From conservation of energy to the principle of least action: A story line

We use the phrase Newtonian mechanics to mean nonrelativistic, nonquantum mechanics. The alternative phrase classical mechanics also applies to relativity, both special and general, and is thus inappropriate in the context of this paper.

Sir Isaac Newton’s Mathematical Principles of Natural Philosophy and His System of the World, translated by Andrew Motte and Florian Cajori (University of California, Berkeley, 1946), p. 13.

Herman H. Goldstine, A History of the Calculus of Variations From the 17th Through the 19th Century (Springer-Verlag, New York, 1980), p. 110.

J. L. Lagrange, Analytic Mechanics, translated by Victor N. Vagliente and Auguste Boissonnade (Kluwer Academic, Dordrecht, 2001).

William Rowan

Hamilton

, “On a general method in dynamics, by which the study of the motions of all free systems of attracting or repelling points is reduced to the search and differentiation of one central Relation or characteristic Function,” Philos. Trans. R. Soci. Part II, 247–308 (1834);

“Second essay on a general method in dynamics,” Part I, 95–144 (1835). Both papers are available at 〈http://www.emis.de/classics/Hamilton/〉.

L. D. Landau and E. M. Lifshitz, Mechanics, Course of Theoretical Physics (Butterworth-Heinemann, London, 1976), 3rd ed., Vol. 1, Chap. 1. Their renaming first occurred in the original 1957 Russian edition.

Richard P. Feynman, Robert B. Leighton, and Matthew Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading MA, 1964), Vol. II, pp. 19-8.

The technically correct term is principle of stationary action. See I. M. Gelfand and S. V. Fomin, Calculus of Variations, translated by Richard A. Silverman (Dover, New York, 2000), Chap. 7, Sec. 36.2. However, we prefer the conventional term principle of least action for reasons not central to the argument of the present paper.

Thomas H. Kuhn, “Energy conservation as an example of simultaneous discovery,” in The Conservation of Energy and the Principle of Least Action, edited by I. Bernard Cohen (Arno, New York, 1981).

E. Noether, “Invariante Variationprobleme,” Nach. v.d. Ges. d. Wiss zu Goettingen, Mathphys. Klasse, 235–257 (1918);

English translation by

M. A.

Tavel

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Richard P.

Feynman

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Edwin F.

Taylor

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Jozef

Hanc

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Slavomir

Tuleja

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Martina

Hancova

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Jozef Hanc, Edwin F. Taylor, and Slavomir Tuleja, “Deriving Lagrange’s equations using elementary calculus,” accepted for publication in Am. J. Phys. Preprint available at 〈http://www.eftaylor.com〉.

Jozef Hanc, Slavomir Tuleja, and Martina Hancova, “Symmetries and conservations laws: Consequences of Noether’s theorem,” accepted for publication in Am. J. Phys. Preprint available at 〈http://www.eftaylor.com〉.

Jozef Hanc, “The original Euler’s calculus-of-variations method: Key to Lagrangian mechanics for beginners,” submitted to Am. J. Phys. Preprint available at 〈http://www.eftaylor.com〉.

Thomas A. Moore, “Getting the most action out of least action,” submitted to Am. J. Phys. Preprint available at 〈http://www.eftaylor.com〉.

Student exercise: Show that $B cos ωt$ and $C cos ωt+D sin ωt$ are also solutions. Use the occasion to discuss the importance of relative phase.

Benjamin Crowell bases an entire introductory treatment on Noether’s theorem: Discover Physics (Light and Matter, Fullerton, CA, 1998–2002). Available at 〈http://www.lightandmatter.com〉.

Noether’s theorem10 implies that when the Lagrangian of a system $(L=K−U$ for our simple cases) is not a function of an independent coordinate, x for example, then the function $∂L/∂ẋ$ is a constant of the motion. This statement also can be expressed in terms of Hamiltonian dynamics. See, for example, Cornelius Lanczos, The Variational Principles of Mechanics (Dover, New York, 1970), 4th ed., Chap. VI, Sec. 9, statement above Eq. (69.1). In all the mechanical systems that we consider here, the total energy is conserved and thus automatically does not contain time explicitly. Also the expression for the kinetic energy K is quadratic in the velocities, and the potential energy U is independent of velocities. These conditions are sufficient for the Hamiltonian of the system to be the total energy. Then our limited version of Noether’s theorem is identical to the statement in Lanczos.

Reference 7, Vol. 1, p. 4-5.

Don S. Lemons, Perfect Form (Princeton U.P., Princeton, 1997), Chap. 4.

Slavomir Tuleja and Edwin F. Taylor, Principle of Least Action Interactive, available at 〈http://www.eftaylor.com〉.

Herbert Goldstein, Charles Poole, and John Safko, Classical Mechanics (Addison-Wesley, San Francisco, 2002), 3rd ed., Sec. 2.7.

Tomas

Tyc

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An extended version of this derivation is available at 〈http://www.eftaylor.com〉.

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw–Hill, New York, 1965), p. 29.

Reference 6, Chap. 1; Ref. 24, Sec. 2.3; D. Gerald Jay Sussman and Jack Wisdom, Structure and Interpretation of Classical Mechanics (MIT, Cambridge, 2001), Chap. 1.

Ya. B. Zeldovich and A. D. Myskis, Elements of Applied Mathematics, translated by George Yankovsky (MIR, Moscow, 1976), pp. 499–500.