American Journal of Physics, Vol. 72, No. 4, pp. 522–527, April 2004
©2004 American Association of Physics Teachers. All rights reserved.
Getting the most^{ }action out of least action: A proposal
Thomas A. Moore^{a)}
Department of Physics^{ }and Astronomy, Pomona College, 610 N. College Avenue, Claremont, California 91711
Received: 8^{ }September 2003; accepted: 12 December 2003Lagrangian methods lie at the foundation of^{ }contemporary theoretical physics. Several recent articles have explored the possibility^{ }of making the principle of least action and Lagrangian methods^{ }a part of the firstyear physics curriculum. I examine some^{ }of this proposal's implications for subsequent courses in the undergraduate^{ }physics major, and focus on the influence that this proposal^{ }might have on the selection of topics and the opportunities^{ }this proposal presents for teaching these courses in a more^{ }contemporary way. Many of these ideas are relevant even if^{ }students first learn Lagrangian methods in a sophomore mechanics course.^{ }© 2004 American Association of Physics Teachers. ^{ }
Contents
I. INTRODUCTION
Hamilton's principle,^{1} more generally known as the^{ }principle of least action (particularly since the publication of Feynman's^{ }lectures^{2}) has played a seminal role in the development of^{ }theoretical physics in the latter part of the 20th century.^{ }Lagrangian methods that extend this principle lie at the heart^{ }of general relativity, quantum field theory, and the standard model^{ }of particle physics, and such methods play a crucial role^{ }in conceptually framing and expressing these theories. ^{ }
Edwin Taylor has^{ }recently argued that this principle provides a simple but powerful^{ }framework for unifying Newtonian mechanics, relativity, and quantum mechanics,^{3} and^{ }he and his collaborators have begun to lay the foundations^{ }for teaching the principle in the introductory course.^{4}^{,}^{5}^{,}^{6}^{,}^{7}^{,}^{8} If we^{ }presume that this proposal is possible and desirable, it has^{ }implications for subsequent courses in the physics major. In this^{ }article, I will examine some of these implications, focusing on^{ }new opportunities that teaching least action in the introductory course^{ }makes possible, as well as on what changes in upperlevel^{ }courses might best support these opportunities in subsequent courses. ^{ }
My^{ }purpose is not to describe a new upperlevel curriculum in^{ }detail. Instead, I hope that by presenting an overview of^{ }the issues and providing references to some available resources, I^{ }will provide some guidance to those who might develop such^{ }curricula. This article also might be interesting to those seeking^{ }to modernize the upperlevel courses that follow an intermediate mechanics^{ }course which discusses Lagrangian methods. ^{ }
II. THE MODERN PHYSICS COURSE
Most upperlevel physics curricula open^{ }with a course in "modern physics," which for the sake^{ }of argument in what follows, I will assume to be^{ }a sophomorelevel class that at least discusses special relativity, some^{ }basic quantum theory, atomic and nuclear physics, and perhaps some^{ }particle physics. In a curriculum where the classical principle of^{ }least action is taught in introductory physics, the modern physics^{ }course might be reworked somewhat to address two important goals:^{ }connect the classical principle of least action with quantum mechanics^{ }and relativity, and build a solid foundation for using the^{ }principle in subsequent courses. I will discuss the link to^{ }quantum mechanics first (for reasons that will become clearer as^{ }we go). ^{ }
Taylor, Vokos, O'Meara, and Thornber have recently published^{ }a curricular plan that connects quantum mechanics with the principle^{ }of least action at a level that seems appropriate for^{ }sophomores.^{9} This plan starts with the students working through the^{ }first half of Feynman's popular book QED.^{10} Feynman's book demonstrates^{ }that it is possible to explain the results of classical^{ }optics in a variety of practical situations using the following^{ }simple model: a photon explores all possible paths between emission^{ }and detection, we imagine the photon traveling along each possible^{ }path to carry an arrow that rotates a number of^{ }times that is proportional to the action along that path,^{ }and the probability that the photon will be observed at^{ }the detection event is proportional to the squared length of^{ }the vector sum of the final arrows for all the^{ }paths that the photon explores. Sophomorelevel majors (unlike Feynman's intended^{ }audience) should be able to understand that the arrows are^{ }visual representations of complex numbers, but this visualization is powerful^{ }and useful even when students can do the calculations with^{ }complex numbers. ^{ }
