American Journal of Physics, Vol. 72, No. 4, pp. 510–513, April 2004
©2004 American Association of Physics Teachers. All rights reserved.
Deriving Lagrange's equations^{ }using elementary calculus
Technical University, Vysokoskolska 4, 042 00 Kosice, Slovakia
Edwin F. Taylor^{b)}
Massachusetts Institute^{ }of Technology, Cambridge, Massachusetts 02139
Slavomir Tuleja^{c)}
Gymnazium arm. gen. L. Svobodu, Komenskeho 4,^{ }066 51 Humenne, Slovakia
Received: 30 December 2002; accepted: 20 June 2003We 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. © 2004 American Association of Physics Teachers.^{ }
Contents
I. INTRODUCTION
The^{ }equations of motion^{1} of a mechanical system can be derived^{ }by two different mathematical methods—vectorial and analytical. Traditionally, introductory mechanics^{ }begins with Newton's laws of motion which relate the force,^{ }momentum, and acceleration vectors. But we frequently need to describe^{ }systems, for example, systems subject to constraints without friction, for^{ }which the use of vector forces is cumbersome. Analytical mechanics^{ }in the form of the Lagrange equations provides an alternative^{ }and very powerful tool for obtaining the equations of motion.^{ }Lagrange's equations employ a single scalar function, and there are^{ }no annoying vector components or associated trigonometric manipulations. Moreover, the^{ }analytical approach using Lagrange's equations provides other capabilities^{2} that allow^{ }us to analyze a wider range of systems than Newton's^{ }second law. ^{ }
The derivation of Lagrange's equations in advanced mechanics^{ }texts^{3} typically applies the calculus of variations to the principle^{ }of least action. The calculus of variation belongs to important^{ }branches of mathematics, but is not widely taught or used^{ }at the college level. Students often encounter the variational calculus^{ }first in an advanced mechanics class, where they struggle to^{ }apply a new mathematical procedure to a new physical concept.^{ }This paper provides a derivation of Lagrange's equations from the^{ }principle of least action using elementary calculus,^{4} which may be^{ }employed as an alternative to (or a preview of) the^{ }more advanced variational calculus derivation. ^{ }
In Sec. II we develop^{ }the mathematical background for deriving Lagrange's equations from elementary calculus.^{ }Section III gives the derivation of the equations of motion^{ }for a single particle. Section IV extends our approach to^{ }demonstrate that the energy and momentum are constants of the^{ }motion. The Appendix expands Lagrange's equations to multiparticle systems and^{ }adds angular momentum as an example of generalized momentum. ^{ }
II. DIFFERENTIAL APPROXIMATION TO THE PRINCIPLE OF^{ }LEAST ACTION
A particle moves along the x axis with potential^{ }energy V(x) which is time independent. For this special case^{ }the Lagrange function or Lagrangian L has the form:^{5}
The^{ }action S along a world line is defined as
The^{ }principle of least action requires that between a fixed initial^{ }event and a fixed final event the particle follow a^{ }world line such that the action S is a minimum.^{ }^{ }
The action S is an additive scalar quantity, and is^{ }the sum of contributions Lt from each segment along the^{ }entire world line between two events fixed in space and^{ }time. Because S is additive, it follows that the principle^{ }of least action must hold for each individual infinitesimal segment^{ }of the world line.^{6} This property allows us to pass^{ }from the integral equation for the principle of least action,^{ }Eq. (2), to Lagrange's differential equation, valid anywhere along the^{ }world line. It also allows us to use elementary calculus^{ }in this derivation. ^{ }
We approximate a small section of the^{ }world line by two straightline segments connected in the middle^{ }(Fig. 1) and make the following approximations: The average position^{ }coordinate in the Lagrangian along a segment is at the^{ }midpoint of the segment.^{7} The average velocity of the particle^{ }is equal to its displacement across the segment divided by^{ }the time interval of the segment. These approximations applied to^{ }segment A in Fig. 1 yield the average Lagrangian L_{A}^{ }and action S_{A} contributed by this segment:
with similar expressions^{ }for L_{B} and S_{B} along segment B. ^{ }
Figure 1. III.^{ }DERIVATION OF LAGRANGE'S EQUATION
We employ the approximations of Sec. II^{ }to derive Lagrange's equations for the special case introduced there.^{ }As shown in Fig. 2, we fix events 1 and^{ }3 and vary the x coordinate of the intermediate event^{ }to minimize the action between the outer two events. ^{ }
