15
Variational Principles in Physics
lean-Louis Basdevant
Variational Principles in Physics
~ Springer
Professor Jean-Louis Basdevant Physics Department Ecole Poly technique 91128 Palaiseau France jean-louis. [email protected]
Library of Congress Control Number: 2006931784
ISBN 0-387-37747-6 ISBN 978-0-387-37747-6
Printed on acid-free paper.
ISBN 0-387-37748-4 (eBook) ISBN 978-0-387-37748-3 (eBook)
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Preface
Optimization under constraints is part of our daily lives. To live as comfortably as possible given that there exist conflicts such as the chores of everyday life or the desires of each individual in a family or group is a simple example. With the development of computer science, optimization has acquired a major role in the modern world. In the future, it is plausible that optimization will become one of the very first concepts to be taught in an elementary course in mathematics.
It is an amazing observation that laws of nature appear to follow such rules. These are expressed mathematically as variational principles. These principles possess two characteristics. First, they appear to be universal. Second, they express physical laws as the results of optimal equilibrium conditions between conflicting causes. In other words, they present natural phenomena as problems of optimization under constraints. The founding idea in modern physics is due to Fermat and his least time principle in optics. This was further developed in the framework of the calculus of variations of Euler and Lagrange. In 1844, Maupertuis found, with the help of Euler, the least action principle in mechanics.
The philosophical impact of the discovery of such principles of natural economy was considerable in the 18th century. However, if the metaphysical enthusiasm did not last long, it is not because of any lack of intellectual beauty or richness. It is because variational principles have constantly produced more and more profound physical results, many of which underlie contemporary theoretical physics. The ambition of this book is to describe some of their physical applications.
After presenting and analyzing some examples, the core of this book is devoted to the analytical mechanics of Lagrange and Hamilton, which is a must in the culture of any physicist of our time. The tools that we will develop will also be used to present the principles of Lagrangian field theory. We then study the motion of a particle in a curved space. This allows us to have a simple but rich taste of general relativity and its first applications. These have had a spectacular revival of interest in recent years, for instance in the
vi Preface
development of gravitational optics which allows us to probe the universe at very far distances. Another unexpected spinoff lies in the accuracy of the global positioning system.
In the last chapter, we present the theory of Feynman path integrals in quantum mechanics. This allows us to discover general structures common to different domains of physics that may seem, a priori, quite far apart.
This book resulted from the last course I delivered in the Ecole Polytechnique, for three years starting in 2001. I was struck by the interest that students found in this aspect of physics. They discovered a cultural component of science that they did not expect. For that reason, teaching this was a very rewarding piece of work.
I have deliberately chosen to develop as few mathematical techniques as possible in order to concentrate on the physical aspects. Mathematical developments can be found in the bibliography.
I am indebted to Andre Rouge for all his useful comments and suggestions. I profited considerably from his great culture.
I want to pay a tribute to the memory of Gilbert Grynberg. He should have been in charge of teaching this course at the Ecole Poly technique. His tremendous fight against a brain tumor prevented him from doing so. I admire his courage, his human qualities, and his intellectual elevation.
I am very grateful to James Rich, who was able to extract me from the traditional French academism and make me share his creative enthusiasm for physics. I hope he doesn't mind some of my mathematically minded remarks. Part of Chapter 6 was directly inspired by his work in a different context.
I thank my friends Adel Bilal, Fran~ois Jacquet, Christoph Kopper, David Langlois and Jean-Fran~ois Roussel for all their comments and suggestions when we were teaching this matter and having fun together.
Finally, I want to thank my students, in particular Claire Biot, Amelie Deslandes, Juan Luis Astray Riveiro, Clarice Aiello Demarchi, Joime Barral, Zoe Fournier, Celine Vallot, and Julien Boudet, for their questions and their kind comments. They have provided this book with a flavor and a spirit of youth that would have been absent without them.