The fundamental problem with the "explore all paths"^{ }model is that actually summing the arrows over all possible^{ }paths is a daunting task. Taylor and his collaborators make^{ }this task simpler by providing computer programs that compute the^{ }sums for various simple paths so that students can explore^{ }the implications of the model. Building on this foundation, Taylor^{ }and his collaborators (aided by more programs) then extend Feynman's^{ }description to help students discover methods for handling free electrons^{ }and then electrons with potential energy, the concept of a^{ }wave function, the concept of the freeparticle propagator, and ultimately^{ }the concept of a boundstate wave function, all with very^{ }little mathematics. ^{ }
The method Taylor and his collaborators use to^{ }develop the freeparticle propagator illustrates their general approach to making^{ }difficult ideas more accessible. The key to making the "explore^{ }many paths" approach practical is to get rid of the^{ }summation over all possible paths. The world line through spacetime^{ }between a given starting event a and a given ending^{ }event b that has the least action is by definition^{ }the world line along which the particle's arrow undergoes the^{ }fewest turns from start to finish. With the help of^{ }the computer programs, a student can find that the only^{ }paths that contribute significantly to the arrow representing the final^{ }sum at event b are those contributing final arrows that^{ }make an angle of less than with the arrow^{ }contributed by the world line of least action; neither the^{ }length nor the direction of the sum is much affected^{ }if one ignores all other paths. Indeed, one finds that^{ }for a free particle, the direction (in the complex plane)^{ }of the arrow representing the sum at b is always^{ }rotated by 45° relative to the direction of the arrow^{ }contributed by the leastaction world line at b (which in^{ }turn is simply a rotated version of the arrow at^{ }the initial event a), and the sum's magnitude depends on^{ }how far a path must deviate from the leastaction world^{ }line to yield a contributed arrow that makes an angle^{ }of with the leastaction arrow. ^{ }
Therefore it should be^{ }possible in principle to forego the sum entirely and calculate^{ }the arrow representing the sum over all paths by rotating^{ }the direction of the arrow contributed by the single leastaction^{ }path by 45° and multiplying by a factor that specifies^{ }the degree to which small deviations from this path affect^{ }the angle of the path's contributed arrow. For a free^{ }particle, this factor can only be a function of the^{ }particle's mass m, Planck's constant h, the time interval between^{ }the initial and final events, and the spatial separation of^{ }those events. Taylor and his collaborators^{9} argue that we can^{ }determine the correct expression for this factor by assuming that^{ }a freeparticle wave function which is uniform over space at^{ }a certain time must remain uniform as time passes (a^{ }result required by symmetry). We can consider any wave function^{ }at a given time to be a set of arrows^{ }(that is, complex numbers) distributed over space. Assume that we^{ }know the wave function arrows (x_{i},t_{0}) at various positions x_{i}^{ }at some initial time t_{0}. The arrow (x,t) at a^{ }different position x and later time t is determined by^{ }determining the sum of the arrows contributed by all paths^{ }starting from the arrow (x_{i},t_{0}) at a given x_{i} at^{ }time t_{0} and arriving at position x and time t,^{ }and then summing over all x_{i} (see Fig. 1). For^{ }the free particle, we can do the sum over all^{ }paths by calculating the arrow contributed by the leastaction path^{ }from x_{i},t_{0} to x, t (which is a straight world^{ }line for a free particle) and use a formula (rule)^{ }involving h, m, t–t_{0}, and x–x_{i} to convert this arrow^{ }to an arrow representing the sum over all paths. By^{ }using a program constructed for this purpose, students can experiment^{ }with different rules until they find one that preserves the^{ }uniform wave function. The process is quite intuitive and requires^{ }very little mathematics. ^{ }
Figure 1. Once we know how to generate a^{ }future wave function from a past one, we can generalize^{ }to particles that are not free and begin to explore^{ }both stationary and dynamic states of bound particles.^{9} After developing^{ }the general concept of a stationary state, we might introduce^{ }the Schrödinger equation and explore bound states of other systems^{ }in a more conventional manner. ^{ }
