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1 From Conservation of to the Principle of Least Action: A Story Line 2 3 Jozef Hanca) 4 Technical University, Vysokoskolska 4, 042 00 Kosice, Slovakia 5 6 Edwin F. Taylorb) 7 Massachusetts Institute of Technology, Cambridge, MA 02139 8 9 ABSTRACT 10 A schematic story line outlines an introduction to Newtonian mechanics that starts by 11 employing the conservation of energy to predict the motion of a particle in a one-dimensional 12 potential. Incorporating constraints and constants of the motion into the energy expression 13 allows analysis of some more complicated systems. A heuristic transition embeds kinetic and 14 in the still more powerful principle of least action and Lagrange's equations. 15 16 NOTES TO EDITORS AND REFEREES: 17 1. Preprints of some papers referenced here but not yet published and some papers published 18 elsewhere are available at the website http://www.eftaylor.com/leastaction.html 19 20 2. Comments and suggestions can conveniently be indexed by page and line number in this 21 pdf file. 22 23 I. INTRODUCTION 24 25 It is sometimes claimed that nature has a "purpose", in that it seeks to take the path that produces the 26 minimum action. . . . In fact [according to Feynman's version of ] nature does 27 exactly the opposite. It takes every path, treating them all on [an] equal footing. We simply end up 28 seeing the path with the stationary action, due to the way the quantum mechanical phases add. It 29 would be a harsh requirement, indeed, to demand that nature make a "global" decision (that is, to 30 compare paths that are separated by large distances), and to choose the one with the smallest action. 31 Instead, we see that everything takes place on a "local” scale. Nearby phases simply add and 32 everything works out automatically. 33 1 34 — David Morin 35 2 36 Here is a nine-sentence history of Newtonian mechanics: In the second half of the 1600s 3 37 Newton proposed his three laws of motion, including what has become F = ma. In the mid- 38 1700s devised and applied one version of the principle of least action using 39 mostly geometrical methods. On 12 August 1755 the 19-year-old Ludovico de la Grange 40 Tournier of Turin (whom we call Joseph Louis Lagrange) sent Euler a letter with a 41 mathematical attachment that streamlined Euler's methods into algebraic form. "[A]fter 42 seeing Lagrange's Euler dropped his own method, espoused that of Lagrange, and 4 43 renamed the subject the calculus of variations." Lagrange, in his path-breaking 1788 Analytical 5 44 Mechanics, introduced what we call the Lagrangian function and Lagrange's equations of 6 45 motion. Almost half a century later (1834-35) William Rowan Hamilton developed

1 7 8 1 Hamilton's principle, to which Landau and Lifshitz and Feynman more recently reassigned 9 2 the name principle of least action. Between 1840 and 1860 the conservation of energy was 10 11 3 established in all its generality. In 1918 Emma Noether proved relations between 12 4 symmetries and conserved quantities. Finally, in 1949 Richard Feynman devised the 5 formulation of quantum mechanics that not only explicitly underpins the principle of least 6 action but also demarcates the limits of validity of Newtonian mechanics. 7 13 8 We have proposed that the principle of least action and Lagrange's equations become the 9 basis of introductory Newtonian mechanics. Three recent articles in this Journal demonstrate 10 how to use elementary calculus to derive from the principle of least action (1) Newton's laws 14 15 16 11 of motion, (2) Lagrange's equations, and (3) examples of Noether's theorem. 12 13 How are these concepts and methods to be introduced to students? In the present paper we 14 suggest a reversal of the historical order: Begin with conservation of energy and graduate to 15 the principle of least action followed by Lagrange's equations. 16 17 We start by using the conservation of energy to analyze particle motion in a one-dimensional 18 potential. Much of the power of the principle of least action and its logical offspring 19 Lagrange's equations results from the fact that they are based on energy, a scalar. When we 20 start with the conservation of energy, we not only preview more advanced concepts and 21 procedures but also invoke some of their power. For example, expressions for energy 22 conforming to constraints automatically eliminate corresponding of constraint from 23 the analysis of motion. Applying a limited version of Noether's theorem expressed in terms 24 of energy identifies constants of the motion. Using constraints and constants of the motion 25 permits us to reduce to one coordinate the description of some important multi-dimensional 26 systems, whose motion can then be solved completely using the conservation of energy. 27 Equilibrium and statics also derive from the conservation of energy. 28 29 Prerequisites for the proposed introduction include elementary trigonometry, polar 30 coordinates, introductory differential calculus and partial derivatives, and the idea of the 31 integral as a sum of increments. In preparation for later formalism, we symbolize the time 32 derivative with a dot over the variable. 33 34 The story line presented in this article omits most details and is offered for discussion, 35 correction, and elaboration. The authors have no present intent to write an introductory text 36 on Newtonian mechanics that follows the story line of this article. 37 38 II. 39 Begin with a discursive description of the forms of energy and the law of global conservation 40 of energy. Introductory treatments of the conservation of energy and conversion among its 17, 18, 19 41 various forms are available in the literature. How far one goes into engineering 20, 21 42 applications of energy and its environmental consequences is a matter of choice. In 43 addition to expressions for kinetic and potential energy of a particle, this introduction should 44 include angular , moment of inertia, and the of a rigid body rotating 45 about a fixed or parallel-moving axis.

