Lagrangian 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 first-year 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.

## REFERENCES

1.

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.2.

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.3.

Edwin F.

Taylor

, “A call to action

,” Am. J. Phys.

71

, 423

–425

(2003

).4.

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

).5.

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〉.

6.

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〉.

7.

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〉.

8.

Jozef Hanc, “The original Euler’s calculus-of-variations method: Key to Lagrangian mechanics for beginners,” Am. J. Phys. (submitted). See 〈www.eftaylor.com/leastaction.html〉.

9.

Edwin F.

Taylor

, Stamatis

Vokos

, John M.

O’Meara

, and Nora S.

Thornber

, “Teaching Feynman’s sum-over-paths 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〉.10.

Richard S. Feynman,

*QED: The Strange Theory of Light and Matter*(Princeton U.P., Princeton, 1985).11.

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.

12.

Edwin F. Taylor and John Archibald Wheeler,

*Spacetime Physics*(Freeman, New York, 1992), 2nd ed., p. 149ff.13.

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 Frame-Independent*(McGraw–Hill, New York, 2003), 2nd ed.14.

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.

15.

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); andGrant R. Fowles,

*Analytical Mechanics*(Saunders, Philadelphia, 1986), 4th ed.16.

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).17.

Ramamurti Shankar,

*Principles of Quantum Mechanics*(Plenum, New York, 1980), Sec. 8.5, pp. 240–241.18.

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.19.

I regularly teach a junior-level 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).20.

We can conveniently combine the advantages of Gaussian and SI units by defining $B\u2261cBconv,$ where $Bconv$ 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 4π become $4\pi k,$ 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).21.

For example,

**A**can be given a more physical meaning than often is supposed. In a static situation where $\phi =0$ 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 φ 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 φ represents potential energy per unit charge,**A**represents “potential momentum” per unit charge.22.

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 four-potential 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.23.

The general argument for the least-action 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. 12.7.24.

Reference 23, Landau and Lifschitz, p. 68.

25.

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 $Ax)$ to see how the calculation goes.

26.

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. 15.27.

Davison E. Soper,

*Classical Field Theory*(Wiley, New York, 1976).28.

D. A.

Van Baak

, “Variational alternatives to Kirchoff’s loop theorem

,” Am. J. Phys.

67

, 36

–44

(1999

).29.

Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler,

*Gravitation*(Freeman, San Francisco, 1973), Chap. 21.
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