The approach in Ref. 9^{ }is plausibly accessible to sophomorelevel physics majors, and has the^{ }advantage of giving these students a deeper, more intuitive, and^{ }perhaps more engaging understanding of quantum mechanics than one typically^{ }gets in a modern physics course. Moreover, this approach is^{ }the only way that I know in which we might^{ }plausibly link the classical principle of least action and quantum^{ }mechanics at this level. This approach, however, will take a^{ }fair amount of class time, and thus will probably displace^{ }some other topics usually covered in such a course.^{11} ^{ }
Next^{ }I would like to discuss the treatment of special relativity^{ }in the modern physics course. The argument about the propagator^{ }assumes that the reader understands what events, world lines, and^{ }spacetime diagrams are (Fig. 1 is essentially a spacetime diagram).^{ }Therefore, a careful treatment of these concepts in the relativity^{ }portion of the course is essential for the success of^{ }the quantum section. My experience is that taking the time^{ }to teach students to use spacetime diagrams and the geometric^{ }analogy to relativity before teaching the Lorentz transformation equations greatly^{ }improves their understanding. Students understand much better the meaning of^{ }the Lorentz transformation equations after they have seen a spacetime^{ }diagram that shows the axes for two different reference frames,^{ }and after they have understood the crucial differences between coordinate^{ }measurements and the invariant spacetime interval. ^{ }
The other topic that^{ }needs to be explored is the concept of a fourvector.^{ }This concept not only makes the relationship between energy, momentum,^{ }and mass much easier to understand, but it provides an^{ }essential foundation for any future application of Lagrangian methods to^{ }special relativity, general relativity, or electricity and magnetism. This course^{ }is not where we should introduce index notation and the^{ }Einstein summation convention, but most students at this level understand^{ }column vectors and matrix multiplication, and we can go a^{ }long way with these tools and explore the most crucial^{ }characteristics of fourvectors (such as their transformation properties, the invariance^{ }of a fourvector's magnitude, the invariance of the dot product^{ }of fourvectors, and the frameindependence of fourvector equations). ^{ }
This part^{ }of the course also should link the classical principle of^{ }least action with the principle that a straight world line^{ }is the world line of longest proper time between two^{ }given events (the latter is easily proved using an elementary^{ }argument^{12}^{,}^{13} and should be a part of the development of^{ }the concept of proper time). The action S for a^{ }relativistic free particle for a given world line can be^{ }written as
where c is the speed of light, and^{ }the integral yields the total proper time measured along the^{ }path. The minus sign ensures that the action is a^{ }minimum for whatever path has maximal proper time, and the^{ }factor mc^{2} gives the action the appropriate units and the^{ }correct linear dependence on the particle's mass. ^{ }
We can write^{ }Eq. (1) in the form of a coordinatetime integration over^{ }a Lagrangian as follows:
which implies that
A simple application^{ }of the Euler–Lagrange equations and some basic calculus establishes that^{ }the particle's velocity components must be constant. We see, therefore,^{ }that we can develop a relativistic principle of leastaction for^{ }a free particle and obtain the constantvelocity result that we^{ }know must be true from other arguments. This result supports^{ }the idea (used in the quantum section) that the world^{ }line of least action for even a relativistic free particle^{ }is indeed a straight world line, and Eq. (2b) is^{ }an essential first step in developing an electromagnetic Lagrangian. ^{ }
Such^{ }a discussion would imply a relativity section that is three^{ }to four weeks long, which is more time than is^{ }usually spent on the topic. In what follows, however, I^{ }will show that this discussion would open up significant opportunities^{ }for subsequent courses. Because applications of relativity are increasingly important^{ }in modern technology, a solid understanding of relativity is more^{ }important to physicists and engineers now than it was even^{ }two decades ago.^{14} ^{ }
III. THE INTERMEDIATE MECHANICS COURSE