2 1 2 III. ONE-DIMENSIONAL MOTION: ANALYTIC SOLUTIONS 3 Narrow the focus to conservation of of a particle moving in a one- 4 dimensional potential. The complete description of such one-dimensional motion follows 5 from the conservation of energy. Unfortunately, an explicit function of position vs time can be 6 derived for only a small fraction of such systems. In other cases one is encouraged to guess 7 the analytic solution, as illustrated in the following example. Any proposed analytic solution 8 is easily checked by substitution into the energy equation. Heuristic guesses are assisted by 9 the fact that the first time derivative of position, not the second, appears in the energy 10 conservation equation. The following example introduces the potential energy diagram, a 11 central tool in our story line and important in later study of advanced mechanics, quantum 12 mechanics, , and many other subjects. 13 14 Harmonic oscillator

Energy Total energy of particle F A B C DE

Potential Kinetic energy energy

Position 15 16 Figure 1. The potential energy diagram is central to our treatment. Sample student exercise: For 17 the value of the energy shown, describe the motion of the particle qualitatively but in detail. 18 Compare the magnitudes and directions of each of the following quantities that describe the 19 particle motion at different positions marked A through F: velocity, , acceleration. In order to 20 describe the motion, what initial condition(s) must be specified in addition to an initial starting 21 point, and are these conditions required for all starting points? How do answers to these 22 questions change for different values of the total energy? 23 24 Qualitative analysis of the parabolic potential energy diagram (or indeed any diagram with 25 motion bounded near a single potential energy minimum) tells us that the motion will be 26 periodic. For the parabolic potential the conservation of energy is written 27

3 1 1 E = mx˙ 2 + kx 2 1 2 2 (1) 2 3 Rearrange Eq. (1) to read: 4 m k x˙ 2 =1− x 2 5 2E 2E (2) 6 7 Recalling the periodicity of motion, we are reminded of trigonometric identity 8 2 2 9 cos θ =1− sin θ (3) 10 11 Set θ = ωt and equate corresponding terms in Eqs. (2) and (3), choosing the positive root, 12 which yields:  2E 1/2 x =   sinωt 13  k  (4) 14 15 Take the time derivative of this expression and compare its square with the left side of Eq. 22 16 (2). The result tells us that 17  k 1/2 ω =   18  m (5) 19 20 The fact that ω does not depend on the amplitude of the displacement function (4) means that 21 the period is the same whatever the value of the total energy E. The simple harmonic 22 oscillator has wide application because a large number of potential energy curves can be 23 approximated as parabolas near their minima. 24 25 IV. ACCELERATION AND FORCE 26 In the absence of , force can be defined in terms of energy. In the simple harmonic 27 oscillator example, take the time derivative of both sides of (1) (for constant energy E and 28 m). 29 30 0 = mx˙ x˙˙ + kxx˙ (6) 31 32 or 33 k x˙˙ =− x 34 m (7) 35 36 Define force as m times the acceleration. Then the simple harmonic force is –kx. More 37 generally, take the time derivative of both sides of the conservation of energy equation for a 38 particle moving in a one-dimensional potential: 39 dU() x dU() x dx dU() x 0 = mx˙ x˙˙ + = mx˙ x˙˙ + = mx˙ x˙˙ + x˙ 40 dt dx dt dx (8)

4 1 2 from which 3 dU() x mx˙˙ ≡ F =− 4 dx (9) 5 6 Thus for motion without dissipation force is defined in terms of energy. 7 8 V. NON-INTEGRABLE MOTIONS IN ONE DIMENSION 9 We cannot solve most one-dimensional motions in closed form, as we did for the simple 10 harmonic oscillator. Our strategy will be to ask the student to begin every solution with a 11 qualitatively analysis using the potential energy diagram (as exemplified in the caption of 12 Fig. 1), then to complete the exact solution employing an interactive computer display (Fig. 13 2). Such a computer display, yet to be programmed but easily specified, numerically 14 integrates the particle motion in a given potential. On this display the student sets up initial 15 conditions by dragging the horizontal energy line up or down and the position dot left or 16 right, both in the classically permitted region (lower panel of Fig. 2). The computer then 17 moves the dot back and forth along the E-line at a rate proportional to that at which the 18 particle will move, while simultaneously drawing the worldline (upper panel of Fig. 2).

