Variational Principles in Physics

Transcription

1 Variational Principles in Physics

2 lean-louis Basdevant Variational Principles in Physics ~ Springer

3 Professor Jean-Louis Basdevant Physics Department Ecole Poly technique Palaiseau France jean-louis. Library of Congress Control Number: ISBN ISBN ISBN (ebook) ISBN (ebook) Printed on acid-free paper 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 springer.com

4 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

5 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 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, F r a n Jacquet, ~ o i s Christoph Kopper, David Langlois and J e a n - F rroussel a n ~ o for i s 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

6 Contents Preface... v 1 Introduction Esthetics and Physics Metaphysics and Science Numbers, Music, and Quantum Physics... " The Age of Enlightenment and the Principle of the Best The Fermat Principle and Its Consequences Variational Principles The Modern Era, from Lagrange to Einstein and Feynman Variational Principles... " The Fermat Principle and Variational Calculus Least Time Principle Variational Calculus of Euler and Lagrange Mirages and Curved Rays Examples of the Principle of Natural Economy Maupertuis Principle Shape of a Massive String Kirchhoff's Laws Electrostatic Potential Soap Bubbles... " Thermodynamic Equilibrium: Principle of Maximal Disorder Principle of Equal Probability of States Most Probable Distribution and Equilibrium... " Lagrange Multipliers Boltzmann Factor Equalization of Temperatures The Ideal Gas Boltzmann's Entropy... " Heat and Work... 42

7 viii Contents 2.4 Problems The Analytical Mechanics of Lagrange Lagrangian Formalism and the Least Action Principle Least Action Principle..., Lagrange-Euler Equations Operation of the Optimization Principle Invariances and Conservation Laws Conjugate Momenta and Generalized Momenta Cyclic Variables Energy and Translations in Time Momentum and Translations in Space Angular Momentum and Rotations Dynamical Symmetries..., Velocity-Dependent Forces..., Dissipative Systems Lorentz Force Gauge Invariance Momentum Lagrangian of a Relativistic Particle Free Particle Energy and Momentum Interaction with an Electromagnetic Field Problems Hamilton's Canonical Formalism Hamilton's Canonical Formalism Canonical Equations Dynamical Systems Poincare and Chaos in the Solar System The Butterfly Effect and the Lorenz Attractor Poisson Brackets and Phase Space Time Evolution and Constants of the Motion Canonical Transformations Phase Space; Liouville's Theorem Analytical Mechanics and Quantum Mechanics Charged Particle in an Electromagnetic Field Hamiltonian Gauge Invariance The Action and the Hamilton-Jacobi Equation The Action as a Function of the Coordinates and Time The Hamilton-Jacobi Equation and Jacobi Theorem Conservative Systems, the Reduced Action, and the Maupertuis Principle Analytical Mechanics and Optics

8 Contents ix Geometric Limit of Wave Optics Semiclassical Approximation in Quantum Mechanics Problems Lagrangian Field Theory Vibrating String Field Equations Generalized Lagrange-Euler Equations Hamiltonian Formalism Scalar Field Electromagnetic Field Equations of First Order in Time Diffusion Equation Schrodinger Equation Problems Motion in a Curved Space Curved Spaces Generalities Metric Tensor Examples Free Motion in a Curved Space Lagrangian Equations of Motion Simple Examples Conjugate Momenta and the Hamiltonian Geodesic Lines Definition Equation of the Geodesics Examples..., Maupertuis Principle and Geodesics Gravitation and the Curvature of Space-Time Newtonian Gravitation and Relativity The Schwarzschild Metric Gravitation and Time Flow Precession of Mercury's Perihelion Gravitational Deflection of Light Rays Gravitational Optics and Mirages Gravitational Lensing Gravitational Mirages Baryonic Dark Matter Problems

9 x Contents 7 Feynman's Principle in Quantum Mechanics Feynman's Principle Recollections of Analytical Mechanics Quantum Amplitudes Superposition Principle and Feynman's Principle Path Integrals Amplitude of Successive Events Free Particle Propagator of a Free Particle Evolution Equation of the Free Propagator Normalization and Interpretation of the Propagator Fourier and Schrodinger Equations Energy and Momentum Interference and Diffraction Wave Function and the Schrodinger Equation Free Particle Particle in a Potential Concluding Remarks Classical Limit Energy and Momentum Optics and Analytical Mechanics The Essence of the Phase Problems Solutions References Index