The Principle of Least Action
History and Physics

This text brings history and the key fields of physics together to present a unique technical discussion of the principles of least action.

Alberto Rojo (Author), Anthony Bloch (Author)

9780521869027, Cambridge University Press

Hardback, published 29 March 2018

266 pages, 96 b/w illus. 1 table
25.4 x 17.9 x 1.6 cm, 0.67 kg

'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 Schr?dinger'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

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.

1. Introduction
2. Prehistory of variational principles
3. An excursio 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 MacCullagh's ether
Appendix D. Characteristic function for a parabolic Keplerian orbit
Appendix E. Saddle paths for reections 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.

Subject Areas: Classical mechanics [PHD], History of science [PDX], Philosophy of science [PDA], History of mathematics [PBX]