5 time

Position

E Energy

U

Position 1 2 Figure 2. Interactive computer display of the worldline derived numerically from the initial 3 energy and the potential energy function. 4 5 VI. REDUCTION TO ONE COORDINATE 6 The analysis of one-dimensional motion using the conservation of energy is powerful but 7 limited. In some important cases we can use constraints and constants of the motion to 8 reduce the description of a wider class of motions to one coordinate. To these we apply our 9 standard procedure: qualitative analysis using the diagram of the (effective) potential energy 10 followed by interactive computer solution. Here are some examples. 11 12 (a) Object rolling without slipping 13 A marble with mass m, radius r, and moment of inertia Imarble rolls without slipping along a 14 curved ramp that lies in the xy plane. A uniform gravitational acceleration g acts in the 15 negative y-direction. The non-slip constraint tells us that v = rω . Conservation of energy leads 16 to the expression: 17

6 2 1 2 1 2 1 2 1  v  E = mv + Imarbleω + mgy = mv + Imarble  + mgy 2 2 2 2  r  1 (10) 1 I  2 1 2 =  m + marble v + mgy = Mv + mgy 2 r2  2 2 3 where I M = m + marble 4 r2 (11) 5 6 Constraints are used twice in this example: explicitly in rolling without slipping and 7 implicitly in the relation between height y and displacement along the curve. 8 9 (b) Projectile motion 10 For projectile motion in a vertical plane subject to a vertical uniform vertical gravitational 11 field, the total energy is 1 1 E = mx˙ 2 + my˙ 2 + mgy 12 2 2 (12) 13 14 The potential energy is not a function of x; therefore, as shown below, x- px is a 15 constant of the motion. The energy equation becomes: 16 2 1 2 p 17 E = my˙ + x + mgy (13) 2 2m 18 19 In Newtonian mechanics the zero-point of energy is arbitrary, so we can reduce the analysis 20 to one dimension by making the substitution: 21 2 p 1 2 22 E'= E − x = my˙ + mgy (14) 2m 2 23 24 In analyzing projectile motion, we have already applied a limited version of the powerful 23 25 Noether's theorem. The special version of Noether's theorem used here, expressed in terms 26 of energy, says that when the total energy E is not an explicit function of an independent 24 27 coordinate, x for example, then the function ∂E ∂x˙ is a constant of the motion. In some 28 important cases we can use the resulting constants of the motion, together with constraints, 29 to describe some two- and even three-dimensional motions with just one coordinate. 30 Projectile motion, above, is one example. Here is another. 31

7 1 (c) Marble, ramp, and turntable m x θ

φ

2 3 4 Figure 3. System of marble, ramp, and turntable. 5 [NOTE: This figure will be redrawn professionally after any referee suggestions.] 6 m I 7 A marble of mass and moment of inertia marble rolls without slipping along a slot on an 8 inclined ramp fixed rigidly to a turntable which rotates freely so that its angular velocity is 9 not necessarily constant (Fig. 3). The moment of inertia of the combined turntable and ramp I x 10 is rot. Assuming that the marble stays on the ramp, find its position as a function of time. 11 12 Start with a qualitative analysis, assuming the marble initially moves outward along the 13 ramp. This outward rolling will slow down the rotation of the turntable, perhaps leading the 14 marble to reverse direction. One possibility is oscillatory motion of the marble along the 15 ramp; another possibility is an equilibrium location of the marble on the ramp with simple 16 circular motion of the marble around the axis of the turntable. Now the analysis. The square 17 of the velocity of the marble is: 18 2 2 ˙ 2 2 2 19 v = x˙ + φ x cos θ (15) 20 21 Conservation of energy yields the equation 22 1 1 1 E = Mx˙ 2 + I φ˙ 2 + Mx 2φ˙ 2 cos2 θ + mgxsinθ 23 2 2 rot 2 (16) 24 25 Here M indicates the marble's mass augmented by the energy effects of its rotation, Eq. (11). 26 The expression (16) is not an explicit function of the angle of rotation φ. Therefore our ˙ 27 version of Noether's theorem tells us that ∂E ∂φ is a constant of the motion, which we 28 recognize as the total angular momentum J: 29 ∂E = ()I + Mx 2 cos2 θ φ˙ = J 30 ∂φ˙ rot (17) 31