The^{ }next course a typical physics major might encounter would be^{ }one in intermediate classical mechanics, which typically discusses subjects such^{ }as orbital motion, damped and driven harmonic oscillators, rotation of^{ }rigid bodies, and perhaps even some chaos and nonlinear dynamics.^{ }Texts for this course commonly include a discussion of the^{ }principle of least action and Lagrangian methods.^{15} If these ideas^{ }are thoroughly discussed in the introductory course, then some time^{ }would become available in this course. My recommendation is that^{ }at least some of this extra time be spent exploring^{ }the application of Lagrangian methods to continuous media. This application^{ }is important because the same methods apply to fields, so^{ }this discussion of continuous media would provide essential background for^{ }any subsequent application of Lagrangian methods to the electromagnetic field.^{ }Reference 16 presents a very nice discussion of continuous media.^{ }^{ }
IV. QUANTUM MECHANICS
Most undergraduate major programs^{ }include a quantum mechanics course in the junior or senior^{ }year. I will assume that students in this course are^{ }familiar with partial derivatives, complex numbers, looking up integrals, and^{ }Taylorseries expansions. ^{ }
A crucial first step in this course would^{ }be to firmly and formally connect the explore all paths^{ }model presented in the sophomore course with the timedependent Schrödinger^{ }equation. Once this connection has been made, the rest of^{ }the course can be taught in the standard way. In^{ }what follows, I will briefly sketch the logic of the^{ }argument: more details can be found in Ref. 17. ^{ }
In^{ }the sophomorelevel course, students should have discovered that for a^{ }free particle, the propagator function that species the contribution to^{ }the quantum amplitude (arrow) (x,t) made by arrows of the^{ }particle's wave function within a sufficiently small range x_{i} around^{ }the position x_{i} at an earlier time t_{0} is given^{ }by
where S_{direct} is the action measured along the straight^{ }worldline from x_{i},t_{0} to x, t. For a free particle^{ }moving in one dimension with a constant potential energy V,^{ }the value of S_{direct} is simply
where tt–t_{0} is the^{ }(coordinate) time difference between the events and ux_{i}–x. So in^{ }this case, we have
To find the complete wave function^{ }amplitude (x,t), we must sum K(x_{i},t_{0})x_{i} over all possible initial^{ }positions x_{i}, as schematically shown in Fig. 1. Note that^{ }the middle factor in Eq. (5) is the only thing^{ }that varies as x_{i} varies, because it will cause u^{ }to vary, and this term rotates the phase angle of^{ }the resulting complex amplitude. As discussed, arrows rotated by an^{ }angle greater than relative to the arrow for u = 0^{ }do not contribute significantly to the result, and we really^{ }only need to be concerned about the contributions from the^{ }initial positions x_{i} close enough to the final position x^{ }so that
Equation (6) proves to be the key to^{ }using the explore all paths approach to derive the Schrödinger^{ }equation. Note that if we choose the time step t = t–t_{0}^{ }between the initial and final wave functions to be infinitesimal,^{ }then u also must be infinitesimal, which means that the^{ }positions of points along all the paths in Fig. 1^{ }that contribute significantly will not be much different from x.^{ }Therefore, even if the particle's potential energy varies with position,^{ }its value over the range of interest for calculating (x,t)^{ }will be essentially equal to V(x), its value at x,^{ }so Eqs. (3,4,5) apply even to the case of nonuniform^{ }V(x) in the limit t0. The sum over all x_{i}^{ }in this limit therefore becomes
because x_{i} = x + u and dx_{i} = du. If^{ }we expand the exponential involving V to order t, (x + u,t_{0})^{ }to order u^{2}, and do some integrals of the form^{ } u^{n}e^{–au2}du,^{18} we find that
If we subtract (x,t_{0}) from both^{ }sides, multiply through by i/t, and take the limit t0,^{ }we find the timedependent Schrödinger equation for one dimension. (It^{ }is not very difficult to generalize this derivation to three^{ }dimensions, but it does not yield any deeper understanding.) ^{ }
V.^{ }ELECTRICITY AND MAGNETISM
The undergraduate curriculum also typically includes a course^{ }in electricity and magnetism offered at the sophomore, junior, or^{ }senior level. I will assume that this course is offered^{ }for juniors and/or seniors and that students have taken a^{ }modern physics course and intermediate mechanics course of the type^{ }already described. ^{ }
The first task in this course would be^{ }to discuss index notation and the Einstein summation convention, the^{ }Lorentz transformation properties of scalars, vectors, and covectors, and the^{ }fourgradient. My experience is that juniors and seniors can become^{ }comfortable with this material within four to five class sessions^{ }if the material is taught carefully.^{19} The relativistic Lorentz force^{ }law provides a good physical context for practicing the notation.^{ }In appropriate units,^{20} this law can be written as
