1

From Conservation of Energy to the Principle of Least Action: A Story Line12

Jozef Hanca)3Technical University, Vysokoskolska 4, 042 00 Kosice, Slovakia4

5Edwin F. Taylorb)6

Massachusetts Institute of Technology, Cambridge, MA 0213978

ABSTRACT9A schematic story line outlines an introduction to Newtonian mechanics that starts by10employing the conservation of energy to predict the motion of a particle in a one-dimensional11potential. Incorporating constraints and constants of the motion into the energy expression12allows analysis of some more complicated systems. A heuristic transition embeds kinetic and13potential energy in the still more powerful principle of least action and Lagrange's equations.14

15NOTES TO EDITORS AND REFEREES:161. Preprints of some papers referenced here but not yet published and some papers published17elsewhere are available at the website http://www.eftaylor.com/leastaction.html18

192. Comments and suggestions can conveniently be indexed by page and line number in this20pdf file.21

22I. INTRODUCTION23

24It is sometimes claimed that nature has a "purpose", in that it seeks to take the path that produces the25minimum action. . . . In fact [according to Feynman's version of quantum mechanics] nature does26exactly the opposite. It takes every path, treating them all on [an] equal footing. We simply end up27seeing the path with the stationary action, due to the way the quantum mechanical phases add. It28would be a harsh requirement, indeed, to demand that nature make a "global" decision (that is, to29compare paths that are separated by large distances), and to choose the one with the smallest action.30Instead, we see that everything takes place on a "local” scale. Nearby phases simply add and31everything works out automatically.32

33 — David Morin134

35Here is a nine-sentence history of Newtonian mechanics:2 In the second half of the 1600s36Newton proposed his three laws of motion,3 including what has become F = ma. In the mid-371700s Leonhard Euler devised and applied one version of the principle of least action using38mostly geometrical methods. On 12 August 1755 the 19-year-old Ludovico de la Grange39Tournier of Turin (whom we call Joseph Louis Lagrange) sent Euler a letter with a40mathematical attachment that streamlined Euler's methods into algebraic form. "[A]fter41seeing Lagrange's work Euler dropped his own method, espoused that of Lagrange, and42renamed the subject the calculus of variations."4 Lagrange, in his path-breaking 1788 Analytical43Mechanics,5 introduced what we call the Lagrangian function and Lagrange's equations of44motion. Almost half a century later (1834-35) William Rowan Hamilton6 developed45

2

Hamilton's principle, to which Landau and Lifshitz7 and Feynman8 more recently reassigned1the name principle of least action.9 Between 1840 and 1860 the conservation of energy was2established in all its generality.10 In 1918 Emma Noether11 proved relations between3symmetries and conserved quantities. Finally, in 1949 Richard Feynman12 devised the4formulation of quantum mechanics that not only explicitly underpins the principle of least5action but also demarcates the limits of validity of Newtonian mechanics.6

7We have proposed13 that the principle of least action and Lagrange's equations become the8basis of introductory Newtonian mechanics. Three recent articles in this Journal demonstrate9how to use elementary calculus to derive from the principle of least action (1) Newton's laws10of motion,14 (2) Lagrange's equations,15 and (3) examples of Noether's theorem.1611

12How are these concepts and methods to be introduced to students? In the present paper we13suggest a reversal of the historical order: Begin with conservation of energy and graduate to14the principle of least action followed by Lagrange's equations.15

16We start by using the conservation of energy to analyze particle motion in a one-dimensional17potential. Much of the power of the principle of least action and its logical offspring18Lagrange's equations results from the fact that they are based on energy, a scalar. When we19start with the conservation of energy, we not only preview more advanced concepts and20procedures but also invoke some of their power. For example, expressions for energy21conforming to constraints automatically eliminate corresponding forces of constraint from22the analysis of motion. Applying a limited version of Noether's theorem expressed in terms23of energy identifies constants of the motion. Using constraints and constants of the motion24permits us to reduce to one coordinate the description of some important multi-dimensional25systems, whose motion can then be solved completely using the conservation of energy.26Equilibrium and statics also derive from the conservation of energy.27

28Prerequisites for the proposed introduction include elementary trigonometry, polar29coordinates, introductory differential calculus and partial derivatives, and the idea of the30integral as a sum of increments. In preparation for later formalism, we symbolize the time31derivative with a dot over the variable.32

33The story line presented in this article omits most details and is offered for discussion,34correction, and elaboration. The authors have no present intent to write an introductory text35on Newtonian mechanics that follows the story line of this article.36