Paris January 2006
lean-Louis Basdevant
Contents
Preface........................................................ v
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Esthetics and Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Metaphysics and Science ................................. 3 1.3 Numbers, Music, and Quantum Physics .................. " 4 1.4 The Age of Enlightenment and the Principle of the Best. . . . .. 7 1.5 The Fermat Principle and Its Consequences. . . . . . . . . . . . . . . .. 8 1.6 Variational Principles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9 1.7 The Modern Era, from Lagrange to Einstein and Feynman .... 12
2 Variational Principles . ................................... " 21 2.1 The Fermat Principle and Variational Calculus. . . . . . . . . . . . .. 22
2.1.1 Least Time Principle .............................. 22 2.1.2 Variational Calculus of Euler and Lagrange ........... 26 2.1.3 Mirages and Curved Rays .......................... 27
2.2 Examples of the Principle of Natural Economy .............. 30 2.2.1 Maupertuis Principle .............................. 30 2.2.2 Shape of a Massive String . . . . . . . . . . . . . . . . . . . . . . . . .. 31 2.2.3 Kirchhoff's Laws ...... . . . . . . . . . . . . . . . . . . . . . . . . . . .. 32 2.2.4 Electrostatic Potential ........... . . . . . . . . . . . . . . . . .. 33 2.2.5 Soap Bubbles ................................... " 34
2.3 Thermodynamic Equilibrium: Principle of Maximal Disorder .. 35 2.3.1 Principle of Equal Probability of States ... . . . . . . . . . .. 35 2.3.2 Most Probable Distribution and Equilibrium ........ " 36 2.3.3 Lagrange Multipliers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37 2.3.4 Boltzmann Factor ................................. 38 2.3.5 Equalization of Temperatures. . . . . . . . . . . . . . . . . . . . . .. 39 2.3.6 The Ideal Gas .................................... 40 2.3.7 Boltzmann's Entropy ............................ " 41 2.3.8 Heat and Work ................................... 42
viii Contents
2.4 Problems............................................... 43
3 The Analytical Mechanics of Lagrange. . . . . . . . . . . . . . . . . . . .. 47 3.1 Lagrangian Formalism and the Least Action Principle. . . . . . .. 49
3.1.1 Least Action Principle ............................ , 49 3.1.2 Lagrange-Euler Equations .......................... 50 3.1.3 Operation of the Optimization Principle. . . . . . . . . . . . .. 52
3.2 Invariances and Conservation Laws ...... . . . . . . . . . . . . . . . . .. 53 3.2.1 Conjugate Momenta and Generalized Momenta ....... 53 3.2.2 Cyclic Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 54 3.2.3 Energy and Translations in Time .................... 54 3.2.4 Momentum and Translations in Space. . . . . . . . . . . . . . .. 56 3.2.5 Angular Momentum and Rotations .................. 57 3.2.6 Dynamical Symmetries ............................ , 57
3.3 Velocity-Dependent Forces ............................... , 58 3.3.1 Dissipative Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 58 3.3.2 Lorentz Force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 59 3.3.3 Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 60 3.3.4 Momentum....................................... 61
3.4 Lagrangian of a Relativistic Particle . . . . . . . . . . . . . . . . . . . . . .. 61 3.4.1 Free Particle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 61 3.4.2 Energy and Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . .. 62 3.4.3 Interaction with an Electromagnetic Field. . . . . . . . . . .. 63
3.5 Problems............................................... 65
4 Hamilton's Canonical Formalism. . . . . . . . . . . . . . . . . . . . . . . . . .. 67 4.1 Hamilton's Canonical Formalism .......................... 68
4.1.1 Canonical Equations ............................... 69 4.2 Dynamical Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 70
4.2.1 Poincare and Chaos in the Solar System .............. 71 4.2.2 The Butterfly Effect and the Lorenz Attractor ........ 71
4.3 Poisson Brackets and Phase Space. . . . . . . . . . . . . . . . . . . . . . . .. 73 4.3.1 Time Evolution and Constants of the Motion ......... 74 4.3.2 Canonical Transformations ......................... 75 4.3.3 Phase Space; Liouville's Theorem. . . . . . . . . . . . . . . . . . .. 78 4.3.4 Analytical Mechanics and Quantum Mechanics. . . . . . .. 80
4.4 Charged Particle in an Electromagnetic Field .... . . . . . . . . . .. 81 4.4.1 Hamiltonian...................................... 81 4.4.2 Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 82
4.5 The Action and the Hamilton-Jacobi Equation .............. 82 4.5.1 The Action as a Function of the Coordinates and Time 83 4.5.2 The Hamilton-Jacobi Equation and Jacobi Theorem. .. 85 4.5.3 Conservative Systems, the Reduced Action, and the
Maupertuis Principle .............................. 87 4.6 Analytical Mechanics and Optics . . . . . . . . . . . . . . . . . . . . . . . . .. 89
Contents ix
4.6.1 Geometric Limit of Wave Optics .................... 89 4.6.2 Semiclassical Approximation in Quantum Mechanics ... 91
4.7 Problems............................................... 92