8 ˙ 1 Substitute for φ from Eq. (17) into (16): 2 2 1 2 J 1 2 E = Mx˙ + + mgx sinθ = Mx˙ + U ()x 3 2 2 2 2 eff (18) 2()Irot + Mx cos θ 4 5 Values of E and J are determined by initial conditions. The effective potential energy Ueff ()x 6 can have a minimum as a function of x. If the marble starts at rest at the position of this 7 minimum, it will move in a circle. The general motion of this system is difficult to analyze 8 using F = ma. 9 10 (d) Motion in a central potential 11 Analyze satellite motion in a central inverse-square gravitational field. Choose coordinates r 12 and φ in the plane of the orbit. Then the expression for total energy is: 13 1 GMm 1 GMm E = mv 2 − = m()r˙ 2 + r2φ˙ 2 − 14 2 r 2 r (19) 15 16 The angle φ does not appear in this equation. Therefore we expect that a constant of the ˙ 17 motion is given by ∂E ∂φ , which is the angular momentum J: 18 ∂E = mr2φ˙ = J 19 ∂φ˙ (20) 20 ˙ 21 Substitute the resulting expression for φ in Eq. (20) into the energy equation (19). 22 1 J 2 GMm 1 E = mr˙ 2 + − = mr˙ 2 + U ()r 23 2 2mr 2 r 2 eff (21) 24 25 The student employs the plot of effective potential energy to carry out the usual qualitative 26 analysis of radial motion followed by computer integration. Emphasize the distinction 27 between bound orbits, unbound orbits, and the intermediate limiting case. With additional 28 use of Eq. (20), the computer can plot the trajectory in the plane. 29 30 VII. EQUILIBRIUM AND STATICS 25 31 is an important feature of motion in everyday life, but it is complicated to analyze. 26 32 Frictional forces are often ignored in advanced mechanics texts. The tendency of a system 33 toward increased and the statistical mechanics and that account for 34 this tendency are not part of our story line here. Nevertheless, in common experience motion 35 usually slows down and stops. Our use of potential energy diagrams makes straightforward 36 the formulation of "stopping" as a tendency of systems to reach equilibrium at a local 37 minimum of the potential energy. (In principle the system could come to equilibrium 38 perched at a maximum of the potential energy, or even at a zero-slope inflection point.) In the 39 absence of dissipative processes, equilibrium is really an application of conservation of 40 energy; a system released from rest in a configuration at a minimal (or stationary) value of 41 the potential energy will remain at rest. (Proof: An infinitesimal displacement results in zero

9 1 change in potential energy. Because of energy conservation, the change in kinetic energy 2 must also remain zero. Since the system is initially at rest, the zero change in kinetic energy 3 forbids displacement.) 4 5 Here, as usual, Feynman is ahead of us. Figure 4 shows an example from his treatment of 27 6 statics. The problem is to find the value of the weight W that keeps the structure at rest. 7 Using units of feet, inches, and pounds, Feynman balances the decrease in potential energy 8 when the weight W drops 4 inches with increases in potential energy for the corresponding 9 2-inch rise of the 60 pound weight plus the 1-inch rise in the 100 pound weight. He demands 10 that the net potential energy change of the system be zero, yielding 11

12 13 Figure 4. Feynman's example of 14 the principle of 15 16 ()−4 inch W + (2 inch)(60 pounds) + (1 inch)(100 pounds) = 0 (22) 17 18 or W = 55 pounds. This method, known formally as the principle of virtual work, is simply an 19 application of the conservation of energy. 20 21 There are many examples of the principle of virtual work, including a mass hanging on a 22 spring, the lever, hydrostatic balance, uniform chain suspended at both ends, vertically 28 23 hanging "slinky," and a ball perched on top of a large sphere, not to mention every static 24 engineering and natural structure in the universe. 25 26 VIII. FREE PARTICLE. PRINCIPLE OF LEAST AVERAGE KINETIC ENERGY 27 The conservation of energy is limited in predicting the motion of mechanical systems. Now 28 we begin the transition to the more powerful principle of least action by looking at the time 29 average of the kinetic energy of a free particle, a particle that moves in a region of zero 30 potential energy or potential energy that has the same value everywhere in the region. In this 31 and the following sections we seek a principle that is more fundamental than the 32 conservation of energy. We will test any fundamental principle we uncover by demanding 33 that, at least, it lead back to the conservation of energy. For this reason the trial worldlines we 34 explore hereafter do not necessarily satisfy conservation of energy.

10 C fixed ttotal

ttotal – t 2

t B

1

A fixed d 2d x 1 2 3 Figure 5. Varying the time of the middle event to determine the path 4 of minimum time-averaged kinetic energy. 5 6 The line in Fig. 5 shows the trial worldline of a particle that moves freely between fixed 7 events A and C. We know that a free particle travels with constant velocity, that is, along a 8 straight worldline. Verify that this is the actual motion by examining a trial worldline for 9 which the central event B in the worldline is displaced in time (and therefore kinetic energy is 10 not the same on legs 1 and 2). Whether the potential energy is zero or uniform in this region, 11 its contribution to the time average energy does not change with the time-displacement of 12 event B, so we ignore the potential energy. Let K represent the kinetic energy. Then the time 13 average of the kinetic energy along the two segments 1 and 2 of the worldline is given by the 14 expression. 15 K t + K t   K = 1 1 2 2 = 1 1 mv 2t + 1 mv 2t 16 average t t  1 1 2 2 (23) total total 2 2 17 18 Multiply through by the fixed total time and expand the velocity expressions using the 19 notation displayed in Fig. 5: 20 d 2 d 2 K t = 1 m t + 1 m ()t − t average total 2 t 2 2 ()t − t 2 total 21 total (24) 1 d 2 1 d 2 = m + m t ()t − t 2 2 total 22