Figure 2. For^{ }simplicity, but without loss of generality, we choose the time^{ }increment t to be the same for each segment, which^{ }also equals the time between the midpoints of the two^{ }segments. The average positions and velocities along segments A and^{ }B are
The expressions in Eq. (4) are all functions^{ }of the single variable x. For later use we take^{ }the derivatives of Eq. (4) with respect to x:
Let^{ }L_{A} and L_{B} be the values of the Lagrangian on^{ }segments A and B, respectively, using Eq. (4), and label^{ }the summed action across these two segments as S_{AB}:
The^{ }principle of least action requires that the coordinates of the^{ }middle event x be chosen to yield the smallest value^{ }of the action between the fixed events 1 and 3.^{ }If we set the derivative of S_{AB} with respect to^{ }x equal to zero^{8} and use the chain rule, we^{ }obtain
We substitute Eq. (5) into Eq. (7), divide through^{ }by t, and regroup the terms to obtain
^{ }
To first^{ }order, the first term in Eq. (8) is the average^{ }value of L/x on the two segments A and B.^{ }In the limit t0, this term approaches the value of^{ }the partial derivative at x. In the same limit, the^{ }second term in Eq. (8) becomes the time derivative of^{ }the partial derivative of the Lagrangian with respect to velocity^{ }d(L/v)/dt. Therefore in the limit t0, Eq. (8) becomes the^{ }Lagrange equation in x:
We did not specify the location^{ }of segments A and B along the world line. The^{ }additive property of the action implies that Eq. (9) is^{ }valid for every adjacent pair of segments. ^{ }
An essentially identical^{ }derivation applies to any particle with one degree of freedom^{ }in any potential. For example, the single angle tracks^{ }the motion of a simple pendulum, so its equation of^{ }motion follows from Eq. (9) by replacing x with ^{ }without the need to take vector components. ^{ }
IV. MOMENTUM AND ENERGY^{ }AS CONSTANTS OF THE MOTION
A. Momentum
We consider the case in^{ }which the Lagrangian does not depend explicitly on the x^{ }coordinate of the particle (for example, the potential is zero^{ }or independent of position). Because it does not appear in^{ }the Lagrangian, the x coordinate is "ignorable" or "cyclic." In^{ }this case a simple and wellknown conclusion from Lagrange's equation^{ }leads to the momentum as a conserved quantity, that is,^{ }a constant of motion. Here we provide an outline of^{ }the derivation. ^{ }
For a Lagrangian that is only a function^{ }of the velocity, L = L(v), Lagrange's equation (9) tells us that^{ }the time derivative of L/v is zero. From Eq. (1),^{ }we find that L/v = mv, which implies that the x momentum,^{ }p = mv, is a constant of the motion. ^{ }
This usual consideration^{ }can be supplemented or replaced by our approach. If we^{ }repeat the derivation in Sec. III with L = L(v) (perhaps as^{ }a student exercise to reinforce understanding of the previous derivation),^{ }we obtain from the principle of least action
We substitute^{ }Eq. (5) into Eq. (10) and rearrange the terms to^{ }find:
or
Again we can use the arbitrary location of^{ }segments A and B along the world line to conclude^{ }that the momentum p is a constant of the motion^{ }everywhere on the world line. ^{ }
B. Energy
Standard texts^{9} obtain conservation^{ }of energy by examining the time derivative of a Lagrangian^{ }that does not depend explicitly on time. As pointed out^{ }in Ref. 9, this lack of dependence of the Lagrangian^{ }implies the homogeneity of time: temporal translation has no influence^{ }on the form of the Lagrangian. Thus conservation of energy^{ }is closely connected to the symmetry properties of nature.^{10} As^{ }we will see, our elementary calculus approach offers an alternative^{ }way^{11} to derive energy conservation. ^{ }
Consider a particle in a^{ }timeindependent potential V(x). Now we vary the time of the^{ }middle event (Fig. 3), rather than its position, requiring that^{ }this time be chosen to minimize the action. ^{ }
Figure 3. For simplicity,^{ }we choose the x increments to be equal, with the^{ }value x. We keep the spatial coordinates of all three^{ }events fixed while varying the time coordinate of the middle^{ }event and obtain
These expressions are functions of the single^{ }variable t, with respect to which we take the derivatives^{ }
and
Despite the form of Eq. (13), the derivatives of^{ }velocities are not accelerations, because the x separations are held^{ }constant while the time is varied. ^{ }
As before [see Eq.^{ }(6)],
Note that students sometimes misinterpret the time differences in^{ }parentheses in Eq. (14) as arguments of L. ^{ }
We find^{ }the value of the time t for the action to^{ }be a minimum by setting the derivative of S_{AB} equal^{ }to zero:
If we substitute Eq. (13) into Eq. (15)^{ }and rearrange the result, we find
