Variational Principles in Physics

190 Pages·2007·5.66 MB·English

by  Professor Jean-Louis Basdevant (auth.)

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 0-387-37748-4 (eBook)
ISBN 978-0-387-37747-6 ISBN 978-0-387-37748-3 (eBook)
Printed on acid-free paper.
© 2007 Springer Science+ Business Media, LLC
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street,
New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly
analysis. Use in connection with any form of information storage and retrieval, electronic
adaptation, computer software, or by similar or dissimilar methodology now known or hereafter
developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if they
are not identified as such, is not to be taken as an expression of opinion as to whether or not
they are subject to proprietary rights.
9 8 7 6 543 2 1
springer.com
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 princi
ples 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 fur
ther 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 Poly
technique, for three years starting in 2001. I was struck by the interest that
students found in this aspect of physics. They discovered a cultural compo
nent 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 devel
opments 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 lean-Louis Basdevant
January 2006
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 consider
ations. 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 verifica
tions 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 prin
ciple, 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 correc
tions 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.
J.-L. Basdevant, Variational Principles in Physics
© Springer Science+Business Media, LLC 2007