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Deriving Lagrange’s equations using elementary calculus
a)
Jozef Hanc
Technical University, Vysokoskolska 4, 042 00 Kosice, Slovakia
b)
Edwin F. Taylor
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
c)
Slavomir Tuleja
Gymnazium arm. gen. L. Svobodu, Komenskeho 4, 066 51 Humenne, Slovakia
~Received 30 December 2002; accepted 20 June 2003!
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. © 2004 American Association of Physics Teachers.
@DOI: 10.1119/1.1603270#
I. INTRODUCTION The action S along a world line is defined as
1
The equations of motion of a mechanical system can be S5E L~x,v!dt. ~2!
derived by two different mathematical methods—vectorial along the
and analytical. Traditionally, introductory mechanics begins world line
with Newton’s laws of motion which relate the force, mo- The principle of least action requires that between a fixed
mentum, and acceleration vectors. But we frequently need to initial event and a fixed final event the particle follow a
describe systems, for example, systems subject to constraints world line such that the action S is a minimum.
without friction, for which the use of vector forces is cum- The action S is an additive scalar quantity, and is the sum
bersome. Analytical mechanics in the form of the Lagrange of contributions LDt from each segment along the entire
equations provides an alternative and very powerful tool for world line between two events fixed in space and time. Be-
obtaining the equations of motion. Lagrange’s equations em- cause S is additive, it follows that the principle of least action
ploy a single scalar function, and there are no annoying vec- must hold for each individual infinitesimal segment of the
6
tor components or associated trigonometric manipulations. world line. This property allows us to pass from the integral
Moreover, the analytical approach using Lagrange’s equa- equation for the principle of least action, Eq. ~2!,to
2 Lagrange’s differential equation, valid anywhere along the
tions provides other capabilities that allow us to analyze a
wider range of systems than Newton’s second law. world line. It also allows us to use elementary calculus in
The derivation of Lagrange’s equations in advanced me- this derivation.
3 We approximate a small section of the world line by two
chanics texts typically applies the calculus of variations to
the principle of least action. The calculus of variation be- straight-line segments connected in the middle ~Fig. 1! and
longs to important branches of mathematics, but is not make the following approximations: The average position
widely taught or used at the college level. Students often coordinate in the Lagrangian along a segment is at the mid-
7
encounter the variational calculus first in an advanced me- point of the segment. The average velocity of the particle is
chanics class, where they struggle to apply a new mathemati- equal to its displacement across the segment divided by the
cal procedure to a new physical concept. This paper provides time interval of the segment. These approximations applied
a derivation of Lagrange’s equations from the principle of to segment A in Fig. 1 yield the average Lagrangian LA and
4
least action using elementary calculus, which may be em- action SA contributed by this segment:
ployed as an alternative to ~or a preview of! the more ad- x 1x x 2x
vanced variational calculus derivation. L [LS 1 2 , 2 1D, ~3a!
In Sec. II we develop the mathematical background for A 2 Dt
deriving Lagrange’s equations from elementary calculus.
Section III gives the derivation of the equations of motion S 'L Dt5LSx11x2,x22x1DDt, ~3b!
for a single particle. Section IV extends our approach to A A 2 Dt
demonstrate that the energy and momentum are constants of with similar expressions for L and S along segment B.
the motion. The Appendix expands Lagrange’s equations to B B
multiparticle systems and adds angular momentum as an ex-
ample of generalized momentum.
III. DERIVATION OFLAGRANGE’SEQUATION
II. DIFFERENTIALAPPROXIMATION TO THE We employ the approximations of Sec. II to derive
PRINCIPLE OFLEASTACTION Lagrange’s equations for the special case introduced there.
A particle moves along the x axis with potential energy As shown in Fig. 2, we fix events 1 and 3 and vary the x
V(x) which is time independent. For this special case the coordinate of the intermediate event to minimize the action
5 between the outer two events.
Lagrange function or Lagrangian L has the form: For simplicity, but without loss of generality, we choose
L~x,v!5T2V51mv22V~x!. ~1! the time increment Dt to be the same for each segment,
2
510 Am. J. Phys. 72 ~4!, April 2004 http://aapt.org/ajp ©2004AmericanAssociation of Physics Teachers 510
S 5L Dt1L Dt. ~6!
AB A B
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
8
derivative of SAB with respect to x equal to zero and use the
chain rule, we obtain
dS ]L dx ]L dv ]L dx
AB505 A A Dt1 A A Dt1 B B Dt
dx ]xA dx ]vA dx ]xB dx
1]LB dvBDt. ~7!