11 1 Find the minimum value of the average kinetic energy by taking the derivative with respect 2 to the time t of the central event and setting the result equal to zero: 3  dK  1 d 2 1 d 2 1 1 4  average  t =− m + m =− mv 2 + mv 2 = 0 (25)  dt  total t 2 ()t − t 2 1 2 2 2 total 2 2 5 6 The right-hand equation leads to the equality: 7 1 1 mv 2 = mv 2 8 2 2 2 1 (26) 9 10 As expected, when the time-average kinetic energy has a minimum value the kinetic energy 11 for a free particle is the same on both segments; the worldline is straight between the fixed 12 initial and final events A and C. In summary, we have illustrated the fact that for the special 13 case of a particle moving in a region of zero (or uniform) potential energy, kinetic energy is 14 conserved if we demand that the time average of the kinetic energy has a minimum value. 15 The general expression for this average is: 16 1 17 K = ∫ Kdt (27) average t total 18 19 In an introductory text one might insert at this point a sidebar on Fermat's principle of least 20 time for the propagation of rays, which predicts every central feature of ray . Now 21 we are ready to develop a similar law that predicts every central feature of mechanics. 22 23 IX. PRINCIPLE OF LEAST ACTION 24 Section VIII examined the principle of least average kinetic energy, which predicts the motion 25 of a particle moving in a region of zero (or uniform) potential energy. Along the way we 26 mentioned Fermat's principle of least time that describes ray optics. Now we move on to the 27 more general principle which includes the principle of least average kinetic energy as a 28 special case. Can we find some quantity whose time average value is a minimum when both 29 kinetic energy and potential energy are functions of position? 30 31 Let K represent kinetic energy and U represent potential energy. One might guess 32 (incorrectly) that the time averaged total K + U would have a minimum value between fixed 33 initial and final events. (Remember: In calculating the value of this average we do not assume 34 the conservation of total energy, but rather demand that energy conservation emerge as one 35 result of our analysis.) 36 37 It turns out that the time averaged difference K – U between kinetic and potential energy has a 38 minimum value for the path taken by the particle. How can this be? To sort this out, think of 39 a baseball pitched in a uniform gravitational field, with the two events of pitch and catch 40 fixed in both location and time (Fig. 6). How will the ball move between these two fixed 41 events? 42

12 y E ()K −U minimum D avg

C

pitch catch B x

A ()K +U minimum? 1 avg 2 3 Figure 6. Alternative trial trajectories of a pitched baseball. 4 5 To begin with, forget about averages and ask why the baseball does not simply move steadily 6 along the straight horizontal trajectory B in the figure. Moving from pitch to catch with 7 constant kinetic energy and constant potential energy certainly satisfies conservation of 8 energy. But the straight horizontal trajectory is excluded because of the importance of the 9 averaged potential energy. 10 11 To see why the horizontal trajectory will not occur, go back to time averages and try the 12 (incorrect) assumption that the trajectory taken by the baseball is the one that has a minimum 13 value of the time average of the sum K + U. That trajectory would fall and then rise, like curve 14 A in Fig. 6. Here is why: By moving downward the ball decreases the average value of the 15 potential energy U (while increasing K a little because of the longer path). This is clearly not 16 what occurs when a baseball is pitched! 17 18 As an alternative hypothesis, try the assumption that the trajectory taken by the baseball is 19 the one that leads to a minimum value of the time average of the difference K – U. Which of 20 the trajectories C, D, or E can satisfy this criterion? By following a trajectory such as C that 21 rises and then falls, the baseball increases the time average value of its potential energy U. 22 Since U enters the average of K – U with a minus sign, a higher trajectory decreases the overall 23 contribution of the potential energy to the average (while increasing kinetic energy 24 somewhat). A minimum value of the time average of K – U is achieved along some trajectory 25 such as D in Fig. 6. For a still higher trajectory E the average value of the potential energy U 26 increases further, but the kinetic energy (proportional to the square of the ) increases 27 even faster, because the baseball must complete in the same time the longer path between the 28 fixed events of pitch and catch. For higher and higher trajectories the average kinetic energy 29 increases without limit, leading to no maximum value. The actual path, indicated 30 schematically by D in Fig. 6, represents a compromise that makes the time-average value of K

13 29 1 – U along the path a minimum. We will see that total energy is automatically conserved 2 along this path of minimum time average of the quantity (K – U) . 3 4 Check this result analytically in the simplest case we can imagine (Fig. 7). In a uniform m I 5 vertical gravitational field a marble of mass and momentum of inertia marble rolls from 6 one horizontal surface to another via a smooth ramp so narrow that we neglect its width. In 7 this system the potential energy changes just once, halfway between initial and final 8 positions. 12 C v A 1 z2 z1 B x d d 9 10 Figure 7. Motion across a potential energy step. 11 12 The worldline of the particle will be bent, corresponding to the reduced speed after the 13 marble mounts the ramp, as shown in Fig. 8.

