American Journal of Physics, Vol. 71, No. 5, pp. 423–425, May 2003
©2003 American Association of Physics Teachers. All rights reserved.
A call to^{ }action
Edwin F. Taylor
Department of Physics Massachusetts Institute of Technology ^{ }Cambridge, Massachusetts 02139
Contents
Would you like to begin the^{ }study of Newtonian mechanics using no vectors and no F = ma?^{ }How about starting quantum mechanics with no complex numbers and^{ }no Schrödinger equation? Would you and your students enjoy exploring^{ }general relativity with no tensors and no field equations? ^{ }
Suppose,^{ }further, that along the way your students learn concepts and^{ }methods central to contemporary physics research. Finally, what if particle^{ }motions described by Newtonian mechanics, general relativity, and quantum mechanics^{ }were summarized in three brief but powerful commands of nature^{ }that turn out to be variations of the same command?^{ }^{ }
Physics is already being taught this way, and students respond^{ }with enthusiasm and understanding. Moreover, the theoretical background for this^{ }curriculum has been around for a long time and is^{ }well developed and deeply understood by the physics and mathematics^{ }communities. ^{ }
Here are nature's commands to the stone and electron:^{ }At the stone moving with nonrelativistic speed in a region^{ }of small space–time curvature, nature shouts: Follow the path of^{ }least action! ^{ }
At the stone moving with any possible speed^{ }in a region of any finite space–time curvature, nature shouts:^{ }Follow the path of maximal aging! ^{ }
At the electron, nature^{ }shouts: Explore all paths! That's it. We now examine each^{ }of these commands in turn. ^{ }
The principle of least action^{ }in Newtonian mechanics^{1}^{,}^{2} defines the action S for a particle^{ }under the influence of a potential: S = L dt = (T–V)dt, where T is^{ }the kinetic energy, V is the potential energy, and L^{ }is the Lagrangian (L = T–V in simple cases) along each segment^{ }of the path.^{3} The quantities T, V, L, and S^{ }are scalars. The principle of least action demands that the^{ }action S be a minimum for the path taken by^{ }the particle.^{4} For onedimensional motion in a conservative potential, students^{ }can use interactive graphics software to seek out by trial^{ }and error the path that minimizes the action.^{5} ^{ }
The principle^{ }of least action appears in Chapter 1 of some advanced^{ }texts in classical mechanics.^{6} By applying the calculus of variations^{ }to the action, these texts derive Lagrange's equations, the equations^{ }that generalize F = ma. It is not necessary to use the^{ }calculus of variations with introductory students; the action is additive,^{ }so the principle of least action must apply to each^{ }small segment along the path. Therefore elementary calculus suffices to^{ }derive Lagrange's equations:^{7}
where q is any independent coordinate, such^{ }as the position x or angle , and is^{ }the time derivative of q. Lagrange's scalar equations allow us^{ }to analyze with equal ease the motion of a particle^{ }or a system subject to constraints, such as a compound^{ }pendulum. In contrast, F = ma requires us to determine every single^{ }(vector!) force of constraint before we can describe the motion^{ }of even the simplest compound system. As stressed by Max^{ }Jammer, " modern particle physics, just like general relativity, seems^{ }to support the thesis that the concept of force has^{ }reached the end of its lifecycle—its disbarment from the inventory^{ }of fundamental concepts in physics ." ^{8} ^{ }
Two criticisms can be leveled^{ }against the principle of least action as an introduction to^{ }Newtonian mechanics: the concept of action is less concrete and^{ }farther from common experience than the concept of force, and^{ }the principle of least action and Lagrange's equations deal awkwardly^{ }with dissipative and frictional forces. In response to the first^{ }criticism, I believe that the comparison should not be between^{ }force and action, but rather between vector force and scalar^{ }energy. One way to make the concept of action concrete^{ }is to introduce it through the conservation of energy. Some^{ }introductory texts already begin mechanics with the conservation of energy.^{9}^{ }An analysis of different forms of energy widens the initial^{ }scope of the subject and allows discussion of current environmental^{ }issues. Early introduction of the potential energy diagram (central to^{ }the later study of advanced mechanics, quantum mechanics, and general^{ }relativity) permits the qualitative analysis of motion before the use^{ }of mathematical formalism. ^{ }
Then we discover that, given the initial^{ }conditions, conservation of energy alone is sufficient to describe completely^{ }the onedimensional motion of a particle. However, conservation of energy^{ }typically fails to specify the motion of a particle in^{ }more than one dimension. Adding conservation of angular momentum for^{ }twodimensional motion in a central potential puts off the inevitable.^{ }But finally we must move beyond the powerful conservation of^{ }energy to the still more powerful principle of least action.^{ }^{ }
The comment that the principle of least action deals awkwardly^{ }with dissipative and frictional forces is certainly valid. Under our^{ }proposal, force is not eliminated but becomes a secondary concept,^{ }available to analyze friction and dissipative forces. Making force secondary^{ }clears center stage for the principle of least action to^{ }predict motions of a huge variety of simple and complex^{ }systems. ^{ }