5 Lagrangian Field Theory .................................. 97 5.1 Vibrating String. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 98 5.2 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 99
5.2.1 Generalized Lagrange-Euler Equations ............... 99 5.2.2 Hamiltonian Formalism ............................ 100
5.3 Scalar Field ............................................. 101 5.4 Electromagnetic Field .................................... 102 5.5 Equations of First Order in Time .......................... 104
5.5.1 Diffusion Equation ................................ 104 5.5.2 Schrodinger Equation .............................. 104
5.6 Problems ............................................... 105
6 Motion in a Curved Space ................................. 107 6.1 Curved Spaces .......................................... 108
6.1.1 Generalities ....................................... 108 6.1.2 Metric Tensor ..................................... 110 6.1.3 Examples ........................................ 111
6.2 Free Motion in a Curved Space ............................ 112 6.2.1 Lagrangian ....................................... 113 6.2.2 Equations of Motion ............................... 113 6.2.3 Simple Examples .................................. 114 6.2.4 Conjugate Momenta and the Hamiltonian ............ 117
6.3 Geodesic Lines .......................................... 117 6.3.1 Definition ........................................ 117 6.3.2 Equation of the Geodesics .......................... 118 6.3.3 Examples ........ , ............................... 119 6.3.4 Maupertuis Principle and Geodesics ................. 121
6.4 Gravitation and the Curvature of Space-Time ............... 122 6.4.1 Newtonian Gravitation and Relativity ................ 122 6.4.2 The Schwarzschild Metric .......................... 124 6.4.3 Gravitation and Time Flow ......................... 125 6.4.4 Precession of Mercury's Perihelion ................... 125 6.4.5 Gravitational Deflection of Light Rays ............... 130
6.5 Gravitational Optics and Mirages .......................... 133 6.5.1 Gravitational Lensing .............................. 133 6.5.2 Gravitational Mirages .............................. 134 6.5.3 Baryonic Dark Matter ............................. 139
6.6 Problems ............................................... 144
x Contents
7 Feynman's Principle in Quantum Mechanics ............... 145 7.1 Feynman's Principle ..................................... 146
7.1.1 Recollections of Analytical Mechanics ................ 146 7.1.2 Quantum Amplitudes .............................. 147 7.1.3 Superposition Principle and Feynman's Principle ...... 147 7.1.4 Path Integrals .................................... 148 7.1.5 Amplitude of Successive Events ..................... 150
7.2 Free Particle ............................................ 152 7.2.1 Propagator of a Free Particle ....................... 152 7.2.2 Evolution Equation of the Free Propagator ........... 154 7.2.3 Normalization and Interpretation of the Propagator .... 155 7.2.4 Fourier and Schrodinger Equations .................. 155 7.2.5 Energy and Momentum ............................ 156 7.2.6 Interference and Diffraction ......................... 157
7.3 Wave Function and the Schrodinger Equation ............... 157 7.3.1 Free Particle ...................................... 158 7.3.2 Particle in a Potential .............................. 159
7.4 Concluding Remarks ..................................... 161 7.4.1 Classical Limit .................................... 161 7.4.2 Energy and Momentum ............................ 162 7.4.3 Optics and Analytical Mechanics .................... 163 7.4.4 The Essence of the Phase ........................... 163
7.5 Problems ............................................... 164
Solutions ...................................................... 167
References ..................................................... 179
Index .......................................................... 181
1
Introduction
Since mysteries are beyond us, let us make believe we organized them.
Jean Cocteau
Art cannot be dissociated from metaphysics and philosophy. In his Aesthetics: Lectures on Fine A rt, in order to answer the question "Why does man have the need to produce works of art?" Georg Wilhelm Friedrich Hegel says that "The general need for art is ... the rational need which drives man to become aware of the inside and outside worlds and to make an object in which he can recognize himself."
1.1 Esthetics and Physics
The same need explains why physics is deeply filled with esthetic considerations. In fact, the beauty of a theory has very often been considered as a decisive argument in its favor. Albert Einstein's general relativity is a famous example. It was formulated in 1916 but only got its true experimental verifications 70 years later.l Nevertheless, nobody seriously thought that the theory could really be disproved. 2 As Lev Davidovich Landau says ([1], Section 82), "[It] is probably the most beautiful of existing theories. It is remarkable that Einstein constructed it purely by deductive arguments and that it is only afterwards that it was confirmed by astronomical observations."
