The Principle of Least Action

History and Physics

Alberto Rojo, Anthony Bloch

• Publisher: Cambridge University Press
• Year: 2018
• Print ISBN: 9780521869027
• eBook ISBN: 9781108299862
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

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.

Table of contents

• Cover
• Half-title page
• Title page
• Copyright page
• Contents
• List of illustrations
• Acknowledgments
• 1 Introduction
• 2 Prehistory of Variational Principles
• 3 An Excursion to Newton’s Principia
• 4 The Optical-Mechanical Analogy, Part I
• 5 D’Alembert, Lagrange, and the Statics-Dynamics Analogy
• 6 The Optical-Mechanical Analogy, Part II: The Hamilton-Jacobi Equation
• 7 Relativity and Least Action
• 8 The Road to Quantum Mechanics
• Appendix A Newton’s Solid of Least Resistance, Using Calculus
• Appendix B Original Statement of d’Alembert’s Principle
• Appendix C Equations of Motion of McCullagh’s Ether
• Appendix D Characteristic Function for a Parabolic Keplerian Orbit
• Appendix E Saddle Paths for Reflections on a Mirror
• Appendix F Kinetic Caustics from Quantum Motion in One Dimension
• Appendix G Einstein’s Proof of the Covariance of Maxwell’s Equations
• Appendix H Relativistic Four-Vector Potential
• Appendix I Ehrenfest’s Proof of the Adiabatic Theorem
• References
• Index