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Principle of Least Action: History and Physics [Kõva köide]

3.00/5 (3 hinnangut Goodreads-ist)
(University of Michigan, Ann Arbor), (Oakland University, Michigan)
  • Formaat: Hardback, 266 pages, kõrgus x laius x paksus: 254x179x16 mm, kaal: 670 g, 1 Tables, black and white; 2 Halftones, black and white; 94 Line drawings, black and white
  • Ilmumisaeg: 29-Mar-2018
  • Kirjastus: Cambridge University Press
  • ISBN-10: 0521869021
  • ISBN-13: 9780521869027
Teised raamatud teemal:
Principle of Least Action: History and PhysicsSuurem pilt
  • Formaat: Hardback, 266 pages, kõrgus x laius x paksus: 254x179x16 mm, kaal: 670 g, 1 Tables, black and white; 2 Halftones, black and white; 94 Line drawings, black and white
  • Ilmumisaeg: 29-Mar-2018
  • Kirjastus: Cambridge University Press
  • ISBN-10: 0521869021
  • ISBN-13: 9780521869027
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780521869027.html
Märksõnad:
The principle of least action originates in the idea that, if nature has a purpose, it should follow a minimum or critical path. This simple principle, and its variants and generalizations, applies to optics, mechanics, electromagnetism, relativity, and quantum mechanics, and provides an essential guide to understanding the beauty of physics. This unique text provides an accessible introduction to the action principle across these various fields of physics, and examines its history and fundamental role in science. It includes - with varying levels of mathematical sophistication - explanations from historical sources, discussion of classic papers, and original worked examples. The result is a story that is understandable to those with a modest mathematical background, as well as to researchers and students in physics and the history of physics.

Arvustused

'I recommend this exciting book to readers interested in how a small number of principles can explain modern physical theory. A major part of this excellent physical, mathematical, and historical study of the principle of least action is the exposition of the optical-mechanical analogy that led to the invention of Schrdinger's wave mechanics in 1926  Containing the mathematical steps of numerous detailed derivations with clearly designed figures to aid the reader, the text is physically and mathematically rigorous and serves as a guide to the next step: the reading of the original papers cited in the references.' Barry R. Masters, Optics & Photonics News