The ingredients of esthetics have many origins. Of course, the beauty of an idea in itself is difficult if not impossible to define in general. However,
lOne usually makes a distinction between the verifications of the equivalence principle, such as the deviation of light rays by a gravitational field, the variation of the pace of a clock in a gravitational potential, or the general relativistic corrections to celestial mechanics, and the true predictions of Einstein's equations, such as the radiation of gravitational waves.
2 This does not mean that one should give up finding experimental proofs.
2 1 Introduction
two factors are easier to identify. These are the simplicity of a theory and its unifying nature. Below, we will mention the archetype of such an intellectual achievement, the Pythagorean musical scale. There are numerous other examples.
After extensive work, both observational3 and calculational,4 Johannes Kepler founded his famous laws on the motion of planets in the solar system. The discovery that, from a Copernican viewpoint, the orbits are the pure and legendary ellipses of the geometry of Apollonius, Euclid and Archimedes has a beauty and a simplicity that Kepler could not resist. He naturally conceived the universe as being constructed in a mathematical esthetic that exhibits both purity and unity. He expressed his emotion in his celebrated phrase "Nature likes simplicity."5 It was both a triumph and a wonder when in the framework of his Principia Isaac Newton was able to deduce Kepler's laws.
The same thing happened in the unification of electricity and magnetism by Andre-Marie Ampere, followed by that of electromagnetism and light by James Clerk Maxwell. This amazing adventure of the 19th century lasted for a long time. The mathematical structure of Maxwell's equations revealed relativity. The unification of electro-weak interactions by Sheldon Glashow, Steven Weinberg and Abdus Salam in the 1960s was the following stage of this fascinating endeavor. It led to the perspective of unifying all fundamental interactions, including gravitation. At each step, simplicity, unity, and esthetics are dominant features.
Simplicity does not mean that things can become understandable by the layman. It is quite the contrary. The simplicity appears within the mathematical language. Galileo was the first to realize that:
Philosophy is written in the immense book which is constantly open in front of us (I mean the universe), however, one cannot understand it without first learning the language and the characters in which it is written. It is written in the mathematical language and its characters are triangles, circles and other geometrical figures in the absence of which it is not possible for a human being to understand a single word of it.
It is tempting to recall the words of Leonardo da Vinci in his Treatise on Painting: "Non mi legga chi non e matematico, nelli mia principi.,,6 Simplicity lies in the mysterious possibility of representing natural phenomena by more and more general mathematical structures. If one can say that one of the most fundamental mathematical structures of quantum mechanics lies in the simplest of the four basic operations (i.e. addition), it is a consequence
3 Galileo's telescope was invented in 1609, ten years later than Kepler's works. 4 Kepler dedicated his memoir Mysterium cosmographicum to John Napier, who
invented logarithms. Kepler said that without logarithms he never would have been able to perform his accurate and difficult calculations.
5 Natura simplicitatem amat. 6 Those who are not mathematicians should not read my principles.
1.2 Metaphysics and Science 3
of an enormous amount of effort toward simplicity and synthesis that was made by physicists and mathematicians in the mid 1920s. The superposition principle, which is the first and most important concept for anyone starting to learn quantum mechanics, is what goes completely against common sense and first physical intuition. However, if the mathematical expression of the superposition principle is very simple, that simplicity can only be appreciated after a long and unavoidable mathematical travel.
1.2 Metaphysics and Science
Philosophical thinking is frequently related to scientific progress, which is understandable. It is interesting, however, to note that truly metaphysical considerations have frequently followed the same paths as physics. When Edmund Halley studied the trajectory of the comet that now bears his name and had been known since at least 240 B.C., he showed that the orbit was an ellipse and, by applying Newton's laws of motion, he was able to predict that the comet would reappear in 1758. Celestial motion was deeply interlinked with the notion of time, that most mysterious physical concept 7 that had been preying on people's minds since they started observing the sky. With the use of Newton's laws, people had become capable of predicting the state of the sky with great accuracy. Newton was amazed by that accuracy, and he found there an argument for the existence of God. Since the system was so finely tuned, since one could predict the future state of the sky, and since, by solving equations, one could recover the state of planets at any previous moment, one had to admit that the solar system, as well as all the cosmos, had been conceived by some superior intelligence.