]vB dx
Wesubstitute Eq. ~5! into Eq. ~7!, divide through by Dt, and
regroup the terms to obtain
Fig. 1. An infinitesimal section of the world line approximated by two 1 S ]LA1]LBD2 1 S]LB2]LAD50. ~8!
straight line segments. 2 ]x ]x Dt ]v ]v
A B B A
To first order, the first term in Eq. ~8! is the average value
which also equals the time between the midpoints of the two of ]L/]x on the two segments A and B. In the limit Dt
segments. The average positions and velocities along seg- →0, this term approaches the value of the partial derivative
ments A and B are at x. In the same limit, the second term in Eq. ~8! becomes
x 1x x2x the time derivative of the partial derivative of the Lagrangian
x 5 1 , v 5 1 , ~4a! with respect to velocity d(]L/]v)/dt. Therefore in the limit
A 2 A Dt Dt→0, Eq. ~8! becomes the Lagrange equation in x:
x 5x1x3, v 5x32x. ~4b! ]L d ]L
B 2 B Dt ]x2dtS]vD50. ~9!
The expressions in Eq. ~4! are all functions of the single We did not specify the location of segments A and B along
variable x. For later use we take the derivatives of Eq. ~4! the world line. The additive property of the action implies
with respect to x: that Eq. ~9! is valid for every adjacent pair of segments.
dxA 1 dvA 1 An essentially identical derivation applies to any particle
dx 52, dx 51Dt, ~5a! with one degree of freedom in any potential. For example,
the single angle w tracks the motion of a simple pendulum,
dx 1 dv 1 so its equation of motion follows from Eq. ~9! by replacing x
B5 , B52 . ~5b! with w without the need to take vector components.
dx 2 dx Dt
Let LA and LB be the values of the Lagrangian on segments IV. MOMENTUMANDENERGYASCONSTANTS
A and B, respectively, using Eq. ~4!, and label the summed OFTHEMOTION
action across these two segments as SAB:
A. Momentum
We consider the case in which the Lagrangian does not
depend explicitly on the x coordinate of the particle ~for ex-
ample, the potential is zero or independent of position!. Be-
cause it does not appear in the Lagrangian, the x coordinate
is ‘‘ignorable’’ or ‘‘cyclic.’’ In this case a simple and well-
knownconclusion from Lagrange’s equation leads to the mo-
mentum as a conserved quantity, that is, a constant of mo-
tion. Here we provide an outline of the derivation.
For a Lagrangian that is only a function of the velocity,
L5L(v), Lagrange’s equation ~9! tells us that the time de-
rivative of ]L/]v is zero. From Eq. ~1!, we find that
]L/]v5mv, whichimplies that the x momentum, p5mv,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
L5L(v) ~perhaps as a student exercise to reinforce under-
Fig. 2. Derivation of Lagrange’s equations from the principle of least action. standing of the previous derivation!, we obtain from the prin-
Points 1 and 3 are on the true world line. The world line between them is ciple of least action
approximated by two straight line segments ~as in Fig. 1!. The arrows show dS ]L dv ]L dv
that the x coordinate of the middle event is varied. All other coordinates are AB505 A A Dt1 B B Dt. ~10!
fixed. dx ]vA dx ]vB dx
511 Am. J. Phys., Vol. 72, No. 4, April 2004 Hanc, Taylor, and Tuleja 511
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!#,
S 5L ~t2t !1L ~t 2t!. ~14!
AB A 1 B 3
Note that students sometimes misinterpret the time differ-
ences 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 SAB equal to zero:
dS ]L dv ]L dv
AB505 A A ~t2t !1L 1 B B ~t 2t!2L .
dt ]v dt 1 A ]v dt 3 B
A B ~15!
Fig. 3. A derivation showing that the energy is a constant of the motion. If we substitute Eq. ~13! into Eq. ~15! and rearrange the
Points 1 and 3 are on the true world line, which is approximated by two result, we find
straight line segments ~as in Figs. 1 and 2!. The arrows show that the t ]L ]L
coordinate of the middle event is varied. All other coordinates are fixed. A v 2L 5 B v 2L . ~16!
]vA A A ]vB B B
Wesubstitute Eq. ~5! into Eq. ~10! and rearrange the terms to Because the action is additive, Eq. ~16! is valid for every
find: segment of the world line and identifies the function
]L ]L v]L/]v2L as a constant of the motion. By substituting Eq.
A5 B ~1! for the Lagrangian into v]L/]v2L and carrying out the
]vA ]vB partial derivatives, we can show that the constant of the mo-
or tion corresponds to the total energy E5T1V.
pA5pB. ~11!
