavepropagating without deformation

2.14 Vibrational modes ofan atomic chain

� � � 75 System composed of an infinitenumber of coupled oscillators

Synoptic Tables of the Problems 3

Chapter 3. Hamilton’s Principle


3.1 The Lorentz force �� 116 Hamilton’s principle applied to anelectromagnetic problem

3.2 Relativistic particle ina central force field

� � � 117 Relativistic Binet’s equation

3.3 Principle of leastaction?

� � � 118 Justification of the concept of“least action”

3.4 Minimum ormaximum action?

�� 119 Why the action is not alwaysminimal

3.5 Is there only onesolution which makesthe action stationary?

�� 120 Hamilton’s principle. Throughtwo points may pass severaltrajectories

3.6 The principle ofMaupertuis

�� 121 Alternative to the Hamiltonprinciple for the determinationof the trajectories

3.7 Fermat’s principle �� 122 Hamilton’s principle in thedomain of optics

3.8 The skier strategy � � � 122 Calculus of variations for thebrachistochrone

3.9 Free motion on anellipsoid

�� 123 Calculus of variations with aholonomic constraint.Lagrange multipliers

3.10 Minimum area for afixed volume

�� 124 Calculus of variations with anintegral constraint.Lagrange multipliers

3.11 The form of soap films � � � 125 Amusing application ofHamilton’s principle.Calculus of variations

3.12 Laplace’s law forsurface tension

� � � 127 Hamilton’s principle applied tohydrostatics

3.13 Chain of pendulums �� 128 Hamilton’s principle for acontinuous system

3.14 Wave equation for aflexible blade

�� 128 Building a Lagrangian density

3.15 Precession ofMercury’s orbit

� � � 128 Hamilton’s principle in thecontext general relativity


Chapter 4. Hamiltonian Formalism

No Title level P. Features

4.1 Electric chargestrapped in conductors

�� 171 Electrostatic image. Hamilton’sequations. First integral

4.2 Symmetry of thetrajectory

� 171 Binet’s equation. Its use fortreating symmetries

4.3 Hamiltonian in arotating frame

�� 172 Change of frame.Legendre transform

4.4 Identical Hamiltonianflows

� 173 Hamilton’s equations and theirflow

4.5 The Runge–Lenzvector

�� 173 Building a constant vector.Relationship with otherconstants of motion

4.6 Quicker and moreecologic than a plane

�� 174 Hamilton’s equations in agravitational field

4.7 Hamiltonian of acharged particle

� � � 176 Hamilton’s equations. Covariantrelativistic formalism

4.8 The first integralinvariant

�� 177 Integral invariant in theHamiltonian formalism

4.9 What about non-autonomous systems?

� 178 To render autonomous anon-autonomous system.Corresponding flow

4.10 The reverse pendulum � � � 178 Hamilton’s equations.Propagator. Stability conditions.Arnold’s tongue

4.11 The Paul trap � � � 180 Electromagnetic system.Propagator. Stability

4.12 Optical Hamilton’sequations

� � � 181 Snell–Descartes law obtained fromHamilton’s equations

4.13 Application to billiardballs

�� 183 Choice of variables for trajectorieson a billiard table.Conservation of the area

4.14 Parabolic double well �� 184 Simple motion and phase portrait

4.15 Stability of circulartrajectories in acentral potential

�� 185 Motion for a power-law potential.Stability conditions



4.16 The bead on the hoop �� 186 Phase portrait. Stability.Bifurcation

4.17 Trajectories in acentral force field

�� 188 Relativistic Binet’s equation.Phase portrait

Chapter 5. Hamilton–Jacobi Formalism


5.1 How to manipulate theaction and the reducedaction

�� 239 Relation between the totalaction and the reduced action.Hamilton–Jacobi equation