37II. ENERGY CONSERVATION38Begin with a discursive description of the forms of energy and the law of global conservation39of energy. Introductory treatments of the conservation of energy and conversion among its40various forms are available in the literature.17, 18, 19 How far one goes into engineering41applications of energy and its environmental consequences is a matter of choice.20, 21 In42addition to expressions for kinetic and potential energy of a particle, this introduction should43include angular velocity, moment of inertia, and the kinetic energy of a rigid body rotating44about a fixed or parallel-moving axis.45

3

1III. ONE-DIMENSIONAL MOTION: ANALYTIC SOLUTIONS2Narrow the focus to conservation of mechanical energy of a particle moving in a one-3dimensional potential. The complete description of such one-dimensional motion follows4from the conservation of energy. Unfortunately, an explicit function of position vs time can be5derived for only a small fraction of such systems. In other cases one is encouraged to guess6the analytic solution, as illustrated in the following example. Any proposed analytic solution7is easily checked by substitution into the energy equation. Heuristic guesses are assisted by8the fact that the first time derivative of position, not the second, appears in the energy9conservation equation. The following example introduces the potential energy diagram, a10central tool in our story line and important in later study of advanced mechanics, quantum11mechanics, general relativity, and many other subjects.12

13Harmonic oscillator14

Energy

Position

Total energy of particle

Potentialenergy

A B C D E F

Kineticenergy

15Figure 1. The potential energy diagram is central to our treatment. Sample student exercise: For16the value of the energy shown, describe the motion of the particle qualitatively but in detail.17Compare the magnitudes and directions of each of the following quantities that describe the18particle motion at different positions marked A through F: velocity, force, acceleration. In order to19describe the motion, what initial condition(s) must be specified in addition to an initial starting20point, and are these conditions required for all starting points? How do answers to these21questions change for different values of the total energy?22

23Qualitative analysis of the parabolic potential energy diagram (or indeed any diagram with24motion bounded near a single potential energy minimum) tells us that the motion will be25periodic. For the parabolic potential the conservation of energy is written26

27

4

E =12

m˙ x 2 +12

kx 2 (1)1

2Rearrange Eq. (1) to read:3

4m

2E˙ x 2 =1−

k2E

x 2 (2)5

6Recalling the periodicity of motion, we are reminded of trigonometric identity7

8cos2θ =1− sin2θ (3)9

10Set θ = ωt and equate corresponding terms in Eqs. (2) and (3), choosing the positive root,11which yields:12

x =2Ek

1/ 2

sinωt (4)13

14Take the time derivative of this expression and compare its square with the left side of Eq.15(2). The result tells us that2216

17

ω =km

1/ 2

(5)18

19The fact that ω does not depend on the amplitude of the displacement function (4) means that20the period is the same whatever the value of the total energy E. The simple harmonic21oscillator has wide application because a large number of potential energy curves can be22approximated as parabolas near their minima.23

24IV. ACCELERATION AND FORCE25In the absence of dissipation, force can be defined in terms of energy. In the simple harmonic26oscillator example, take the time derivative of both sides of (1) (for constant energy E and27mass m).28

290 = m˙ x ̇ ̇ x + kx˙ x (6)30

31or32

33

˙ ̇ x = −km

x (7)3435

Define force as m times the acceleration. Then the simple harmonic force is –kx. More36generally, take the time derivative of both sides of the conservation of energy equation for a37particle moving in a one-dimensional potential:38

39

0 = m˙ x ̇ ̇ x +dU x( )

dt= m˙ x ̇ ̇ x +

dU x( )dx

dxdt

= m˙ x ̇ ̇ x +dU x( )

dx˙ x (8)40

5

1from which2

3

m˙ ̇ x ≡ F = −dU x( )

dx(9)4

5Thus for motion without dissipation force is defined in terms of energy.6

7V. NON-INTEGRABLE MOTIONS IN ONE DIMENSION8We cannot solve most one-dimensional motions in closed form, as we did for the simple9harmonic oscillator. Our strategy will be to ask the student to begin every solution with a10qualitatively analysis using the potential energy diagram (as exemplified in the caption of11Fig. 1), then to complete the exact solution employing an interactive computer display (Fig.122). Such a computer display, yet to be programmed but easily specified, numerically13integrates the particle motion in a given potential. On this display the student sets up initial14conditions by dragging the horizontal energy line up or down and the position dot left or15right, both in the classically permitted region (lower panel of Fig. 2). The computer then16moves the dot back and forth along the E-line at a rate proportional to that at which the17particle will move, while simultaneously drawing the worldline (upper panel of Fig. 2).18

6

Position

E

U

time

Energy

Position

1Figure 2. Interactive computer display of the worldline derived numerically from the initial2energy and the potential energy function.3