As pointed out by Thomas A. Moore, "Not only^{ }does the leastaction principle offer a means of formulating classical^{ }mechanics that is more flexible and powerful than Newtonian mechanics,^{ }variations on the leastaction principle have proved useful in general^{ }relativity theory, quantum field theory, and particle physics. As a^{ }result, this principle lies at the core of much of^{ }contemporary physics." ^{10} ^{ }
The twin paradox helps us to predict how^{ }a high speed free particle moves in flat space–time. The^{ }traveling twin streaks away to a distant star, returning to^{ }greet her stayathome sister, who has relaxed in an inertial^{ }frame between the events of departure and return. The two^{ }alternative journeys are most clearly described using a world line,^{ }a position versus time plot. When the twins meet at^{ }the end of the trip, the reading on the wristwatch^{ }of the stayathome twin (who follows the natural straight world^{ }line) is greater than the reading on the wristwatch of^{ }the roundtrip twin. The aging between the events of departure^{ }and arrival is greater along the straight world line than^{ }along every nearby alternative world line between these two events.^{ }This example illustrates nature's command to the stone: Follow the^{ }world line of maximal aging! ^{ }
The stone obeys the principle^{ }of maximal aging even in curved space–time (because a local^{ }inertial frame can be defined on every segment of the^{ }freeparticle world line). General relativity calls the resulting world line^{ }a geodesic. In the vicinity of a timeindependent structure such^{ }as a nonspinning or spinning star, neutron star, or black^{ }hole, a timeindependent metric relates aging along a segment of^{ }world line to particle displacement in space and time coordinates^{ }along that segment. By employing the metric and the principle^{ }of maximal aging, students can explore big chunks of general^{ }relativity without tensors or the field equations.^{11} ^{ }
General relativity is^{ }general, so the stone follows the maximalaging geodesic in everyday^{ }motions near the Earth. For low speeds and weak gravitational^{ }effects, however, the principle of maximal aging simplifies to the^{ }nonrelativistic principle of least action.^{12} Newtonian mechanics thus becomes a^{ }limiting case of relativistic mechanics (see Fig. 1). No pedagogical^{ }reason keeps students from a first immersion in general relativity^{ }in their second year of physics instruction. ^{ }
Figure 1. Forms of the^{ }least action principle are useful not only in general relativity,^{ }but also in other contemporary fields of physics.^{10} Why not^{ }prepare our students to deploy this prodigious arsenal? ^{ }
The least^{ }action command for the stone follows seamlessly (in the limiting^{ }case of large mass) from a conceptually simple approach to^{ }quantum mechanics devised by Feynman more than half a century^{ }ago.^{13} Nature's threeword command to the electron is Explore all^{ }paths! Here again path is understood to mean world line.^{ }Feynman's approach is well documented.^{14} In brief, Feynman tells us^{ }how to evaluate the contribution of each alternative world line^{ }to the probability that a particle emitted at event A^{ }will be detected at a later event B. The key^{ }point is that as the mass of a particle increases,^{ }the set of world lines between A and B that^{ }contribute significantly to the probability of detection at B shrinks^{ }to a narrow pencil around the world line of least^{ }action.^{15} In the limit of large mass this pencil contracts^{ }seamlessly to the single world line predicted by the principle^{ }of least action. Newtonian mechanics becomes an obvious special case^{ }of quantum mechanics (see Fig. 1). Using the Feynman approach,^{ }students can wield the power of quantum mechanics earlier in^{ }their careers than is possible when they start with the^{ }Schrödinger equation. ^{ }
How far along are we in utilizing the^{ }principle of least action and its mighty siblings in the^{ }first and second year undergraduate physics curriculum? At least one^{ }introductory physics text, already in its second edition, begins Newtonian^{ }mechanics with the principle of least action.^{16} Another text, for^{ }honors students, introduces Lagrange's equations in an early chapter.^{17} These^{ }treatments provide existence proofs to guide us as we develop^{ }our own story line for the first year of undergraduate^{ }physics. At least one sophomorelevel text explores general relativity using^{ }a variety of metrics together with the principle of maximal^{ }aging.^{11} The Feynman manypaths approach to quantum mechanics exists in^{ }scattered resources, some of them already used in classes.^{5} ^{ }
The^{ }principle of least action is simple, potent, and fundamental. It^{ }spans classical and contemporary physics and introduces deep concepts central^{ }to current research. Now is the time for action! ^{ }
REFERENCES
Citation links [e.g., Phys. Rev. D 40, 2172 (1989)] go to online journal abstracts. Other links (see Reference Information) are available with your current login. Navigation of links may be more efficient using a second browser window.