5.2 Action for aone-dimensionalharmonic oscillator

�� 241 Reduced action. Hamilton–Jacobiequation

5.3 Motion on a surfaceand geodesic

�� 241 Action with a general metric.Principle of Maupertuis

5.4 Wave surface for freefall

�� 242 Hamilton–Jacobi equation. Wavefronts in a gravitational field

5.5 Peculiar wave fronts �� 243 Wave fronts and trajectories ina gravitational field.Hamilton–Jacobi equation

5.6 Electrostatic lens � � � 243 Electromagnetism and theprinciple of Maupertuis for asystem with cylindrical symmetry

5.7 Maupertuis’ principlewith an electromag-netic field

� � � 245 Electromagnetic field and theprinciple of Maupertuis.Cyclotron motion

5.8 Separable Hamilto-nian, separable action

� 246 Separation of the variables in theHamilton–Jacobi equations

5.9 Stark effect � � � 247 Parabolic coordinates whichseparate the variables

5.10 Orbits of Earth’ssatellites

� � � 248 Elliptic coordinates whichseparate the variables

5.11 Phase and groupvelocities

� 251 A notion used in optics which isalso valid in mechanics


Chapter 6. Integrable Systems


6.1 Expression of theperiod for aone-dimensional motion

� 289 Reduced action and angularfrequency

6.2 One-dimensionalparticle in a box

�� 290 Angle-action variables.Quantization

6.3 Ball bouncing on theground

�� 290 Angle-action variables.Quantization

6.4 Particle in a constantmagnetic field

� � � 291 Action variable. Phase portrait.Landau’s levels

6.5 Actions for the Keplerproblem

� � � 292 Energy as a function of actions.Quantization

6.6 The Sommerfeld atom � � � 293 Energy as a function of actions.Relativistic systems.Quantization

6.7 Energy as a function ofactions

� � � 294 Form of the Hamiltonian as afunction of actions

6.8 Invariance of the circu-lation under a continu-ous deformation

� � � 296 Functions in involution.Circulation on a torus

6.9 Ball bouncing on amoving tray

� 297 Time-dependent canonicaltransformation for the free fall

6.10 Harmonicoscillator with avariable frequencys

�� 298 Time-dependent canonicaltransformation for the harmonicoscillator

6.11 Choice of themomentum

�� 298 A particular canonicaltransformation

6.12 Invariance of thePoisson bracketunder a canonicaltransformation

� 299 Poisson brackets. Canonicaltransformation

6.13 Canonicity for acontact transformation

� 299 Poisson brackets. Canonicaltransformation

6.14 One-dimensional freefall

�� 300 Canonical transformation.Trajectory

6.15 One-dimensional freefall again

�� 301 Angle-action variables.Generating function

6.16 Scale dilation as afunction of time

� � � 301 Time dependent canonical trans-formation. Generating function



6.17 From the harmonicoscillator to Coulomb’sproblem

�� 302 Angle-action variables. Canonicaltransformation. Energy as afunction of actions

6.18 Generators forfundamental trans-formations

�� 303 Flow parameters. Generators.Relativity

Chapter 7. Quasi-integrable Systems


7.1 Limits of theperturbative expansion

� 347 Differential equation. Classicalperturbation theory

7.2 Non-canonical versuscanonical perturbativeexpansion

�� 347 Anharmonic oscillator.Canonical and non-canonicalperturbation theories

7.3 First canonical correc-tion for the pendulum

� � � 348 Simple pendulum.Quartic perturbation.Exact and perturbative treatments

7.4 Beyond the first ordercorrection

� � � 349 Canonical perturbation theory.Second order

7.5 Adiabatic invariant inan elevator

�� 350 Action variable.Adiabatic invariant

7.6 Adiabatic invariant andadiabatic relaxation

� � � 351 Action variable. Adiabaticinvariant. Monatomic ideal gaz

7.7 Charge in a slowlyvarying magnetic field

� � � 352 Generating function.Canonical transformation.Adiabatic invariant

7.8 Illuminations concer-ning the aurora borealis

�� 354 Electromagnetism.Adiabatic invariant.Cyclotron motion and drift

7.9 Bead on a rigid wire:Hannay’s phase

� � � 356 Angle-action variables. Adiabaticinvariance. Hannay’s phase


Chapter 8. From Order to Chaos


8.1 Disappearance of reso-nant tori

�� 396 Resonant tori. KAM theorem

8.2 Continuous fractionsor how to play withirrational numbers

�� 396 Continuous fractions.Rational and irrational numbers.Convergence

8.3 Properties of the phasespace of the standardmapping

� 398 Poincaré’s section. Symmetries.Parameters

8.4 Bifurcation of the peri-odic trajectory 1:1 forthe standard mapping

�� 398 Standard mapping. Fixed points.Bifurcation

8.5 Chaos–ergodicity :a slight difference

�� 399 Condition for ergodicity

8.6 Acceleration modes:a curiosity of thestandard mapping

�� 401 Poincaré’s section.Standard mapping.Acceleration of the momentum

8.7 Demonstration of akicked rotor?

� � � 401 Jerky pendulum. Sawtoothmapping

8.8 Anosov’s mapping(or Arnold’s cat)

� � � 403 Fibonacci sequence. Anosov’smapping. Fixed points

8.9 Fermi’s accelerator � � � 405 Moving walls. Conservation of thearea. Ulam’s mapping

8.10 Damped pendulumand standard mapping

� � 407 Non-Hamiltonian system.Friction. Spiral point

8.11 Stability of periodicorbits on billiard table

� � � 409 Mapping on a billiard table.Derivative matrix of the mapping.Stability of periodic trajectories.Examples of billiard tables

8.12 Lagrangian points:Jupiter’s Greeks andTrojans

� � � 412 Restricted three-body problem.Lagrangian points and theirstability

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