4VI. REDUCTION TO ONE COORDINATE5The analysis of one-dimensional motion using the conservation of energy is powerful but6limited. In some important cases we can use constraints and constants of the motion to7reduce the description of a wider class of motions to one coordinate. To these we apply our8standard procedure: qualitative analysis using the diagram of the (effective) potential energy9followed by interactive computer solution. Here are some examples.10

11(a) Object rolling without slipping12A marble with mass m, radius r, and moment of inertia Imarble rolls without slipping along a13curved ramp that lies in the xy plane. A uniform gravitational acceleration g acts in the14negative y-direction. The non-slip constraint tells us that v = rω . Conservation of energy leads15to the expression:16

17

7

E =12mv2 +

12Imarbleω

2 + mgy =12mv 2 +

12Imarble

vr

2

+ mgy

=12m +

Imarble

r2

v 2 + mgy =

12Mv2 + mgy

(10)1

2where3

M = m +Imarbler2

(11)4

5Constraints are used twice in this example: explicitly in rolling without slipping and6implicitly in the relation between height y and displacement along the curve.7

8(b) Projectile motion9For projectile motion in a vertical plane subject to a vertical uniform vertical gravitational10field, the total energy is11

E =12

m˙ x 2 +12

m˙ y 2 + mgy (12)1213

The potential energy is not a function of x; therefore, as shown below, x-momentum px is a14constant of the motion. The energy equation becomes:15

16

E =12

m˙ y 2 +px

2

2m+ mgy (13)17

18In Newtonian mechanics the zero-point of energy is arbitrary, so we can reduce the analysis19to one dimension by making the substitution:20

21

E '= E −px

2

2m=

12

m˙ y 2 + mgy (14)22

23In analyzing projectile motion, we have already applied a limited version of the powerful24Noether's theorem.23 The special version of Noether's theorem used here, expressed in terms25of energy, says that when the total energy E is not an explicit function of an independent26coordinate, x for example, then the function ∂E ∂˙ x is a constant of the motion.24 In some27important cases we can use the resulting constants of the motion, together with constraints,28to describe some two- and even three-dimensional motions with just one coordinate.29Projectile motion, above, is one example. Here is another.30

31

8

(c) Marble, ramp, and turntable1

x

φ

θ

m

23

Figure 3. System of marble, ramp, and turntable.4[NOTE: This figure will be redrawn professionally after any referee suggestions.]5

6A marble of mass m and moment of inertia Imarble rolls without slipping along a slot on an7inclined ramp fixed rigidly to a turntable which rotates freely so that its angular velocity is8not necessarily constant (Fig. 3). The moment of inertia of the combined turntable and ramp9is Irot. Assuming that the marble stays on the ramp, find its position x as a function of time.10

11Start with a qualitative analysis, assuming the marble initially moves outward along the12ramp. This outward rolling will slow down the rotation of the turntable, perhaps leading the13marble to reverse direction. One possibility is oscillatory motion of the marble along the14ramp; another possibility is an equilibrium location of the marble on the ramp with simple15circular motion of the marble around the axis of the turntable. Now the analysis. The square16of the velocity of the marble is:17

18v 2 = ˙ x 2 + ˙ φ 2x 2 cos2 θ (15)19

20Conservation of energy yields the equation21

22

E =12

M˙ x 2 +12

Irot˙ φ 2 +

12

Mx 2 ˙ φ 2 cos2 θ + mgx sinθ (16)23

24Here M indicates the marble's mass augmented by the energy effects of its rotation, Eq. (11).25The expression (16) is not an explicit function of the angle of rotation φ. Therefore our26version of Noether's theorem tells us that ∂E ∂ ˙ φ is a constant of the motion, which we27recognize as the total angular momentum J:28

29∂E∂ ˙ φ

= Irot + Mx 2 cos2 θ( ) ˙ φ = J (17)30

31

9

Substitute for ˙ φ from Eq. (17) into (16):12

E =12

M˙ x 2 +J 2

2 Irot + Mx 2 cos2 θ( )+ mgx sinθ =

12

M˙ x 2 + Ueff x( ) (18)3

4Values of E and J are determined by initial conditions. The effective potential energy Ueff x( )5can have a minimum as a function of x. If the marble starts at rest at the position of this6minimum, it will move in a circle. The general motion of this system is difficult to analyze7using F = ma.8

9(d) Motion in a central potential10Analyze satellite motion in a central inverse-square gravitational field. Choose coordinates r11and φ in the plane of the orbit. Then the expression for total energy is:12

