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A call to action
1.Richard P. Feynman, The Feynman Lectures on Physics (Addison–Wesley, Reading, MA, 1964), Vol. II, Chap. 19. Feynman writes that “When I was in high school, my physics teacher—whose name was Mr. Bader—called me down one day after physics class and said, ‘You look bored; I want to tell you something interesting.’ Then he told me something that I found absolutely fascinating, and have, since then, always found fascinating… the principle of least action.”
2.We use the phrase Newtonian mechanics to mean nonrelativistic, nonquantum mechanics. Some texts use the term classical mechanics to include both Newtonian and relativistic mechanics.
3.An editorial is not the setting in which to discuss technical details of these statements. See the mechanics texts in Ref. 6 and Herbert Goldstein, Charles Poole, and John Safko, Classical Mechanics (Addison–Wesley, Reading, MA 2002), 3rd ed. For further background, see Ref. 5.
4.The action for an actual path may be a saddle point rather than a minimum. So the most general, but rather esoteric, term for our basic law is the principle of stationary action.
5.Sample software and student materials for both general relativity and quantum mechanics appropriate for the second year of study are available at 〈www.eftaylor.com/〉.
6.L. D. Landau and E. M. Lifshitz, Mechanics, Course of Theoretical Physics (Butterworth-Heinenann, London, 1976), 3rd ed., Vol. 1, Chap. 1. Gerald Jay Sussman and Jack Wisdom, Structure and Interpretation of Classical Mechanics (MIT, Cambridge, 2001), Chap. 1.
7.Jozef Hanc, Edwin F. Taylor, and Slavomir Tuleja, “Deriving Lagrange’s equations using elementary calculus,” Am. J. Phys. (submitted). See also Cornelius Lanczos, The Variational Principles of Mechanics (Dover, New York, 1986), 4th ed., pp. 49–54.
8.Max Jammer, Concepts of Force (Dover, New York, 1999), preface.
9.Roger A. Hinrichs and Merlin Kleinbach, Energy: Its Use and the Environment (Harcourt Brace, New York, 2002), 3rd ed.;
9.Gordon Aubrecht, Energy (Prentice–Hall College Division, Englewood Cliffs, NJ, 1994), 2nd ed.;
9.Thomas A. Moore begins the study of mechanics with conservation laws in Six Ideas That Shaped Physics, Unit C: Conservation Laws Constrain Interactions (McGraw–Hill, New York, 2003), 2nd ed.
10.Thomas A. Moore in the entry on “least-action principle” in Macmillan Encyclopedia of Physics (Simon & Schuster Macmillan, New York, 1996), Vol. 2, pp. 840–842.
11.Edwin F. Taylor and John Archibald Wheeler, Exploring Black Holes: Introduction to General Relativity (Addison–Wesley Longman, Reading, MA, 2000).
12.A brief derivation is available at Ref. 5.
13.Freeman Dyson in Some Strangeness in the Proportion, edited by Harry Woolf (Addison–Wesley, Reading, MA, 1980), p. 376, writes that “Thirty-one years ago (1949), Dick Feynman told me about his ‘sum over histories’ version of quantum mechanics. ‘The electron does anything it likes,’ he said. ‘It just goes in any direction at any speed, forward or backward in time, however it likes, and then you add up the amplitudes and it gives you the wave-function.’ I said to him, ‘You’re crazy.’ But he wasn’t.”
14.Some background references are available at Ref. 5.
15.R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw–Hill, New York, 1965), p. 29.
16.Jorge Dias de Deus, Mário Pimenta, Ana Noronha, Teresa Peña, and Pedro Brogueira, Introducción al la Fı́sica (Spanish) (McGraw–Hill, New York, 2001), ISBN 84-481-3190-8 and Introdução à Fı́sica (Portuguese) (McGraw–Hill, New York, 2000), 2nd ed., ISBN 972-773-035-3.
17.David Morin introduces Lagrange’s equations in Chap. 5 of his honors introductory physics text. A draft is available at 〈http://www.courses.fas.harvard.edu/∼phys16/handouts/textbook/ch5.pdf〉.
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