where^{ }
and u^{} is the charged particle's fourvelocity with components u^{t} = [1–^{2}/c^{2}]^{–1/2},^{ }u^{i} = ^{i}/c, p^{µ} = mc u^{µ} is the particle's fourmomentum, q is its charge,^{ } is the proper time measured along its world line^{ }and I am using a metric with a timelike signature^{ }(+–––). Equation (9) involves scalars, vectors, covectors, and tensors and^{ }yet when the sums are written out explicitly, the three^{ }spatial components reduce to the Lorentz law taught in introductory^{ }physics and the time component reduces to conservation of energy.^{ }By examining the transformation properties of all the pieces, students^{ }can demonstrate that Eq. (9) must have the same form^{ }in all reference frames. It also is a good exercise^{ }for students to show that the antisymmetric nature of F^{µ}^{ }ensures that d(p^{µ}p_{µ})/d= 0, meaning that the particle's rest mass m = p^{µ}p_{µ}^{ }is fixed. ^{ }
To fully connect electricity and magnetism with the^{ }principle of least action, we also must develop the concept^{ }of the magnetic potential A. Textbooks at this level avoid^{ }or marginalize the magnetic potential, partly because when it is^{ }presented in the usual way, it can be a tricky^{ }and abstract concept. However, there are ways to make the^{ }magnetic potential more accessible,^{21} and there are some good reasons^{ }to discuss it fully even if we ignore the principle^{ }of least action.^{22} ^{ }
One possible story line for introducing the^{ }fourpotential is made possible by the principle of least action.^{ }The action for a nonrelativistic particle moving in a static^{ }electric field is
Our goal is to see if we^{ }can guess the appropriate relativistic action for this case. We^{ }already know how to generalize the kinetic energy part; the^{ }action for a free particle is given in Eq. (2a).^{ }Like this part, whatever we add to the action to^{ }account for the field must be a relativistic scalar. But^{ }is the electric potential a relativistic scalar or something^{ }else? By considering the field between the plates of a^{ }parallelplate capacitor when viewed in a frame moving parallel to^{ }the plates, it can be quickly argued that must^{ }transform like the time component of a fourvector. So in^{ }a fully relativistic expression for the action, the electromagnetic field^{ }must appear in the form of a fourvector that we^{ }will call A^{µ}. However, the term we add to the^{ }Lagrangian must be a relativistic scalar, so the term must^{ }be the dot product of A^{µ} and some other fourvector.^{ }The only available fourvector in the case of a point^{ }particle is the particle's own fourvelocity u^{µ}. So we propose^{ }a relativistic action of the form
where the components of^{ }A are the spatial components of A^{µ}. We can easily^{ }show that S in Eq. (11b) reduces to Eq. (10)^{ }in the nonrelativistic limit (except for an extra rest energy^{ }term that does not affect the motion). ^{ }
What kind of^{ }motion does this principle imply? Although we can quickly give^{ }the result in index notation, let me demonstrate the argument^{ }in a form that might be more accessible to a^{ }junior physics major. Consider the x component of the Euler–Lagrange^{ }equation. The partial derivatives of the Lagrangian in this case^{ }are
where p^{x} is the relativistic momentum. The Euler–Lagrange equations^{ }in this case therefore imply that
which implies that
The^{ }usual definition of the electric field is the force per^{ }unit charge on a test charge at rest, so we^{ }have
If we identify
we can easily see that Eq.^{ }(13b) is equivalent to the x component of the Lorentz^{ }force law given by Eq. (9a). We also can see^{ }quite generally that
and that Faraday's law and div B = 0 are^{ }identities implied by Eq. (15). ^{ }