13

E =12

mv2 −GMm

r=

12

m ˙ r 2 + r2 ˙ φ 2( ) −GMm

r(19)14

15The angle φ does not appear in this equation. Therefore we expect that a constant of the16motion is given by ∂E ∂ ˙ φ , which is the angular momentum J:17

18∂E∂ ˙ φ

= mr2 ˙ φ = J (20)19

20Substitute the resulting expression for ˙ φ in Eq. (20) into the energy equation (19).21

22

E =12

m˙ r 2 +J 2

2mr2 −GMm

r=

12

m˙ r 2 + Ueff r( ) (21)23

24The student employs the plot of effective potential energy to carry out the usual qualitative25analysis of radial motion followed by computer integration. Emphasize the distinction26between bound orbits, unbound orbits, and the intermediate limiting case. With additional27use of Eq. (20), the computer can plot the trajectory in the plane.28

29VII. EQUILIBRIUM AND STATICS30Friction is an important feature of motion in everyday life, but it is complicated to analyze.2531Frictional forces are often ignored in advanced mechanics texts.26 The tendency of a system32toward increased entropy and the statistical mechanics and thermodynamics that account for33this tendency are not part of our story line here. Nevertheless, in common experience motion34usually slows down and stops. Our use of potential energy diagrams makes straightforward35the formulation of "stopping" as a tendency of systems to reach equilibrium at a local36minimum of the potential energy. (In principle the system could come to equilibrium37perched at a maximum of the potential energy, or even at a zero-slope inflection point.) In the38absence of dissipative processes, equilibrium is really an application of conservation of39energy; a system released from rest in a configuration at a minimal (or stationary) value of40the potential energy will remain at rest. (Proof: An infinitesimal displacement results in zero41

10

change in potential energy. Because of energy conservation, the change in kinetic energy1must also remain zero. Since the system is initially at rest, the zero change in kinetic energy2forbids displacement.)3

4Here, as usual, Feynman is ahead of us. Figure 4 shows an example from his treatment of5statics.27 The problem is to find the value of the weight W that keeps the structure at rest.6Using units of feet, inches, and pounds, Feynman balances the decrease in potential energy7when the weight W drops 4 inches with increases in potential energy for the corresponding82-inch rise of the 60 pound weight plus the 1-inch rise in the 100 pound weight. He demands9that the net potential energy change of the system be zero, yielding10

11

12Figure 4. Feynman's example of13the principle of virtual work14

15−4 inch( )W + (2 inch)(60 pounds) + (1 inch)(100 pounds) = 0 (22)16

17or W = 55 pounds. This method, known formally as the principle of virtual work, is simply an18application of the conservation of energy.19

20There are many examples of the principle of virtual work, including a mass hanging on a21spring, the lever, hydrostatic balance, uniform chain suspended at both ends, vertically22hanging "slinky,"28 and a ball perched on top of a large sphere, not to mention every static23engineering and natural structure in the universe.24

25VIII. FREE PARTICLE. PRINCIPLE OF LEAST AVERAGE KINETIC ENERGY26The conservation of energy is limited in predicting the motion of mechanical systems. Now27we begin the transition to the more powerful principle of least action by looking at the time28average of the kinetic energy of a free particle, a particle that moves in a region of zero29potential energy or potential energy that has the same value everywhere in the region. In this30and the following sections we seek a principle that is more fundamental than the31conservation of energy. We will test any fundamental principle we uncover by demanding32that, at least, it lead back to the conservation of energy. For this reason the trial worldlines we33explore hereafter do not necessarily satisfy conservation of energy.34

11

t

ttotal

A fixed

C fixed

x

B

1

2ttotal – t

d 2d12

Figure 5. Varying the time of the middle event to determine the path3of minimum time-averaged kinetic energy.4

5The solid line in Fig. 5 shows the trial worldline of a particle that moves freely between fixed6events A and C. We know that a free particle travels with constant velocity, that is, along a7straight worldline. Verify that this is the actual motion by examining a trial worldline for8which the central event B in the worldline is displaced in time (and therefore kinetic energy is9not the same on legs 1 and 2). Whether the potential energy is zero or uniform in this region,10its contribution to the time average energy does not change with the time-displacement of11event B, so we ignore the potential energy. Let K represent the kinetic energy. Then the time12average of the kinetic energy along the two segments 1 and 2 of the worldline is given by the13expression.14

15

Kaverage =K1t1 + K2t2

ttotal=1ttotal

12mv1

2t1 +12mv2

2t2

(23)16

17Multiply through by the fixed total time and expand the velocity expressions using the18notation displayed in Fig. 5:19

20

Kaveragettotal =12m d2

t 2 t +12m d2

ttotal − t( )2 ttotal − t( )

=12m d2

t+

12m d2

t total − t( )