Muu info

This text brings history and the key fields of physics together to present a unique technical discussion of the principles of least action.
List of Illustrations
ix
Acknowledgments xii
1 Introduction
1(5)
2 Prehistory of Variational Principles
6(32)
2.1 Queen Dido and the Isoperimetric Problem
6(7)
2.1.1 Zenodorus's Solution*
7(6)
2.2 Hero of Alexandria and the Law of Reflection
13(1)
2.3 Galileo and the Curve of Swiftest Descent
14(4)
2.4 Bending of Light Rays and Fermat's Minimum Principle
18(9)
2.4.1 Fermat's Method of Maxima and Minima
21(3)
2.4.2 Huygens' Simplified Derivation of Snell's Law
24(3)
2.5 Newton and the Solid of Least Resistance*
27(11)
2.5.1 The Sphere and the Cylinder
28(2)
2.5.2 An Application "in the Building of Ships"
30(4)
2.5.3 The First Genuine Variational Calculation
34(4)
3 An Excursion to Newton's Principia
38(13)
3.1 Newton's Propositions on the Laws of Motion
38(1)
3.2 Geometrical Derivation of Kepler's Laws of Planetary Motion
39(12)
3.2.1 Proposition 1: Equal Areas are Swept Out in Equal Times
39(2)
3.2.2 Proposition 6: The Force Law and the Geometry of the Orbit*
41(2)
3.2.3 Circular Orbits
43(2)
3.2.4 Proposition 10: Elliptical Orbit with the Center of Force at the Center of the Ellipse
45(3)
3.2.5 Proposition 11: Center of Force at the Focus of the Ellipse*
48(3)
4 The Optical-Mechanical Analogy, Part I
51(28)
4.1 Bernoulli's Challenge and the Brachistochrone
51(8)
4.1.1 Huygens and the Horologium Oscillatorium
52(3)
4.1.2 Leibniz's Solution of the Brachistochrone
55(2)
4.1.3 Bernoulli's Solution: Particle Paths as Light Rays
57(2)
4.2 Maupertuis, Least Action, and Metaphysical Mechanics
59(3)
4.3 Euler and the Method of Maxima and Minima*
62(6)
4.3.1 Euler's Derivation of Orbits from the Least Action Principle
65(3)
4.4 Examples of the Optical-Mechanical Analogy
68(4)
4.4.1 Conservation of "Angular Momentum" for Light Rays
69(2)
4.4.2 The Terrestrial Brachistochrone
71(1)
4.5 The String Analogy and the Principle of Least Action
72(7)
4.5.1 The Least Action Principle and Stretchable Strings
75(4)
5 D'Alembert, Lagrange, and the Statics-Dynamics Analogy
79(33)
5.1 The Principle of Virtual Work
79(3)
5.2 Statics Meets Dynamics: Bernoulli's Calculation of the Center of Oscillation
82(3)
5.3 D'Alembert's Principle
85(5)
5.4 Lagrange's Dynamics
90(10)
5.4.1 Lagrange's "Scientific Poem"
93(3)
5.4.2 Symmetries
96(4)
5.5 Lagrange versus d'Alembert: Dissipative and Nonholonomic Systems
100(5)
5.5.1 Dissipation in a Reversible System: Lamb's Model
101(2)
5.5.2 Nonholonomic Systems
103(2)
5.6 Gauss's Principle of Least Constraint
105(2)
5.7 Least Action with a Twist: the Elasticity of the Ether and Maxwell's Equations*
107(5)
6 The Optical-Mechanical Analogy, Part II: The Hamilton-Jacobi Equation
112(50)
6.1 Hamilton's "Theory of Systems of Rays"
112(7)
6.2 Conical Refraction*
119(9)
6.2.1 Fresnel's Equations for Anisotropic Crystals
119(3)
6.2.2 Analytical Derivation of the Wave Surface
122(2)
6.2.3 Hamilton's Derivation of the Conical Cusp
124(1)
6.2.4 Internal Conical Refraction: "The Plum Laid Down on a Table"
125(3)
6.3 Hamilton's Law of Varying Action*
128(3)
6.4 An Example from Hamilton: The Characteristic Function V for a Parabolic Orbit*
131(3)
6.5 Hamilton's "Second Essay on a General Method in Dynamics"*
134(8)
6.5.1 Example: Particle in a Uniform Gravitational Field
140(2)
6.6 Hamilton-Jacobi and Huygens' Principle*
142(1)
6.7 Applications and Examples
143(6)
6.7.1 The Equation of a Light Ray
143(2)
6.7.2 Hamiltonian of the Harmonic Oscillator
145(1)
6.7.3 Hamilton-Jacobi Equation for a Particle in a Magnetic Field
146(3)
6.8 When the Principle of Least Action Loses its "Least"
149(13)
6.8.1 Focus and Kinetic Focus
149(2)
6.8.2 Kinetic Focus for a Free Particle on a Sphere
151(1)
6.8.3 Saddle Paths for the Harmonic Oscillator
152(2)
6.8.4 Kinetic Focus of Elliptic Planetary Orbits
154(1)
6.8.5 Gouy's Phase and Critical Action
155(3)
6.8.6 Caustics
158(4)
7 Relativity and Least Action
162(27)
7.1 Simultaneity and the Relativity of Time
163(3)
7.2 The Relativistic "F = ma"
166(6)
7.2.1 The Energy-Momentum Four-Vector
169(2)
7.2.2 Invariance of the Relativistic Action
171(1)
7.3 Hamilton-Jacobi Equation for a Relativistic Particle
172(1)
7.4 The Principle of Equivalence
173(7)
7.4.1 Bending of Light Rays According to the Equivalence Principle
176(2)
7.4.2 Bending of Light Rays, Newtonian Calculation
178(2)
7.5 Space-Time is Curved
180(2)
7.6 Weak Gravity around a Static, Spherical Star
182(4)
7.6.1 Precession of the Perihelion of Mercury
182(3)
7.6.2 Bending of the Light Rays in the General Theory
185(1)
7.7 Hilbert's Least Action Principle for General Relativity*
186(3)
8 The Road to Quantum Mechanics
189(32)
8.1 The Need for a New Mechanics
189(3)
8.2 Bohr's `Trilogy" of 1913 and Sommerfeld's Generalization
192(7)
8.2.1 Sommerfeld and the Kepler Problem
195(3)
8.2.2 The Fine Structure of the Hydrogen Spectrum
198(1)
8.3 Adiabatic Invariants
199(4)
8.4 De Broglie's Matter Waves
203(2)
8.5 Schrodinger's Wave Mechanics
205(4)
8.5.1 The Eikonal Equation
206(1)
8.5.2 Schrodinger's Derivation
207(2)
8.6 Dirac's Lagrangian View of Quantum Mechanics
209(2)
8.7 Feynman's Thesis and Path Integrals
211(3)
8.8 Huygens' Principle in Optics and Quantum Mechanics*
214(7)
8.8.1 First-Order (in Time) Propagator for the Wave Equation
215(2)
8.8.2 Huygens' Principle and Spherical Wavelets
217(1)
8.8.3 Cancellation of the Backwards Wave
218(3)
Appendix A Newton's Solid of Least Resistance, Using Calculus 221(2)
Appendix B Original Statement of d'Alembert's Principle 223(1)
Appendix C Equations of Motion of McCullagh's Ether 224(1)
Appendix D Characteristic Function for a Parabolic Keplerian Orbit 225(2)
Appendix E Saddle Paths for Reflections on a Mirror 227(2)
Appendix F Kinetic Caustics from Quantum Motion in One Dimension 229(4)
Appendix G Einstein's Proof of the Covariance of Maxwell's Equations 233(2)
Appendix H Relativistic Four-Vector Potential 235(3)
Appendix I Ehrenfest's Proof of the Adiabatic Theorem 238(3)
References 241(13)
Index 254
Alberto Rojo is Associate Professor at Oakland University, Michigan. He is a Fulbright specialist in Physics Education and was awarded the Jack Williams Endowed Chair in Science and Humanities from the University of Eastern New Mexico. His research focuses primarily on theoretical condensed matter and he has previously published books in the popular science field. Anthony Bloch is the Alexander Ziwet Collegiate Professor of Mathematics at the University of Michigan, Ann Arbor. He has received various awards including a Presidential Young Investigator Award, a Guggenheim Fellowship, a Simons Fellowship, and he is Fellow of the Institute of Electrical and Electronics Engineers (IEEE), the Society for Industrial and Applied Mathematics (SIAM), and the American Mathematical Society (AMS). He has served on the editorial boards of various journals and is currently Editor-in-Chief of the SIAM Journal of Control and Optimization.