Once we have gone this^{ }far, we can derive the sourcedependent Maxwell equations from a^{ }plausible principle of least action.^{23} Students should know from the^{ }treatment of continuous media in the intermediate mechanics course that^{ }a leastaction principle for the electromagnetic field will involve integrating^{ }a Lagrangian density over all space and time. This Lagrangian^{ }density must be a relativistic scalar and must involve a^{ }term that is quadratic in the field quantities. These requirements^{ }imply that the resulting Euler–Lagrange equations will produce linear differential^{ }equations in the field, which is required for the field^{ }to obey the superposition principle. The only plausible candidates for^{ }such terms are A_{µ}A^{µ} and F_{µ}F^{µ}. The first of these^{ }leads to absurd results, for example, the resulting field equations^{ }in the electrostatic case involve directly, not the derivatives^{ }of , which does not match Gauss' law. For the^{ }second case, we can argue that the sign of the^{ }integral has to be negative for the quantity to have^{ }a plausible minimum,^{24} and that we must have a factor^{ }of 1/k (where k is Coulomb's constant) to make the^{ }units come out right. The Lagrangian density also must involve^{ }a term that is linear in the fourcurrent J^{µ} = [,j/c], where^{ }j is the ordinary current density, so that the sources^{ }will appear linearly in the field equation. The only plausible^{ }term with the right units in this case is A_{µ}J^{µ}.^{ }Therefore, the leastaction principle for the electromagnetic field must be^{ }something like
where g^{µ} is the inverse flatspace metric and^{ }b is some unitless constant that specifies the relative magnitude^{ }and sign of the two terms. The field quantities A_{µ}^{ }play the role of coordinates and the gradients _{µ}A_{} play^{ }the role of "velocities." With only a bit of work,^{25}^{ }the Euler–Lagrange equations yield
If we choose b = –16, the time^{ }component of Eq. (17) matches Gauss' law. By writing them^{ }out, students can discover that the other components spell out^{ }the Ampere–Maxwell relation. ^{ }
VI. OTHER^{ }INTERESTING APPLICATIONS OF LEAST ACTION
Because many electromagnetic circuits have direct^{ }mechanical analogs, we often can use Lagrangian methods to find^{ }equations of motion for such circuits, even for very complicated^{ }electromechanical circuits. It turns out that we can even handle^{ }realistic resistors by treating them as generalized external forces. These^{ }issues (along with many other applications of Lagrangian techniques) are^{ }beautifully discussed by Wells.^{26} ^{ }
Another interesting source of applications of^{ }the principle of least action to fields at a fairly^{ }advanced level is a book written some time ago by^{ }Soper.^{27} This book even includes a discussion of dissipative effects^{ }that might be appropriate in an upperlevel course. ^{ }
Once students^{ }are used to the principle of least action, other variational^{ }calculations become conceptually simpler. Several years ago, Van Baak discussed^{ }a variational technique that enables one to solve complicated steadystate^{ }circuits without invoking Kirchoff's loop rule.^{28} Because applying the loop^{ }rule requires careful attention to signs, it is a common^{ }source of student errors. Van Baak's approach avoids this problem.^{ }^{ }
Finally, I point out that if students have studied special^{ }relativity in some depth and have seen index notation and^{ }know about fourvectors, covectors, and tensors, they have a background^{ }that provides a great springboard for studying general relativity. The^{ }geodesic equations of motion can be treated as a leastaction^{ }principle. One can even use a Lagrangian to find equations^{ }of motion for the gravitational field,^{29} a method widely used^{ }by researchers in the field (particularly those doing numerical simulations).^{ }^{ }
VII. CONCLUSIONS
My^{ }goal has been to reflect on what kinds of changes^{ }to the upperlevel curriculum might help students take full advantage^{ }of an introductorylevel exploration of the principle of least action.^{ }I have only provided a broad sketch; there is much^{ }work to be done before these suggestions can become anything^{ }approaching a practical curriculum. The proposed changes would in some^{ }cases mean shifting priorities to allow sufficient time for the^{ }development of some of the techniques, and I have no^{ }doubt that some of the changes would present problems that^{ }would have to be worked out. ^{ }
However, the proposed changes^{ }could create a very exciting upperlevel curriculum that could more^{ }clearly display the deep underlying connections between mechanics, relativity, electrodynamics,^{ }and quantum mechanics. These changes would give us a thoroughly^{ }21stcentury physics curriculum that teaches viewpoints and techniques currently used^{ }by researchers. The principle of least action is among the^{ }most beautiful and powerful physical principles ever envisioned. With some^{ }vision and effort, the least action principle could become a^{ }greater part of the common background of physics undergraduates. ^{ }
ACKNOWLEDGMENTS
I would^{ }like to thank E. F. Taylor, the editors of this^{ }special issue, and the reviewers for making valuable suggestions about^{ }how to improve this article. ^{ }
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.