(24)21

22

12

Find the minimum value of the average kinetic energy by taking the derivative with respect1to the time t of the central event and setting the result equal to zero:2

3dKaverage

dt

t total = −

12m d2

t 2+12m d2

t total − t( )2= −

12mv1

2 +12mv2

2 = 0 (25)4

5The right-hand equation leads to the equality:6

712mv2

2 =12mv1

2 (26)8

9As expected, when the time-average kinetic energy has a minimum value the kinetic energy10for a free particle is the same on both segments; the worldline is straight between the fixed11initial and final events A and C. In summary, we have illustrated the fact that for the special12case of a particle moving in a region of zero (or uniform) potential energy, kinetic energy is13conserved if we demand that the time average of the kinetic energy has a minimum value.14The general expression for this average is:15

16

Kaverage =1ttotal

Kdt∫ (27)17

18In an introductory text one might insert at this point a sidebar on Fermat's principle of least19time for the propagation of light rays, which predicts every central feature of ray optics. Now20we are ready to develop a similar law that predicts every central feature of mechanics.21

22IX. PRINCIPLE OF LEAST ACTION23Section VIII examined the principle of least average kinetic energy, which predicts the motion24of a particle moving in a region of zero (or uniform) potential energy. Along the way we25mentioned Fermat's principle of least time that describes ray optics. Now we move on to the26more general principle which includes the principle of least average kinetic energy as a27special case. Can we find some quantity whose time average value is a minimum when both28kinetic energy and potential energy are functions of position?29

30Let K represent kinetic energy and U represent potential energy. One might guess31(incorrectly) that the time averaged total K + U would have a minimum value between fixed32initial and final events. (Remember: In calculating the value of this average we do not assume33the conservation of total energy, but rather demand that energy conservation emerge as one34result of our analysis.)35

36It turns out that the time averaged difference K – U between kinetic and potential energy has a37minimum value for the path taken by the particle. How can this be? To sort this out, think of38a baseball pitched in a uniform gravitational field, with the two events of pitch and catch39fixed in both location and time (Fig. 6). How will the ball move between these two fixed40events?41

42

13

pitch catchB

A

C

D

E

x

y

K −U( ) avg minimum

K +U( )avg minimum?12

Figure 6. Alternative trial trajectories of a pitched baseball.34

To begin with, forget about averages and ask why the baseball does not simply move steadily5along the straight horizontal trajectory B in the figure. Moving from pitch to catch with6constant kinetic energy and constant potential energy certainly satisfies conservation of7energy. But the straight horizontal trajectory is excluded because of the importance of the8averaged potential energy.9

10To see why the horizontal trajectory will not occur, go back to time averages and try the11(incorrect) assumption that the trajectory taken by the baseball is the one that has a minimum12value of the time average of the sum K + U. That trajectory would fall and then rise, like curve13A in Fig. 6. Here is why: By moving downward the ball decreases the average value of the14potential energy U (while increasing K a little because of the longer path). This is clearly not15what occurs when a baseball is pitched!16

17As an alternative hypothesis, try the assumption that the trajectory taken by the baseball is18the one that leads to a minimum value of the time average of the difference K – U. Which of19the trajectories C, D, or E can satisfy this criterion? By following a trajectory such as C that20rises and then falls, the baseball increases the time average value of its potential energy U.21Since U enters the average of K – U with a minus sign, a higher trajectory decreases the overall22contribution of the potential energy to the average (while increasing kinetic energy23somewhat). A minimum value of the time average of K – U is achieved along some trajectory24such as D in Fig. 6. For a still higher trajectory E the average value of the potential energy U25increases further, but the kinetic energy (proportional to the square of the speed) increases26even faster, because the baseball must complete in the same time the longer path between the27fixed events of pitch and catch. For higher and higher trajectories the average kinetic energy28increases without limit, leading to no maximum value. The actual path, indicated29schematically by D in Fig. 6, represents a compromise that makes the time-average value of K30

14

– U along the path a minimum.29 We will see that total energy is automatically conserved1along this path of minimum time average of the quantity (K – U) .2

3Check this result analytically in the simplest case we can imagine (Fig. 7). In a uniform4vertical gravitational field a marble of mass m and momentum of inertia Imarble rolls from5one horizontal surface to another via a smooth ramp so narrow that we neglect its width. In6this system the potential energy changes just once, halfway between initial and final7positions.8

d d

AC

v1

Bz1z2

x

1 2

9Figure 7. Motion across a potential energy step.10

11The worldline of the particle will be bent, corresponding to the reduced speed after the12marble mounts the ramp, as shown in Fig. 8.13