Because the action is^{ }additive, Eq. (16) is valid for every segment of the^{ }world line and identifies the function vL/v–L as a constant^{ }of the motion. By substituting Eq. (1) for the Lagrangian^{ }into vL/v–L and carrying out the partial derivatives, we can^{ }show that the constant of the motion corresponds to the^{ }total energy E = T + V. ^{ }
V. SUMMARY
Our derivation and the extension to multiple degrees^{ }of freedom in the Appendix allow the introduction of Lagrange's^{ }equations and its connection to the principle of least action^{ }without the apparatus of the calculus of variations. The derivations^{ }also may be employed as a preview of Lagrangian mechanics^{ }before its more formal derivation using variational calculus. ^{ }
One of^{ }us (ST) has successfully employed these derivations and the resulting^{ }Lagrange equations with a small group of talented high school^{ }students. They used the equations to solve problems presented in^{ }the Physics Olympiad. The excitement and enthusiasm of these students^{ }leads us to hope that others will undertake trials with^{ }larger numbers and a greater variety of students. ^{ }
ACKNOWLEDGMENT
The authors would like^{ }to express thanks to an anonymous referee for his or^{ }her valuable criticisms and suggestions, which improved this paper. ^{ }
APPENDIX: EXTENSION TO MULTIPLE DEGREES OF FREEDOM
We^{ }discuss Lagrange's equations for a system with multiple degrees of^{ }freedom, without pausing to discuss the usual conditions assumed in^{ }the derivations, because these can be found in standard advanced^{ }mechanics texts.^{3} ^{ }
Consider a mechanical system described by the following^{ }Lagrangian:
where the q are independent generalized coordinates and the^{ }dot over q indicates a derivative with respect to time.^{ }The subscript s indicates the number of degrees of freedom^{ }of the system. Note that we have generalized to a^{ }Lagrangian that is an explicit function of time t. The^{ }specification of all the values of all the generalized coordinates^{ }q_{i} in Eq. (17) defines a configuration of the system.^{ }The action S summarizes the evolution of the system as^{ }a whole from an initial configuration to a final configuration,^{ }along what might be called a world line through multidimensional^{ }space–time. Symbolically we write:
^{ }
The generalized principle of least action^{ }requires that the value of S be a minimum^{ }for the actual evolution of the system symbolized in^{ }Eq. (18). We make an argument similar to that in^{ }Sec. III for the onedimensional motion of a particle^{ }in a potential. If the principle of least action holds^{ }for the entire world line through the intermediate configurations of^{ }L in Eq. (18), it also holds for an infinitesimal^{ }change in configuration anywhere on this world line. ^{ }
Let the^{ }system pass through three infinitesimally close configurations in the ordered^{ }sequence 1, 2, 3 such that all generalized coordinates remain^{ }fixed except for a single coordinate q at configuration^{ }2. Then the increment of the action from configuration^{ }1 to configuration 3 can be considered to be a^{ }function of the single variable q. As a consequence, for^{ }each of the s degrees of freedom, we can^{ }make an argument formally identical to that carried out^{ }from Eq. (3) through Eq. (9). Repeated s times, once^{ }for each generalized coordinate q_{i}, this derivation leads to s^{ }scalar Lagrange equations that describe the motion of the system:^{ }
The inclusion of time explicitly in the Lagrangian (17) does^{ }not affect these derivations, because the time coordinate is held^{ }fixed in each equation. ^{ }
Suppose that the Lagrangian (17) is^{ }not a function of a given coordinate q_{k}. An argument^{ }similar to that in Sec. IV A tells us that the^{ }corresponding generalized momentum L/_{k} is a constant of the motion.^{ }As a simple example of such a generalized momentum, we^{ }consider the angular momentum of a particle in a central^{ }potential. If we use polar coordinates r, to describe^{ }the motion of a single particle in the plane, then^{ }the Lagrangian has the form L = T–V = m(^{2} + r^{2}^{2})/2–V(r), and the angular momentum^{ }of the system is represented by L/. ^{ }
If the Lagrangian^{ }(17) is not an explicit function of time, then a^{ }derivation formally equivalent to that in Sec. IVB (with time^{ }as the single variable) shows that the function ( _{1}L/_{i})–L, sometimes^{ }called^{12} the energy function h, is a constant of the^{ }motion of the system, which in the simple cases we^{ }cover^{13} can be interpreted as the total energy E of^{ }the system. ^{ }
If the Lagrangian (17) depends explicitly on time,^{ }then this derivation yields the equation dh/dt = –L/t. ^{ }
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.