Again we can use the arbitrary location of segments A and B V. SUMMARY
along the world line to conclude that the momentum p is a Our derivation and the extension to multiple degrees of
constant of the motion everywhere on the world line. freedom in the Appendix allow the introduction of
Lagrange’s equations and its connection to the principle of
B. Energy least action without the apparatus of the calculus of varia-
9 tions. The derivations also may be employed as a preview of
Standard texts obtain conservation of energy by examin- Lagrangian mechanics before its more formal derivation us-
ing the time derivative of a Lagrangian that does not depend ing variational calculus.
explicitly on time. As pointed out in Ref. 9, this lack of One of us ~ST! has successfully employed these deriva-
dependence of the Lagrangian implies the homogeneity of tions and the resulting Lagrange equations with a small
time: temporal translation has no influence on the form of the group of talented high school students. They used the equa-
Lagrangian. Thus conservation of energy is closely con- tions to solve problems presented in the Physics Olympiad.
10
nected to the symmetry properties of nature. As we will The excitement and enthusiasm of these students leads us to
see, our elementary calculus approach offers an alternative hope that others will undertake trials with larger numbers
11
way to derive energy conservation. and a greater variety of students.
Consider a particle in a time-independent 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 mini- ACKNOWLEDGMENT
mize the action. The authors would like to express thanks to an anonymous
For simplicity, we choose the x increments to be equal, referee for his or her valuable criticisms and suggestions,
with the value Dx. We keep the spatial coordinates of all which improved this paper.
three events fixed while varying the time coordinate of the
middle event and obtain
v 5 Dx , v 5 Dx . ~12! APPENDIX: EXTENSION TO MULTIPLE DEGREES
A t2t1 B t32t OFFREEDOM
These expressions are functions of the single variable t, with We discuss Lagrange’s equations for a system with mul-
respect to which we take the derivatives tiple degrees of freedom, without pausing to discuss the
usual conditions assumed in the derivations, because these
dvA Dx vA 3
52 52 , ~13a! can be found in standard advanced mechanics texts.
dt ~t2t1!2 t2t1 Consider a mechanical system described by the following
and Lagrangian:
˙ ˙ ˙
L5L~q1,q2,...,qs,q1,q2,...,qs,t!, ~17!
dvB5 Dx 5 vB . ~13b! where the q are independent generalized coordinates and the
dt ~t32t!2 t32t dot over q indicates a derivative with respect to time. The
512 Am. J. Phys., Vol. 72, No. 4, April 2004 Hanc, Taylor, and Tuleja 512
subscript s indicates the number of degrees of freedom of the a!Electronic mail: jozef.hanc@tuke.sk
system. Note that we have generalized to a Lagrangian that is b!Electronic mail: eftaylor@mit.edu; http://www.eftaylor.com
an explicit function of time t. The specification of all the c!Electronic mail: tuleja@stonline.sk
values of all the generalized coordinates q in Eq. ~17! de- 1We take ‘‘equations of motion’’ to mean relations between the accelera-
i tions, velocities, and coordinates of a mechanical system. See L. D. Lan-
fines a configuration of the system. The action S summarizes dau and E. M. Lifshitz, Mechanics ~Butterworth-Heinemann, Oxford,
the evolution of the system as a whole from an initial con- 1976!, Chap. 1, Sec. 1.
figuration to a final configuration, along what might be called 2Besides its expression in scalar quantities ~such as kinetic and potential
a world line through multidimensional space–time. Symboli- energy!, Lagrangian quantities lead to the reduction of dimensionality of a
cally we write: problem, employ the invariance of the equations under point transforma-
tions, and lead directly to constants of the motion using Noether’s theo-
rem. More detailed explanation of these features, with a comparison of
˙ ˙ ˙ analytical mechanics to vectorial mechanics, can be found in Cornelius
S5Einitialconfiguration L~q1,q2...qs ,q1,q2...qs ,t!dt.
to final configuration Lanczos, The Variational Principles of Mechanics ~Dover, New York,
~18! 1986!, pp. xxi–xxix.
The generalized principle of least action requires that 3Chapter 1 in Ref. 1 and Chap. V in Ref. 2; Gerald J. Sussman and Jack
the value of S be a minimum for the actual evolution of Wisdom, Structure and Interpretation of Classical Mechanics ~MIT, Cam-
the system symbolized in Eq. ~18!. We make an argu- bridge, 2001!, Chap. 1; Herbert Goldstein, Charles Poole, and John Safko,
Classical Mechanics ~Addison–Wesley, Reading, MA, 2002!, 3rd ed.,
ment similar to that in Sec. III for the one-dimensional Chap. 2. An alternative method derives Lagrange’s equations from
motion of a particle in a potential. If the principle of least D’Alambert principle; see Goldstein, Sec. 1.4.