15

12

t

ttotal

A fixed

C fixed

d 2d x

B

1

2ttotal – t

34

Figure 8. Broken worldline of a marble rolling across steps connected5by a narrow ramp. The region on the right is shaded to represent the6higher potential energy of the marble on the second step7

8We demand that the total travel time from position A to position C have a fixed value ttotal9and check whether minimizing the time average of the difference (K – U) leads to the10conservation of energy.11

12time average =

1t total

K −U( )1t1 + K −U( )2 t2[ ] (28)13

14Define a new symbol S, called the action:15

16Action ≡ S ≡ time average × ttotal = K −U( )1t1 + K −U( )2 t2 (29)17

18Use the symbols in Fig. 8 to write Eq. (29) in the form19

20S =

12mv1

2t1 −mgz1t1 +12mv2

2t2 −mgz2t2

=md2

2t−mgz1t +

md2

2 t total − t( )−mgz2 t total − t( )

(30)21

16

1(For simplicity we neglect the marble's kinetic energy of rotation.) Demand that the value of2the action be a minimum with respect to the choice of the intermediate time t:3

4dSdt

= −md2

2t 2−mgz1 +

md2

2 t total − t( )2+ mgz2 = 0 (31)5

6This result can be written:7

8dSdt

= −1

2mv1

2 + mgz1

+

1

2mv2

2 + mgz2

= 0 (32)9

10A simple rearrangement shows that Eq. (32) represents the conservation of energy. Here11energy conservation has been derived from the more fundamental principle of least action.12

13

12

3 4z1

z2z3

z4

d d d d1415

Figure 9. A more complicated potential energy diagram, leading in the16limit to a potential energy curve that varies smoothly with position17

18The action Eq. (29) can be generalized for multiple steps connected by smooth, narrow19transitions, such as the one shown in Fig. 9. The argument leading to conservation of Eq. (32)20then applies to every adjacent pair of steps in the potential energy diagram. The21corresponding generalization of the action is:22

23S = K −U( )1t1 + K −U( )2 t2 + K −U( )3 t3 + K −U( )4 t4 (33)24

25The computer can hunt for and find the minimum value of S directly by varying the values of26the intermediate times, as shown schematically in Fig. 10.27

28

17

12 3 4

t

x

Fixed

Fixed

B

C

DE

A

1Figure 10. Varying the times at the potential energy steps to find a worldline of least2action. The computer program temporarily fixes the end events of a two-segment3section, say events B and D, then varies the middle event C to find the minimum value4of the total S, then varies D while C is kept fixed, and so on. (End-events A and E5always remain fixed.) The computer cycles through this process repeatedly until the6value of S does not change further because this value has reached a minimum for the7worldline as a whole. The resulting worldline approximates the one taken by the8particle.9

10A continuous potential energy curve is the limiting case of that shown in Fig. 9 as the number11of steps increases without limit while the time along each step becomes an increment dt. For12the resulting potential energy curve the general expression for the action S is:13

14Action ≡ S ≡ L

entirepath

∫ dt = K −U( )entirepath

∫ dt (34)15

16Here L (= K – U for the cases we treat) is called the Lagrangian. The principle of least action17says that the value of the action S is a minimum for the actual motion of the particle, a18condition that leads not only to conservation of energy but also to a unique specification of19the worldline.20

21A more general form of the principle of least action also predicts the motion of a particle in22more than one spatial dimension as well as the time development of systems containing23many particles. The principle of least action can even predict the motion of some systems in24which energy is not conserved.30 With generalizations of the Lagrangian L, the principle of25least action describes relativistic motion,31 governs many non-mechanical systems, and can26be used in the derivations of Maxwell's equations, Schroedinger's wave equation, the27diffusion equation, geodesic worldlines in general relativity, and electric currents in steady-28state circuits, among many other applications.29

30

18

The present section has not provided a proof of the principle of least action in Newtonian1mechanics. A fundamental proof rests on nonrelativistic quantum mechanics, for instance2that outlined by Tyc32 which uses the deBroglie relation to show that the phase change of a3quantum wave along any worldline is equal to S h where S is the classical action along that4worldline. Starting with this result, Feynman and Hibbs show33 that his "sum over all paths"5description of quantum motion reduces seamlessly to the Newtonian principle of least action6as the mass of particles increases.7

8Once students have mastered the principle of least action, it is easy to motivate the9introduction of nonrelativistic quantum mechanics: Quantum mechanics simply assumes that10the electron actually explores all the possible worldlines considered in finding the Newtonian11worldline of least action.12