14 1 2

C fixed ttotal

ttotal – t 2

B t

1

A fixed d2dx 3 4 5 Figure 8. Broken worldline of a marble rolling across steps connected 6 by a narrow ramp. The region on the right is shaded to represent the 7 higher potential energy of the marble on the second step 8 t 9 We demand that the total travel time from position A to position C have a fixed value total 10 and check whether minimizing the time average of the difference (K – U) leads to the 11 conservation of energy. 12 1 13 time average = []()K −U t + ()K −U t (28) t 1 1 2 2 total 14 15 Define a new symbol S, called the action: 16 ≡ S ≡ × t = ()K −U t + ()K −U t 17 Action time average total 1 1 2 2 (29) 18 19 Use the symbols in Fig. 8 to write Eq. (29) in the form 20 1 1 S = mv 2t − mgz t + mv 2t − mgz t 2 1 1 1 1 2 2 2 2 2 21 2 2 (30) md md = − mgz1t + − mgz2 ()ttotal − t 2t 2()ttotal − t

15 1 2 (For simplicity we neglect the marble's kinetic energy of rotation.) Demand that the value of 3 the action be a minimum with respect to the choice of the intermediate time t: 4 dS md2 md2 5 =− − mgz1 + + mgz2 = 0 (31) 2 2 2 dt t 2()ttotal − t 6 7 This result can be written: 8 dS  1   1  9 =− mv 2 + mgz  +  mv 2 + mgz  = 0 (32) dt  2 1 1  2 2 2  10 11 A simple rearrangement shows that Eq. (32) represents the conservation of energy. Here 12 energy conservation has been derived from the more fundamental principle of least action. 13 3 4 2 z3 z2 z z1 1 4

d d d d 14 15 16 Figure 9. A more complicated potential energy diagram, leading in the 17 limit to a potential energy curve that varies smoothly with position 18 19 The action Eq. (29) can be generalized for multiple steps connected by smooth, narrow 20 transitions, such as the one shown in Fig. 9. The argument leading to conservation of Eq. (32) 21 then applies to every adjacent pair of steps in the potential energy diagram. The 22 corresponding generalization of the action is: 23 S = ()K −U t + ()K −U t + ()K −U t + ()K −U t 24 1 1 2 2 3 3 4 4 (33) 25 26 The computer can hunt for and find the minimum value of S directly by varying the values of 27 the intermediate times, as shown schematically in Fig. 10. 28

16 t E D Fixed

C B A 2 3 4 Fixed 1 x 1 2 Figure 10. Varying the times at the potential energy steps to find a worldline of least 3 action. The computer program temporarily fixes the end events of a two-segment 4 section, say events B and D, then varies the middle event C to find the minimum value 5 of the total S, then varies D while C is kept fixed, and so on. (End-events A and E 6 always remain fixed.) The computer cycles through this process repeatedly until the 7 value of S does not change further because this value has reached a minimum for the 8 worldline as a whole. The resulting worldline approximates the one taken by the 9 particle. 10 11 A continuous potential energy curve is the limiting case of that shown in Fig. 9 as the number 12 of steps increases without limit while the time along each step becomes an increment dt. For 13 the resulting potential energy curve the general expression for the action S is: 14 15 Action ≡ S ≡ ∫ L dt = ∫ ()K −U dt (34) entire entire path path 16 17 Here L (= K – U for the cases we treat) is called the Lagrangian. The principle of least action 18 says that the value of the action S is a minimum for the actual motion of the particle, a 19 condition that leads not only to conservation of energy but also to a unique specification of 20 the worldline. 21 22 A more general form of the principle of least action also predicts the motion of a particle in 23 more than one spatial dimension as well as the time development of systems containing 24 many particles. The principle of least action can even predict the motion of some systems in 30 25 which energy is not conserved. With generalizations of the Lagrangian L, the principle of 31 26 least action describes relativistic motion, governs many non-mechanical systems, and can 27 be used in the derivations of Maxwell's equations, Schroedinger's wave equation, the 28 diffusion equation, geodesic worldlines in general relativity, and electric currents in steady- 29 state circuits, among many other applications. 30