At the peak of a physical theory, it is not unusual to find that scientists have invoked some "higher power." This can be transformed into a theological argument, as in the case of Newton. It is frequently expressed as a question addressed to the structured "organism" of which natural phenomena appear to consist. Kepler and the planetary orbits are one example. Einstein's famous sayings such as "Subtle is the Lord, but malicious He is not" or "God does not play dice" are in the minds of everyone. However, beyond these questions or assertions, one constantly finds, together with the progress of physics, a metaphysical quest for the causes of the world and the principles of knowledge. This quest is often transformed into seeking a genuine "meta-theory." By itself, the name of "M-Theory," which was born in 1995 with the proof of the equivalence between different superstring theories and led to a spectacular revival of interest in the theory of fundamental interactions, is perhaps revealing in this respect.
7 "What really is time? If no one asks me, I know. But if someone asks the question and that I must explain it, I no longer know." Saint Augustine, Confessions XI, XIV, 17.
4 1 Introduction
1.3 Numbers, Music, and Quantum Physics
The birth of modern physics is commonly placed in the 17th century with Galileo. In fact, he laid its two founding stones: the experimental method and the formulation of the theory in a mathematical language.
However, the starting point of experimental and theoretical physics lies 2200 years before that. In fact, the Pythagorean theorem occults what is, in precisely Galileo's terms, the first modern discovery in physics-the theory of sounds and the musical scale. It is modern in the sense that this discovery possesses the two properties of having an experimental foundation and of being expressed in a mathematical manner.
Music is the first abstract art. It is fascinating because it reaches directly the subconscious. It escapes any attempt to be verbalized. Apart from technical discussions between experts, one cannot tell music. Musical writing is a permanent source of amazement, as one can see in Figure 1.1 .
b 1fJ,----tp~
1~e: .. ~ ~
.. , ... .. 3 frt ... r" •• ~ tI.. b .. o __
dur .. t .. ---l"hlllht1~
5
Fig. 1.1. Sylvano Bussotti, "Piano pieces for David Tudor # 4," excerpt from "Pieces de Chair II" (Pieces of Flesh). (Courtesy of Casa Ricordi-BMG Ricordi Milan; all rights reserved.)
It is difficult to put a date on the birth of this art, but for sure, very quickly, humans in their songs understood the existence of harmony. The octave, which is the simplest example, consists in the amazing discovery that the same sound can be reproduced at a high pitch as well as at a low pitch.
The legend is that Pythagoras discovered the explanation of the musical pitch by noticing that the pitch was directly related to the length of the object that produced the sound. He used to pass daily in front of a blacksmith's
1.3 Numbers, Music, and Quantum Physics 5
workshop in Samos, his native island.8 He observed that rods of iron of different length gave different sounds under the blacksmith's hammer. As Arthur Koestler says ([2], Chapters V and VII), "The ear-splitting crashes and bangs in the workshop which, since the Bronze Age had yielded to the Iron Age, had been regarded by ordinary mortals as a mere nuisance, were suddenly lifted out of their habitual context: the 'bangs' became 'clangs' of music. In the technical language of the communication engineer, Pythagoras had turned '''noise'' into "information."
Back home, Pythagoras proceeded to an experimental verification of his ideas on musical objects, in particular the vibrating strings of a lyre. He understood that if he divided a string by integers belonging to the tetraktys, the set of integers 1, 2, 3, 4 whose sum is the "perfect" number 10, he obtained what had for a long time been named the "harmony" -the octave, the fifth, and the fourth.
As Denis Diderot wrote in the entry Pythagorism of his encyclopedia "L'Encyclopedie" :
Music is a concert of several discordant sounds. One must not restrict one's ideas to sounds only. The purpose of harmony is more general. Harmony has its invariant rules .... The octave, the fifth and the fourth form the basis of harmonic arithmetics.
The way Pythagoras discovered the ratios of these intervals shows he was a man of great genius. . .. There are songs for each kind of passion, whether one wants to temper them, or to excite them. The flute is dull. The philosopher will take the lyre: he will play it in the morning and in the evening.
After studying all subtleties he could find on the harmonics of a sound, and the way he could reduce them to the interval of one octave by dividing them by powers of 2, Pythagoras ended up with musical scales, in particular the one that bears his name, which is shown in Table 1.1. The numbers indicate ratios of frequencies. (The Greek modes were expressed in a decreasing sequence according to the length of the string.)
Table 1.1. Frequency ratios in the musical scale of Pythagoras.
note CDEFGAB C
ratio of frequencies 1 2. §.!. :! 9. TI 243 2 8 64 3 2 16 128
8 Actually, it is unimportant whether the anecdote is true or not, or whether it is Pythagoras himself who made the discovery. In fact, the important points lie in the profoundness of the idea, in the experimental observation that it entailed, and in the resulting integer number theory that reached us.
