Variational Principles in Physics

By: Jean-Louis Basdevant

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Hardcover | 24 January 2007

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Hardcover
196 Pages

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23.39 x 15.6 x 1.27

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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. Prof. 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.

Industry 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)

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

Variational Principles in Physics

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