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 (ButterworthHeinemann, Oxford, 1976), Chap. 1, Sec. 1.
first citation in article

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.
first citation in article

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. first citation in article

Our derivation is a modification of the finite difference technique
employed by Euler in his pathbreaking 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
(SpringerVerlag, 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. first citation in article

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison–Wesley, Reading, MA, 1964), Vol. 2, Chap. 19.
first citation in article

See Ref. 5, p. 198 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). [ISI]
first citation in article

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. first citation in article

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).
first citation in article

Reference 1, Chap. 2 and Ref. 3, Goldstein et al., Sec. 2.7.
first citation in article

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. BobilloAres, "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," ibid. 66 (11), 989–995 (1998).
first citation in article

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). first citation in article

Reference 3, Goldstein et al., Sec. 2.7.
first citation in article

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 in
A. S. de Castro, "Exploring a rheonomic system," Eur. J. Phys. 21, 23–26 (2000) [Inspec]
and C. Ferrario and A. Passerini, "Comment on Exploring a rheonomic system," ibid. 22, L11–L14 (2001). [Inspec] [ISI]
first citation in article
CITING ARTICLES
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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 (5 kB)Fig. 1. An infinitesimal section^{ }of the world line approximated by two straight line segments. First citation in article
Full figure (6 kB)Fig. 2. Derivation^{ }of Lagrange's equations from the principle of least action. Points^{ }1 and 3 are on the true world line. The^{ }world line between them is approximated by two straight line^{ }segments (as in Fig. 1). The arrows show that the^{ }x coordinate of the middle event is varied. All other^{ }coordinates are fixed. First citation in article
Full figure (6 kB)Fig. 3. A derivation showing that the energy is a^{ }constant of the motion. Points 1 and 3 are on^{ }the true world line, which is approximated by two straight^{ }line segments (as in Figs. 1 and 2). The arrows^{ }show that the t coordinate of the middle event is^{ }varied. All other coordinates are fixed. First citation in article
FOOTNOTES
^{a}Electronic mail:^{ }jozef.hanc@tuke.sk
^{b}Electronic mail: eftaylor@mit.edu; http://www.eftaylor.com
^{c}Electronic mail: tuleja@stonline.sk
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