Herbert Goldstein, Charles P. Poole, Jr., and John L. Safko, Classical Mechanics (Addison–Wesley, San Francisco, 2002), 3rd ed., Vol. 1, Chap. 2, pp. 34ff.
first citation in article

Richard P. Feynman, Robert B. Leighton, and Matthew Sands, The Feynman Lectures on Physics (Addison–Wesley, Reading, MA, 1964), Vol. 2, Chap. 19, pp. 19–1ff.
first citation in article

Edwin F. Taylor, "A call to action," Am. J. Phys. 71, 423–425 (2003). [ISI]
first citation in article

Jozef Hanc, Slavomir Tuleja, and Martina Hancova, "Simple derivation of
Newtonian mechanics from the principle of least action," Am. J. Phys. 71, 386–391 (2003). [ISI]
first citation in article

Jozef Hanc, Edwin F. Taylor, and Slavomir Tuleja, "Deriving Lagrange's
equations using elementary calculus," Am. J. Phys. (submitted). See www.eftaylor.com/leastaction.html.
first citation in article

Edwin F. Taylor and Jozef Hanc, "From conservation of energy to the
principle of least action: A story line," Am. J. Phys. (submitted). See
www.eftaylor.com/leastaction.html.
first citation in article

Jozef Hanc, Slavomir Tuleja, and Martina Hancova, "Symmetries and
conservation laws: Consequences of Noether's theorem," Am. J. Phys.
(submitted). See www.eftaylor.com/leastaction.html.
first citation in article

Jozef Hanc, "The original Euler's calculusofvariations method: Key to
Lagrangian mechanics for beginners," Am. J. Phys. (submitted). See www.eftaylor.com/leastaction.html.
first citation in article

Edwin F. Taylor, Stamatis Vokos, John M. O'Meara, and Nora S. Thornber,
"Teaching Feynman's sumoverpaths quantum theory," Comput. Phys. 12, 190–199 (1998). Current versions of the draft teaching materials and computer programs discussed in this article are available online at www.eftaylor.com/download.html#quantum.
first citation in article

Richard S. Feynman, QED: The Strange Theory of Light and Matter (Princeton U.P., Princeton, 1985).
first citation in article

Such modern physics courses often include a discussion of the
historical development of quantum mechanics that would be less relevant
to this approach. Cutting much of this material will help make some
room. first citation in article

Edwin F. Taylor and John Archibald Wheeler, Spacetime Physics (Freeman, New York, 1992), 2nd ed., p. 149ff.
first citation in article

Thomas A. Moore, A Traveler's Guide to Spacetime (McGraw–Hill, New York, 1995), pp. 86–87. The same argument also appears on pp. 83–84 of Moore's introductory textbook, Six Ideas That Shaped Physics, Unit R: The Laws of Physics are FrameIndependent (McGraw–Hill, New York, 2003), 2nd ed.
first citation in article

The relativity of simultaneity has become a very practical engineering
problem for the designers of the global positioning system. Students
can see the delay imposed by light travel time when satellite
communications are used on television. Experimental general relativity
has mushroomed in recent years, and gravitational waves will likely be
discovered in the coming decade. Moreover, aspects of relativistic
cosmology previously considered esoteric are likely to have a large
impact on physics in the next couple of decades. first citation in article

Examples include Jerry B. Marion and Stephen T. Thornton, Classical Dynamics of Particles and Systems (Saunders, Fort Worth, 1995), 4th ed.;
Ralph Baierlein, Newtonian Dynamics (McGraw–Hill, New York, 1983); and
Grant R. Fowles, Analytical Mechanics (Saunders, Philadelphia, 1986), 4th ed.
first citation in article

Herbert Goldstein, Charles P. Poole, Jr., and John L. Safko, Classical Mechanics
(Addison–Wesley, San Francisco, 2002), 3rd ed. Secs. 13.1 and 13.2 (up
to the middle of p. 563) are at a level suitable for sophomores or
juniors. One would probably not need to derive the Euler–Lagrange
equations the way that they do, but rather state the equations
(appealing to analogy) and show that they work for a simple case (as
the authors do at the top of p. 563). first citation in article

Ramamurti Shankar, Principles of Quantum Mechanics (Plenum, New York, 1980), Sec. 8.5, pp. 240–241.
first citation in article

The results for these definite integrals given in standard integral tables assume (usually implicitly) that a is real. However, the same results apply even if a is complex, as long as the real part of a>0. See, for example, Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions
(Dover, New York, 1964), p. 302, where no such assumption is made. I am
not sure that it is necessary to have students worry about this issue
unless they ask. first citation in article

I regularly teach a juniorlevel course in general relativity where
students are required to master this material. I have found that there
are some tricks for teaching index notation at this level that are
beyond the scope of this article to discuss in detail, but it helps
greatly if students are explicitly taught to recognize the difference
between free and summed indices, and if they write out expanded
versions of the equations when necessary. Students also should be
required to calculate the time derivative of a product involving an
implied sum and do other exercises where the correct answer depends on
correctly recognizing the implied sums. J. B. Hartle's Gravity
(Addison–Wesley, San Francisco, 2003) is better than most general
relativity books in teaching the notation (and in presenting the entire
subject of relativity to undergraduates). first citation in article