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."
first citation in article

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. first citation in article

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.
first citation in article

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. first citation in article

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/.
first citation in article

L. D. Landau and E. M. Lifshitz, Mechanics, Course of Theoretical Physics (ButterworthHeinenann, 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.
first citation in article

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.
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Max Jammer, Concepts of Force (Dover, New York, 1999), preface.
first citation in article

Roger A. Hinrichs and Merlin Kleinbach, Energy: Its Use and the Environment (Harcourt Brace, New York, 2002), 3rd ed.;
Gordon Aubrecht, Energy (Prentice–Hall College Division, Englewood Cliffs, NJ, 1994), 2nd ed.;
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.
first citation in article

Thomas A. Moore in the entry on "leastaction principle" in Macmillan Encyclopedia of Physics (Simon & Schuster Macmillan, New York, 1996), Vol. 2, pp. 840–842.
first citation in article

Edwin F. Taylor and John Archibald Wheeler, Exploring Black Holes: Introduction to General Relativity (Addison–Wesley Longman, Reading, MA, 2000).
first citation in article

A brief derivation is available at Ref. 5.
first citation in article

Freeman Dyson in Some Strangeness in the Proportion,
edited by Harry Woolf (Addison–Wesley, Reading, MA, 1980), p. 376,
writes that "Thirtyone 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 wavefunction.' I said to him,
`You're crazy.' But he wasn't." first citation in article

Some background references are available at Ref. 5.
first citation in article

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw–Hill, New York, 1965), p. 29.
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Jorge Dias de Deus, Mário Pimenta, Ana Noronha, Teresa Peña, and Pedro Brogueira, Introducción al la Física (Spanish) (McGraw–Hill, New York, 2001), ISBN 8448131908 and Introdução à Física (Portuguese) (McGraw–Hill, New York, 2000), 2nd ed., ISBN 9727730353.
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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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CITING ARTICLES
This list contains links to other online articles that cite the article currently being viewed.

Action: Forcing Energy to Predict Motion
Dwight E. Neuenschwander et al., Phys. Teach. 44, 146 (2006)

Use of the Einstein–Brillouin–Keller action quantization
Lorenzo J. Curtis et al., Am. J. Phys. 72, 1521 (2004)

Getting the most action out of least action: A proposal
Thomas A. Moore, Am. J. Phys. 72, 522 (2004)

From conservation of energy to the principle of least action: A story line
Jozef Hanc et al., Am. J. Phys. 72, 514 (2004)

Modern mechanics
Ruth W. Chabay et al., Am. J. Phys. 72, 439 (2004)
FIGURES
Full figure (22 kB)Fig. 1. Story line showing the principle of least action^{ }sandwiched between relativity and quantum mechanics. First citation in article
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