We derive Lagrange’s equations of motion from the principle of least action using elementary calculus rather than the calculus of variations. We also demonstrate the conditions under which energy and momentum are constants of the motion.

## REFERENCES

1.

We take “equations of motion” to mean relations between the accelerations, velocities, and coordinates of a mechanical system. See L. D. Landau and E. M. Lifshitz,

*Mechanics*(Butterworth-Heinemann, Oxford, 1976), Chap. 1, Sec. 1.2.

Besides its expression in scalar quantities (such as kinetic and potential energy), Lagrangian quantities lead to the reduction of dimensionality of a problem, employ the invariance of the equations under point transformations, and lead directly to constants of the motion using Noether’s theorem. More detailed explanation of these features, with a comparison of analytical mechanics to vectorial mechanics, can be found in Cornelius Lanczos,

*The Variational Principles of Mechanics*(Dover, New York, 1986), pp. xxi–xxix.3.

Chapter 1 in Ref. 1 and Chap. V in Ref. 2; Gerald J. Sussman and Jack Wisdom,

*Structure and Interpretation of Classical Mechanics*(MIT, Cambridge, 2001), Chap. 1; Herbert Goldstein, Charles Poole, and John Safko,*Classical Mechanics*(Addison–Wesley, Reading, MA, 2002), 3rd ed., Chap. 2. An alternative method derives Lagrange’s equations from D’Alambert principle; see Goldstein, Sec. 1.4.4.

Our derivation is a modification of the finite difference technique employed by Euler in his path-breaking 1744 work, “The method of finding plane curves that show some property of maximum and minimum.” Complete references and a description of Euler’s original treatment can be found in Herman H. Goldstine,

*A History of the Calculus of Variations from the 17th Through the 19th Century*(Springer-Verlag, New York, 1980), Chap. 2. Cornelius Lanczos (Ref. 2, pp. 49–54) presents an abbreviated version of Euler’s original derivation using contemporary mathematical notation.5.

R. P. Feynman, R. B. Leighton, and M. Sands,

*The Feynman Lectures on Physics*(Addison–Wesley, Reading, MA, 1964), Vol. 2, Chap. 19.6.

See Ref. 5, p. 19-8 or in more detail,

J.

Hanc

, S.

Tuleja

, and M.

Hancova

, “Simple derivation of Newtonian mechanics from the principle of least action

,” Am. J. Phys.

71

(4

), 386

–391

(2003

).7.

There is no particular reason to use the midpoint of the segment in the Lagrangian of Eq. (2). In Riemann integrals we can use any point on the given segment. For example, all our results will be the same if we used the coordinates of either end of each segment instead of the coordinates of the midpoint. The repositioning of this point can be the basis of an exercise to test student understanding of the derivations given here.

8.

A zero value of the derivative most often leads to the world line of minimum action. It is possible also to have a zero derivative at an inflection point or saddle point in the action (or the multidimensional equivalent in configuration space). So the most general term for our basic law is the principle of stationary action. The conditions that guarantee the existence of a minimum can be found in I. M. Gelfand and S. V. Fomin,

*Calculus of Variations*(Prentice–Hall, Englewood Cliffs, NJ, 1963).10.

The most fundamental justification of conservation laws comes from symmetry properties of nature as described by Noether’s theorem. Hence energy conservation can be derived from the invariance of the action by temporal translation and conservation of momentum from invariance under space translation. See

N. C.

Bobillo-Ares

, “Noether’s theorem in discrete classical mechanics

,” Am. J. Phys.

56

(2

), 174

–177

(1988

)or

C. M.

Giordano

and A. R.

Plastino

, “Noether’s theorem, rotating potentials, and Jacobi’s integral of motion

,” Am. J. Phys.

66

(11

), 989

–995

(1998

).11.

Our approach also can be related to symmetries and Noether’s theorem, which is the main subject of J. Hanc, S. Tuleja, and M. Hancova, “Symmetries and conservation laws: Consequences of Noether’s theorem,” Am. J. Phys. (to be published).

12.

Reference 3, Goldstein et al., Sec. 2.7.

13.

For the case of generalized coordinates, the energy function

*h*is generally not the same as the total energy. The conditions for conservation of the energy function*h*are distinct from those that identify*h*as the total energy. For a detailed discussion see Ref. 12. Pedagogically useful comments on a particular example can be found inA. S.

de Castro

, “Exploring a rheonomic system

,” Eur. J. Phys.

21

, 23

–26

(2000

)and

C.

Ferrario

and A.

Passerini

, “Comment on Exploring a rheonomic system

,” Eur. J. Phys.

22

, L11

–L14

(2001

).
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