We can conveniently combine the advantages of Gaussian and SI units by defining BcB_{conv}, where B_{conv} is the conventional magnetic field measured in teslas. The redefined B
has units of N/C, just like the electric field (with 300 MN/C
corresponding to 1 T.) All electromagnetic equations then take the same
mathematical form as they would in Gaussian units, except that factors
of 4 become 4k, where k
is the Coulomb constant. However, the units for all quantities other
than the magnetic field are in SI. This system makes the symmetries
between the electric and magnetic fields apparent (and the equations
much more beautiful) without having to deal with Gaussian units. This
unit system also has the advantage of making it easy to show the
connections between electromagnetic field theory and gravitational
field theory (where the gravitational constant G is not typically suppressed as is the corresponding Coulomb constant k in Gaussian units).
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For example, A can be given a more physical meaning than often is supposed. In a static situation where = 0 and a particle moves perpendicular to A, the Euler–Lagrange equations implied by Eq. (11) imply that the quantity p + (q/c)A is constant in time. Just as the scalar potential
at a point in space near a static charge distribution is the total work
per unit charge that one would have to do on a charged test particle to
move it from infinity to that point, the quantity A/c at
a point in space near a static (and neutral) current distribution is
the total momentum per unit charge that one would have to supply to a
charged test particle to keep it moving from infinity to that position
along a path that is always perpendicular to A. Therefore, if represents potential energy per unit charge, A represents "potential momentum" per unit charge.
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For example, the Aharonov–Bohm effect suggests that the magnetic potential is more fundamental than E and B, and is certainly more directly connected to quantum mechanics. See J. J. Sakurai, Modern Quantum Mechanics, edited by San Fu Tuan (Addison–Wesley, Redwood City, CA, 1985), pp. 136–139, or John S. Townsend, A Modern Approach to Quantum Mechanics
(McGraw–Hill, New York, 1992), pp. 399–404 for good discussions of this
effect. The fourpotential also provides significant advantages for
calculating electromagnetic fields: indeed, R. L. Coren of Drexel
University once told me that computer programs used by electrical
engineers almost always calculate the scalar and magnetic potentials
instead of calculating E and B directly.
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The general argument for the leastaction derivation of the field equations comes from L. D. Landau and E. M. Lifschitz, The Classical Theory of Fields (Pergamon, Oxford, 1975), 4th ed., pp. 67–74, and from John David Jackson, Classical Electrodynamics (Wiley, New York, 1999), 3rd ed., Sec. 12.7.
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Reference 23, Landau and Lifschitz, p. 68.
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With students who are still becoming familiar with the index notation,
the easiest way to have them work out the implications of the
electromagnetic Lagrangian is for them to write out the implied sums in
the two terms (because the metric is diagonal, there are not that many
terms to write) and then calculate the Euler–Lagrange equation for a
specific field coordinate (say A^{x}) to see how the calculation goes.
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Dare A. Wells, Shaum's Outline of Theory and Problems of Lagrangian Dynamics (McGraw–Hill, New York, 1967). The section on electrical and electromechanical systems is Chap. 15.
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Davison E. Soper, Classical Field Theory (Wiley, New York, 1976).
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D. A. Van Baak, "Variational alternatives to Kirchoff's loop theorem," Am. J. Phys. 67, 36–44 (1999). [ISI]
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Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler, Gravitation (Freeman, San Francisco, 1973), Chap. 21.
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Hamilton's principle: Why is the integrated difference of the kinetic and potential energy minimized?
Alberto G. Rojo, Am. J. Phys. 73, 831 (2005)
FIGURES
Full figure (9 kB)Fig. 1. We^{ }can calculate the wave function amplitude (x,t) at position x^{ }at time t by using Eq. (1) to calculate the^{ }contribution of the wave function amplitude (x_{i},t_{0}) at a position^{ }x_{i} at an earlier time t_{0} and then summing over^{ }all x_{i}. The diagonal lines show the direct paths that^{ }connect the various points x_{i} with the final point x. First citation in article
FOOTNOTES
^{a}Electronic mail: tmoore@pomona.edu
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