13X. LAGRANGE'S EQUATIONS14Lagrange's equations are conventionally derived from the principle of least action using the15calculus of variations.34 These derivations analyze the worldline as a whole. However, the16action is a summation; if the value of the sum is minimum along the entire worldline, then17the contribution along each incremental segment of the worldline must also be a minimum.18This simplifying insight, due originally to Euler, allows a derivation of Lagrange's equations19using elementary calculus.35 The appendix shows a derivation of Lagrange's equation20directly from F = ma.21

22ACKNOWLEDGMENT23The authors would like to acknowledge useful comments by Don S. Lemons.24

25APPENDIX: DERIVATION OF LAGRANGE'S EQUATIONS DIRECTLY FROM F=ma26Both F = ma and Lagrange's equations can be derived from the more fundamental principle of27least action.36 Here we move laterally from one derived formulation to the other, showing28that Lagrange's equation leads to F = ma for particle motion in one dimension, as adapted29from a similar derivation by Zeldovich and Myskis.37 Using the definition of force in Eq. (9),30we write Newton's second law as:31

32

F = −dUdx

= m˙ ̇ x (A1)33

34Assume that m is constant. Rearrange and recast Eq. (A1) to read:35

36d −U( )

dx−

ddt

m˙ x ( ) =d −U( )

dx−

ddt

dd˙ x

m˙ x 2

2

= 0 (A2)37

38In the right-hand Eq. (A2) add, under the derivative signs, specifically chosen terms whose39partial derivatives are equal to zero:40

41

19

∂∂x

m˙ x 2

2−U x( )

−

ddt

∂∂˙ x

m˙ x 2

2−U x( )

= 0 (A3)1

2Set L = K – U, leading to the Lagrange equation in one dimension:3

4∂L∂x

−ddt

∂L∂˙ x

= 0 (A4)5

6This sequence of steps can be reversed to show that Lagrange's equation leads to Newton's7second law of motion F = ma.8

9FOOTNOTES AND REFERENCES10

11a)email: jozef.hanc@tuke.sk12

13b)email: eftaylor@mit.edu14

151David Morin, "The Lagrangian Method," Chapter 5 of a draft honors introductory mechanics16text. Available at17http://www.courses.fas.harvard.edu/~phys16/handouts/textbook/ch5.pdf (no final slash).18

192We use the phrase Newtonian mechanics to mean non-relativistic, non-quantum mechanics.20The alternative phrase classical mechanics also applies to relativity, both special and general,21and is thus inappropriate in the context of the present paper.22

233Sir Isaac Newton's Mathematical Principles of Natural Philosophy and his System of the World, Tr.24Andrew Motte and Florian Cajori, University of California Press, 1946, page 13.25

264Herman H. Goldstine, A History of the Calculus of Variations from the 17th through the 19th27Century, Springer-Verlag, 1980, page 110.28

295J. L. Lagrange, Analytic Mechanics, Victor N. Vagliente and Auguste Boissonnade Tr.30(Kluwer Academic Publishers, 2001).31

326William Rowan Hamilton, "On a general method in dynamics, by which the study of the33motions of all free systems of attracting or repelling points is reduced to the search and34differentiation of one central Relation or characteristic Function," Philosophical Transactions of35the Royal Society, part II for 1834, pp. 247-308; "Second essay on a general method in36dynamics," Philosophical Transactions of the Royal Society, part I for 1835, pp. 95-144. Both37papers are available at http://www.emis.de/classics/Hamilton/38

39

20

7Landau and E. M. Lifshitz, Mechanics, Course of Theoretical Physics (Butterworth-Heinemann,1London, 1976), 3rd ed., Vol. 1, Chap. 1. Their renaming first occurred in the original 19572Russian edition.3

48Richard P. Feynman, Robert B. Leighton and Matthew Sands, The Feynman Lectures on5Physics (Addison-Wesley, 1964), Vol. II, p. 19-8.6

79The technically correct term is principle of stationary action. See I. M. Gelfand and S. V. Fomin,8Calculus of Variations, Richard A. Silverman, Tr., (1963, Dover Publications 2000) Chapter 7,9Sec. 36.2. However, we have a strong preference for the conventional term principle of least10action for reasons not central to the argument of the present paper.11

1210Thomas H. Kuhn, "Energy Conservation as an Example of Simultaneous Discovery?" in The13Conservation of Energy and the Principle of Least Action, I. Bernard Cohen, ed, Arno Press, 1981.14

1511E. Noether, "Invariante Variationsprobleme," Goett. Nachr. 1918, pp. 235-257.16

1712Richard P. Feynman, Nobel Lecture, December 11, 1965, from Nobel Lectures, Physics 1963-181970 (Elsevier).19