action holds for the entire world line through the inter- 4Our derivation is a modification of the finite difference technique em-
mediate configurations of L in Eq. ~18!, it also holds for an ployed by Euler in his path-breaking 1744 work, ‘‘The method of finding
infinitesimal change in configuration anywhere on this world plane curves that show some property of maximum and minimum.’’Com-
line. plete references and a description of Euler’s original treatment can be
Let the system pass through three infinitesimally close found in Herman H. Goldstine, A History of the Calculus of Variations
from the 17th Through the 19th Century ~Springer-Verlag, New York,
configurations in the ordered sequence 1, 2, 3 such that all 1980!, Chap. 2. Cornelius Lanczos ~Ref. 2, pp. 49–54! presents an abbre-
generalized coordinates remain fixed except for a single viated version of Euler’s original derivation using contemporary math-
coordinate q at configuration 2. Then the increment of ematical notation.
the action from configuration 1 to configuration 3 can be 5R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on
Physics ~Addison–Wesley, Reading, MA, 1964!, Vol. 2, Chap. 19.
considered to be a function of the single variable q.Asa 6See Ref. 5, p. 19-8 or in more detail, J. Hanc, S. Tuleja, and M. Hancova,
consequence, for each of the s degrees of freedom, we ‘‘Simple derivation of Newtonian mechanics from the principle of least
can make an argument formally identical to that carried action,’’ Am. J. Phys. 71 ~4!, 386–391 ~2003!.
out from Eq. ~3! through Eq. ~9!. Repeated s times, once for 7There is no particular reason to use the midpoint of the segment in the
each generalized coordinate qi, this derivation leads to s Lagrangian of Eq. ~2!. In Riemann integrals we can use any point on the
scalar Lagrange equations that describe the motion of the given segment. For example, all our results will be the same if we used the
system: 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
]L d ]L test student understanding of the derivations given here.
2 S D50 ~i51,2,3,...,s!. ~19! 8A zero value of the derivative most often leads to the world line of mini-
˙
]qi dt ]qi mum action. It is possible also to have a zero derivative at an inflection
The inclusion of time explicitly in the Lagrangian ~17! does point or saddle point in the action ~or the multidimensional equivalent in
not affect these derivations, because the time coordinate is configuration space!. So the most general term for our basic law is the
held fixed in each equation. 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
Suppose that the Lagrangian ~17! is not a function of a Variations ~Prentice–Hall, Englewood Cliffs, NJ, 1963!.
given coordinate q . An argument similar to that in Sec. 9Reference 1, Chap. 2 and Ref. 3, Goldstein et al., Sec. 2.7.
k 10The most fundamental justification of conservation laws comes from sym-
IVAtells us that the corresponding generalized momentum metry properties of nature as described by Noether’s theorem. Hence en-
˙
]L/]qk is a constant of the motion. As a simple example of ergy conservation can be derived from the invariance of the action by
such a generalized momentum, we consider the angular mo- temporal translation and conservation of momentum from invariance un-
mentum of a particle in a central potential. If we use polar der space translation. See N. C. Bobillo-Ares, ‘‘Noether’s theorem in dis-
coordinates r, u to describe the motion of a single particle in crete classical mechanics,’’Am. J. Phys. 56 ~2!, 174–177 ~1988! or C. M.
the plane, then the Lagrangian has the form L5T2V Giordano and A. R. Plastino, ‘‘Noether’s theorem, rotating potentials, and
2 2˙2 Jacobi’s integral of motion,’’ ibid. 66 ~11!, 989–995 ~1998!.
˙ 11
5m(r 1r u )/22V(r), and the angular momentum of the 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, ‘‘Sym-
system is represented by ]L/]u. metries and conservation laws: Consequences of Noether’s theorem,’’Am.
If the Lagrangian ~17! is not an explicit function of time, J. Phys. ~to be published!.
then a derivation formally equivalent to that in Sec. IVB 12Reference 3, Goldstein et al., Sec. 2.7.
~with time as the single variable! shows that the function 13For the case of generalized coordinates, the energy function h is generally
˙ ˙ 12 not the same as the total energy. The conditions for conservation of the
((q1]L/]qi)2L, sometimes called the energy function h,
is a constant of the motion of the system, which in the simple energy function h are distinct from those that identify h as the total energy.
13 For a detailed discussion see Ref. 12. Pedagogically useful comments on a
cases we cover can be interpreted as the total energy E of particular example can be found in A. S. de Castro, ‘‘Exploring a rheon-
the system. omic system,’’ Eur. J. Phys. 21, 23–26 ~2000! and C. Ferrario and A.
If the Lagrangian ~17! depends explicitly on time, then Passerini, ‘‘Comment on Exploring a rheonomic system,’’ ibid. 22,L11–
this derivation yields the equation dh/dt52]L/]t. L14 ~2001!.
513 Am. J. Phys., Vol. 72, No. 4, April 2004 Hanc, Taylor, and Tuleja 513
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