Variational Principles in Physics

by Jean-Louis Basdevant

Overview

Optimization under constraints is an essential part of everyday life. Indeed, we routinely solve problems by striking a balance between contradictory interests, individual desires and material contingencies. This notion of equilibrium was dear to thinkers of the enlightenment, as illustrated by Montesquieu's famous formulation: "In all magistracies, the greatness of the power must be compensated by the brevity of the duration."

Astonishingly, natural laws are guided by a similar principle. Variational principles have proven to be surprisingly fertile. For example, Fermat used variational methods to demonstrate that light follows the fastest route from one point to another, an idea which came to be known as Fermat's principle, a cornerstone of geometrical optics. Variational Principles in Physics explains variational principles and charts their use throughout modern physics. The heart of the book is devoted to the analytical mechanics of Lagrange and Hamilton, the basic tools of any physicist. Professor Basdevant also offers simple but rich first impressions of Einstein's General Relativity, Feynman's Quantum Mechanics, and more revealing and amazing interconnections between various fields of physics.

About the Author:
Jean-Louis Basdevant is Professor and former Chair of the Department of Physics at the Ecole Polytechnique, and Director of Research for the CNRS

Product Details

ISBN-13:	9783031216923
Publisher:	Springer-Verlag New York, LLC
Publication date:	02/07/2023
Sold by:	Barnes & Noble
Format:	eBook
File size:	22 MB
Note:	This product may take a few minutes to download.

About the Author

Jean-Louis Basdevant, graduated from Ecole Normale Supérieure (1958-1963), Ph.D. at Strasbourg University (1967), is Assistant Researcher and Director of Research at the CNRS (1963-2007), CERN (1970-1972), Professor at the Ecole Polytechnique (1969-2004), and Nominated Honorary Professor (2004).

Table of Contents

Preface     v
Introduction     1
Esthetics and Physics     1
Metaphysics and Science     3
Numbers, Music, and Quantum Physics     4
The Age of Enlightenment and the Principle of the Best     7
The Fermat Principle and Its Consequences     8
Variational Principles     9
The Modern Era, from Lagrange to Einstein and Feynman     12
Variational Principles     21
The Fermat Principle and Variational Calculus     22
Least Time Principle     22
Variational Calculus of Euler and Lagrange     26
Mirages and Curved Rays     27
Examples of the Principle of Natural Economy     30
Maupertuis Principle     30
Shape of a Massive String     31
Kirchhoff's Laws     32
Electrostatic Potential     33
Soap Bubbles     34
Thermodynamic Equilibrium: Principle of Maximal Disorder     35
Principle of Equal Probability of States     35
Most Probable Distribution and Equilibrium     36
Lagrange Multipliers     37
Boltzmann Factor     38
Equalization of Temperatures     39
The Ideal Gas     40
Boltzmann's Entropy     41
Heat and Work     42
Problems     43
The Analytical Mechanics of Lagrange     47
Lagrangian Formalism and the Least Action Principle     49
Least Action Principle     49
Lagrange-Euler Equations     50
Operation of the Optimization Principle     52
Invariances and Conservation Laws     53
Conjugate Momenta and Generalized Momenta     53
Cyclic Variables     54
Energy and Translations in Time     54
Momentum and Translations in Space     56
Angular Momentum and Rotations     57
Dynamical Symmetries     57
Velocity-Dependent Forces     58
Dissipative Systems     58
Lorentz Force     59
Gauge Invariance     60
Momentum     61
Lagrangian of a Relativistic Particle     61
Free Particle     61
Energy and Momentum     62
Interaction with an Electromagnetic Field     63
Problems     65
Hamilton's Canonical Formalism     67
Hamilton's Canonical Formalism     68
Canonical Equations      69
Dynamical Systems     70
Poincare and Chaos in the Solar System     71
The Butterfly Effect and the Lorenz Attractor     71
Poisson Brackets and Phase Space     73
Time Evolution and Constants of the Motion     74
Canonical Transformations     75
Phase Space; Liouville's Theorem     78
Analytical Mechanics and Quantum Mechanics     80
Charged Particle in an Electromagnetic Field     81
Hamiltonian     81
Gauge Invariance     82
The Action and the Hamilton-Jacobi Equation     82
The Action as a Function of the Coordinates and Time     83
The Hamilton-Jacobi Equation and Jacobi Theorem     85
Conservative Systems, the Reduced Action, and the Maupertuis Principle     87
Analytical Mechanics and Optics     89
Geometric Limit of Wave Optics     89
Semiclassical Approximation in Quantum Mechanics     91
Problems     92
Lagrangian Field Theory     97
Vibrating String     98
Field Equations     99
Generalized Lagrange-Euler Equations     99
Hamiltonian Formalism     100
Scalar Field     101
Electromagnetic Field     102
Equations of First Order in Time     104
Diffusion Equation     104
Schrodinger Equation     104
Problems     105
Motion in a Curved Space     107
Curved Spaces     108
Generalities     108
Metric Tensor     110
Examples     111
Free Motion in a Curved Space     112
Lagrangian     113
Equations of Motion     113
Simple Examples     114
Conjugate Momenta and the Hamiltonian     117
Geodesic Lines     117
Definition     117
Equation of the Geodesics     118
Examples     119
Maupertuis Principle and Geodesics     121
Gravitation and the Curvature of Space-Time     122
Newtonian Gravitation and Relativity     122
The Schwarzschild Metric     124
Gravitation and Time Flow     125
Precession of Mercury's Perihelion     125
Gravitational Deflection of Light Rays     130
Gravitational Optics and Mirages     133
Gravitational Lensing     133
Gravitational Mirages     134
Baryonic Dark Matter      139
Problems     144
Feynman's Principle in Quantum Mechanics     145
Feynman's Principle     146
Recollections of Analytical Mechanics     146
Quantum Amplitudes     147
Superposition Principle and Feynman's Principle     147
Path Integrals     148
Amplitude of Successive Events     150
Free Particle     152
Propagator of a Free Particle     152
Evolution Equation of the Free Propagator     154
Normalization and Interpretation of the Propagator     155
Fourier and Schrodinger Equations     155
Energy and Momentum     156
Interference and Diffraction     157
Wave Function and the Schrodinger Equation     157
Free Particle     158
Particle in a Potential     159
Concluding Remarks     161
Classical Limit     161
Energy and Momentum     162
Optics and Analytical Mechanics     163
The Essence of the Phase     163
Problems     164
Solutions     167
References     179
Index     181

Editorial Reviews

From the reviews:

"This book is based on a course given by Professor Basdevant at the Ecole Polytechnique, near Paris. … it would be an excellent basis for a course, provided one has the right collection of undergraduates. The book could also be used at a more advanced level in many ways … . For anyone with radical tendencies in this direction, Basdevant’s book will provide an elegant launching point." (L. S. Schulman, SIAM Review, Vol. 49 (4), 2007)

"It invites the reader onto the vessel ‘variational principles’ to embark on a journey through a variety of different areas and regions of physics. … In general … independent of individual preferences, Basdevant’s book provides a well-balanced overview of various old and new areas of theoretical physics based on an approach by variational principles. Not only will students profit from this book, but also more advanced readers might get captivated by it." (H. Hogreve, Mathematical Reviews, Issue 2009 f)

From the Publisher