17 1 The present section has not provided a proof of the principle of least action in Newtonian 2 mechanics. A fundamental proof rests on nonrelativistic quantum mechanics, for instance 32 3 that outlined by Tyc which uses the deBroglie relation to show that the phase change of a 4 quantum wave along any worldline is equal to S h where S is the classical action along that 33 5 worldline. Starting with this result, Feynman and Hibbs show that his "sum over all paths" 6 description of quantum motion reduces seamlessly to the Newtonian principle of least action 7 as the mass of particles increases. 8 9 Once students have mastered the principle of least action, it is easy to motivate the 10 introduction of nonrelativistic quantum mechanics: Quantum mechanics simply assumes that 11 the actually explores all the possible worldlines considered in finding the Newtonian 12 worldline of least action. 13 14 X. LAGRANGE'S EQUATIONS 15 Lagrange's equations are conventionally derived from the principle of least action using the 34 16 calculus of variations. These derivations analyze the worldline as a whole. However, the 17 action is a summation; if the value of the sum is minimum along the entire worldline, then 18 the contribution along each incremental segment of the worldline must also be a minimum. 19 This simplifying insight, due originally to Euler, allows a derivation of Lagrange's equations 35 20 using elementary calculus. The appendix shows a derivation of Lagrange's equation 21 directly from F = ma. 22 23 ACKNOWLEDGMENT 24 The authors would like to acknowledge useful comments by Don S. Lemons. 25 26 APPENDIX: DERIVATION OF LAGRANGE'S EQUATIONS DIRECTLY FROM F=ma 27 Both F = ma and Lagrange's equations can be derived from the more fundamental principle of 36 28 least action. Here we move laterally from one derived formulation to the other, showing 29 that Lagrange's equation leads to F = ma for particle motion in one dimension, as adapted 37 30 from a similar derivation by Zeldovich and Myskis. Using the definition of force in Eq. (9), 31 we write Newton's second law as: 32 dU F =− = mx˙˙ 33 dx (A1) 34 35 Assume that m is constant. Rearrange and recast Eq. (A1) to read: 36 d()−U d d()−U d  d  mx˙ 2  37 − ()mx˙ = −    = 0 (A2) dx dt dx dt  dx˙  2  38 39 In the right-hand Eq. (A2) add, under the derivative signs, specifically chosen terms whose 40 partial derivatives are equal to zero: 41

18 ∂  mx˙ 2  d  ∂  mx˙ 2  1  −Ux() −   −Ux() = 0 (A3) ∂x  2  dt ∂ x˙  2  2 3 Set L = K – U, leading to the Lagrange equation in one dimension: 4 ∂L d ∂ L − = 0 5 ∂x dt  ∂x˙  (A4) 6 7 This sequence of steps can be reversed to show that Lagrange's equation leads to Newton's 8 second law of motion F = ma. 9 10 FOOTNOTES AND REFERENCES 11 a) 12 email: [email protected] 13 b) 14 email: [email protected] 15 1 16 David Morin, "The Lagrangian Method," Chapter 5 of a draft honors introductory mechanics 17 text. Available at 18 http://www.courses.fas.harvard.edu/~phys16/handouts/textbook/ch5.pdf (no final slash). 19 2 20 We use the phrase Newtonian mechanics to mean non-relativistic, non-quantum mechanics. 21 The alternative phrase also applies to relativity, both special and general, 22 and is thus inappropriate in the context of the present paper. 23 3 24 Sir 's Mathematical Principles of Natural Philosophy and his System of the World, Tr. 25 Andrew Motte and Florian Cajori, University of California Press, 1946, page 13. 26 4 27 Herman H. Goldstine, A History of the Calculus of Variations from the 17th through the 19th 28 Century, Springer-Verlag, 1980, page 110. 29 5 30 J. L. Lagrange, Analytic Mechanics, Victor N. Vagliente and Auguste Boissonnade Tr. 31 (Kluwer Academic Publishers, 2001). 32 6 33 William Rowan Hamilton, "On a general method in dynamics, by which the study of the 34 motions of all free systems of attracting or repelling points is reduced to the search and 35 differentiation of one central Relation or characteristic Function," Philosophical Transactions of 36 the Royal Society, part II for 1834, pp. 247-308; "Second essay on a general method in 37 dynamics," Philosophical Transactions of the Royal Society, part I for 1835, pp. 95-144. Both 38 papers are available at http://www.emis.de/classics/Hamilton/ 39

19 7 1 Landau and E. M. Lifshitz, Mechanics, Course of Theoretical (Butterworth-Heinemann, 2 London, 1976), 3rd ed., Vol. 1, Chap. 1. Their renaming first occurred in the original 1957 3 Russian edition. 4 8 5 Richard P. Feynman, Robert B. Leighton and Matthew Sands, The Feynman Lectures on 6 Physics (Addison-Wesley, 1964), Vol. II, p. 19-8. 7 9 8 The technically correct term is principle of stationary action. See I. M. Gelfand and S. V. Fomin, 9 Calculus of Variations, Richard A. Silverman, Tr., (1963, Dover Publications 2000) Chapter 7, 10 Sec. 36.2. However, we have a strong preference for the conventional term principle of least 11 action for reasons not central to the argument of the present paper. 12 10 13 Thomas H. Kuhn, "Energy Conservation as an Example of Simultaneous Discovery?" in The 14 Conservation of Energy and the Principle of Least Action, I. Bernard Cohen, ed, Arno Press, 1981. 15 11 16 E. Noether, "Invariante Variationsprobleme," Goett. Nachr. 1918, pp. 235-257. 17 12 18 Richard P. Feynman, Nobel Lecture, December 11, 1965, from Nobel Lectures, Physics 1963- 19 1970 (Elsevier). 20 13 21 Edwin F. Taylor, "A Call to Action," Am. J. Phys., 71 (5), May 2003, 423-425. 22 14 23 Jozef Hanc, Slavomir Tuleja, Martina Hancova, "Simple derivation of Newtonian 24 mechanics from the principle of least action," Am. J. Phys., 71 (4) April 2003, 385-391. 25 15 26 Jozef Hanc, Edwin F. Taylor, and Slavomir Tuleja, "Deriving Lagrange's Equations Using 27 Elementary Calculus," accepted for publication in Am. J. Phys.. 28 16 29 Jozef Hanc, Slavomir Tuleja, Martina Hancova, "Symmetries and Conservations Laws: 30 Consequences of Noether's Theorem," accepted for publication in Am. J. Phys. 31 17 32 Reference 8, Vol. 1, Chapter 4. Feynman's parable about Dennis the Menace and his 33 indestructible blocks is classic. The final statement "There are no blocks" emphasizes that 34 energy is not a thing, cannot be pointed at, and typically cannot even be localized; it is the 35 property of a system. 36 18 37 Benjamin Crowell: Conservation Laws (Light and Matter, Fullerton, CA, 1998-2002). 38 Available at http://www.lightandmatter.com 39 19 40 Alan Van Heuvelen and Xueli Zou, "Multiple representations of work-energy processes," 41 Am. J. Phys. 69 (2), 2001, pp 184-194 42