2013Edwin F. Taylor, "A Call to Action," Am. J. Phys., 71 (5), May 2003, 423-425.21

2214Jozef Hanc, Slavomir Tuleja, Martina Hancova, "Simple derivation of Newtonian23mechanics from the principle of least action," Am. J. Phys., 71 (4) April 2003, 385-391.24

2515Jozef Hanc, Edwin F. Taylor, and Slavomir Tuleja, "Deriving Lagrange's Equations Using26Elementary Calculus," accepted for publication in Am. J. Phys..27

2816Jozef Hanc, Slavomir Tuleja, Martina Hancova, "Symmetries and Conservations Laws:29Consequences of Noether's Theorem," accepted for publication in Am. J. Phys.30

3117Reference 8, Vol. 1, Chapter 4. Feynman's parable about Dennis the Menace and his32indestructible blocks is classic. The final statement "There are no blocks" emphasizes that33energy is not a thing, cannot be pointed at, and typically cannot even be localized; it is the34property of a system.35

3618Benjamin Crowell: Conservation Laws (Light and Matter, Fullerton, CA, 1998-2002).37Available at http://www.lightandmatter.com38

3919Alan Van Heuvelen and Xueli Zou, "Multiple representations of work-energy processes,"40Am. J. Phys. 69 (2), 2001, pp 184-19441

42

21

20Roger A. Hinrich and Merlin Kleinbach, Energy: Its Use and the Environment (Harcourt, Inc.12002).2

321Gordon J. Aubrecht, Energy, 2nd ed. Edition (Prentice-Hall, 1995).4

522Student exercise: Show that B cos ωt and C cos ωt + D sin ωt are also solutions. Use the6occasion to discuss the importance of relative phase.7

823Benjamin Crowell bases an entire introductory treatment on Noether's theorem: Discover9Physics (Light and Matter, Fullerton, CA, 1998-2002). Available at10http://www.lightandmatter.com11

1224Noether's theorem says that when the Lagrangian of a system (L = K – U for our simple13cases) is not a function of an independent coordinate, x for example, then the function ∂L d˙ x 14is a constant of the motion. This statement can also be expressed in terms of Hamiltonian15dynamics. See for example Cornelius Lanczos, The Variational Principles of Mechanics, 4th ed.16(Dover Publications, 1970), Chap VI, Sec. 9, statement above Eq. (69.1). In all the mechanical17systems that we consider here, total energy is conserved and thus automatically does not18contain time explicitly. The expression for kinetic energy K is always quadratic in the19velocities, while the potential energy U is independent of velocities. These conditions are20sufficient for the Hamiltonian of the system to be the total energy. Then our limited version21of Noether's theorem is identical to the statement in Lanczos referenced above. However, we22have prepared for students a simpler, deeper, and more intuitive argument that justifies our23modified Noether theorem.24

2525Ruth Chabay and Bruce Sherwood, Matter & Interactions (John Wiley, 2002) Vol. 1, pp 188-26194 and 244-248. Page 248 lists background references in Am. J. Phys.27

2826Of course on the molecular level there is no friction. Of the current advanced mechanics29texts listed in our references, only Landau and Lifshitz, Ref. 7, mentions sliding friction even30in passing (p. 122); concerning motion in a dissipative medium they say (p. 74) " . . . there are31no equations of motion in the mechanical sense. Thus the problem of the motion of a body in32a medium is not one of mechanics." Their conclusion would surprise Isaac Newton.33

3427Reference 8,Vol. 1, page 4-5.35

3628Don S. Lemons, Perfect Form (Princeton University Press, 1997), Chapter 4.37

3829Slavomir Tuleja, Edwin F. Taylor, Java software: Principle of Least Action Interactive, 2003,39available at the website http://www.eftaylor.com/leastaction.html.40

4130Herbert Goldstein, Charles Poole, and John Safko, Classical Mechanics, 3rd ed.(Addison-42Wesley, Reading, MA 2002), Section2.7.43

22

131Reference 8, Vol. II, p. 19-82

332Tomas Tyc, "The de Broglie hypothesis leading to path integrals," Eur. J. Phys. 17 (May41996), 156-157. Version available at the URL given in reference 29.5

633R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965),7p. 29.8

934Reference 7, Chap. 1; Reference 30, Sect. 2.3; D. Gerald Jay Sussman and Jack Wisdom,10Structure and Interpretation of Classical Mechanics (MIT, Cambridge, 2001), Chap. 1.11

1235Reference 1513

1436References 14 and 15.15

1637Ya. B. Zeldovich and A. D. Myskis, Elements of Applied Mathematics, George Yankovsky Tr.17(MIR Publishers Moscow,1976) pages 499-50018