20 20 1 Roger A. Hinrich and Merlin Kleinbach, Energy: Its Use and the Environment (Harcourt, Inc. 2 2002). 3 21 4 Gordon J. Aubrecht, Energy, 2nd ed. Edition (Prentice-Hall, 1995). 5 22 6 Student exercise: Show that B cos ωt and C cos ωt + D sin ωt are also solutions. Use the 7 occasion to discuss the importance of relative phase. 8 23 9 Benjamin Crowell bases an entire introductory treatment on Noether's theorem: Discover 10 Physics (Light and Matter, Fullerton, CA, 1998-2002). Available at 11 http://www.lightandmatter.com 12 24 13 Noether's theorem says that when the Lagrangian of a system (L = K – U for our simple 14 cases) is not a function of an independent coordinate, x for example, then the function ∂Ldx˙ 15 is a constant of the motion. This statement can also be expressed in terms of Hamiltonian 16 dynamics. See for example Cornelius Lanczos, The Variational Principles of Mechanics, 4th ed. 17 (Dover Publications, 1970), Chap VI, Sec. 9, statement above Eq. (69.1). In all the mechanical 18 systems that we consider here, total energy is conserved and thus automatically does not 19 contain time explicitly. The expression for kinetic energy K is always quadratic in the 20 , while the potential energy U is independent of velocities. These conditions are 21 sufficient for the Hamiltonian of the system to be the total energy. Then our limited version 22 of Noether's theorem is identical to the statement in Lanczos referenced above. However, we 23 have prepared for students a simpler, deeper, and more intuitive argument that justifies our 24 modified Noether theorem. 25 25 26 Ruth Chabay and Bruce Sherwood, Matter & Interactions (John Wiley, 2002) Vol. 1, pp 188- 27 194 and 244-248. Page 248 lists background references in Am. J. Phys. 28 26 29 Of course on the molecular level there is no friction. Of the current advanced mechanics 30 texts listed in our references, only Landau and Lifshitz, Ref. 7, mentions sliding friction even 31 in passing (p. 122); concerning motion in a dissipative medium they say (p. 74) " . . . there are 32 no equations of motion in the mechanical sense. Thus the problem of the motion of a body in 33 a medium is not one of mechanics." Their conclusion would surprise Isaac Newton. 34 27 35 Reference 8,Vol. 1, page 4-5. 36 28 37 Don S. Lemons, Perfect Form (Princeton University Press, 1997), Chapter 4. 38 29 39 Slavomir Tuleja, Edwin F. Taylor, Java software: Principle of Least Action Interactive, 2003, 40 available at the website http://www.eftaylor.com/leastaction.html. 41 30 42 Herbert Goldstein, Charles Poole, and John Safko, Classical Mechanics, 3rd ed.(Addison- 43 Wesley, Reading, MA 2002), Section2.7.

21 1 31 2 Reference 8, Vol. II, p. 19-8 3 32 4 Tomas Tyc, "The de Broglie hypothesis leading to path integrals," Eur. J. Phys. 17 (May 5 1996), 156-157. Version available at the URL given in reference 29. 6 33 7 R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965), 8 p. 29. 9 34 10 Reference 7, Chap. 1; Reference 30, Sect. 2.3; D. Gerald Jay Sussman and Jack Wisdom, 11 Structure and Interpretation of Classical Mechanics (MIT, Cambridge, 2001), Chap. 1. 12 35 13 Reference 15 14 36 15 References 14 and 15. 16 37 17 Ya. B. Zeldovich and A. D. Myskis, Elements of Applied Mathematics, George Yankovsky Tr. 18 (MIR Publishers Moscow,1976) pages 499-500

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