American Journal of Physics, Vol. 72, No. 4, pp. 514–521, April 2004
©2004 American Association of Physics Teachers. All rights reserved.
From conservation of^{ }energy to the principle of least action: A story line
Technical^{ }University, Faculty of Metallurgy, Department of Metal Forming, Vysokoskolska 4,^{ }042 00 Kosice, Slovakia
Edwin F. Taylor^{b)}
Massachusetts Institute of Technology, Department of Physics, Cambridge,^{ }Massachusetts 02139
Received: 27 August 2003; accepted: 5 December 2003We outline a story line that^{ }introduces Newtonian mechanics by employing conservation of energy to predict^{ }the motion of a particle in a onedimensional potential. We^{ }show that incorporating constraints and constants of the motion into^{ }the energy expression allows us to analyze more complicated systems.^{ }A heuristic transition embeds kinetic and potential energy into the^{ }still more powerful principle of least action. © 2004 American^{ }Association of Physics Teachers. ^{ }
Contents
I.^{ }INTRODUCTION
An eight sentence history of Newtonian mechanics^{1} shows how much^{ }the subject has developed since Newton introduced F = dp/dt in the^{ }second half of the 1600s.^{2} In the mid1700s Euler devised^{ }and applied a version of the principle of least action^{ }using mostly geometrical methods. In 1755 the 19yearold JosephLouis Lagrange^{ }sent Euler a letter that streamlined Euler's methods into algebraic^{ }form. "[A]fter seeing Lagrange's work Euler dropped his own method,^{ }espoused that of Lagrange, and renamed the subject the calculus^{ }of variations."^{3} Lagrange, in his 1788 Analytical Mechanics,^{4} introduced what^{ }we call the Lagrangian function and Lagrange's equations of motion.^{ }About half a century later (1834–1835) Hamilton published Hamilton's principle,^{5}^{ }to which Landau and Lifshitz^{6} and Feynman^{7} reassigned the name^{ }principle of least action.^{8} Between 1840 and 1860 conservation of^{ }energy was established in all its generality.^{9} In 1918 Noether^{10}^{ }proved several relations between symmetries and conserved quantities. In the^{ }1940s Feynman^{11} devised a formulation of quantum mechanics that not^{ }only explicitly underpins the principle of least action, but also^{ }shows the limits of validity of Newtonian mechanics. ^{ }
Except for^{ }conservation of energy, students in introductory physics are typically introduced^{ }to the mechanics of the late 1600s. To modernize this^{ }treatment, we have suggested^{12} that the principle of least action^{ }and Lagrange's equations become the basis of introductory Newtonian mechanics.^{ }Recent articles discuss how to use elementary calculus to derive^{ }Newton's laws of motion,^{13} Lagrange's equations,^{14} and examples of Noether's^{ }theorem^{15} from the principle of least action, describe the modern^{ }rebirth of Euler's methods^{16} and suggest ways in which upper^{ }undergraduate physics classes can be transformed using the principle of^{ }least action.^{17} ^{ }
How are these concepts and methods to be^{ }introduced to undergraduate physics students? In this paper we suggest^{ }a reversal of the historical order: Begin with conservation of^{ }energy and graduate to the principle of least action and^{ }Lagrange's equations. The mathematical prerequisites for the proposed course include^{ }elementary trigonometry, polar coordinates, introductory differential calculus, partial derivatives, and^{ }the idea of the integral as a sum of increments.^{ }^{ }
The story line presented in this article omits most details^{ }and is offered for discussion, correction, and elaboration. We do^{ }not believe that a clear story line guarantees student understanding.^{ }On the contrary, we anticipate that trials of this approach^{ }will open up new fields of physics education research. ^{ }
II. ONEDIMENSIONAL MOTION: ANALYTIC^{ }SOLUTIONS
We start by using conservation of energy to analyze particle^{ }motion in a onedimensional potential. Much of the power of^{ }the principle of least action and its logical offspring, Lagrange's^{ }equations, results from the fact that they are based on^{ }energy, a scalar. When we start with conservation of energy,^{ }we not only preview more advanced concepts and procedures, but^{ }also invoke some of their power. For example, expressions for^{ }the energy which are consistent with any constraints automatically eliminate^{ }the corresponding constraint forces from the equations of motion. By^{ }using the constraints and constants of the motion, we often^{ }can reduce the description of multidimensional systems to one coordinate,^{ }whose motion can then be found using conservation of energy.^{ }Equilibrium and statics also derive from conservation of energy. ^{ }
We^{ }first consider onedimensional motion in a uniform vertical gravitational field.^{ }Heuristic arguments lead to the expression mgy for the potential^{ }energy. We observe, with Galileo, that the velocity of a^{ }particle in free fall from rest starting at position y = 0^{ }decreases linearly with time: = –gt, where we have expressed the^{ }time derivative by a dot over the variable. This relation^{ }integrates to the form
We multiply Eq. (1) by mg^{ }and rearrange terms to obtain the first example of conservation^{ }of energy:
where the symbol E represents the total energy^{ }and the symbols K and U represent the kinetic and^{ }potential energy, respectively. ^{ }
A complete description of the motion of^{ }a particle in a general conservative onedimensional potential follows from^{ }the conservation of energy. Unfortunately, an explicit function of the^{ }position versus time can be derived for only a fraction^{ }of such systems. Students should be encouraged to guess analytic^{ }solutions, a powerful general strategy because any proposed solution is^{ }easily checked by substitution into the energy equation. Heuristic guesses^{ }are assisted by the fact that the first time derivative^{ }of the position, not the second, appears in the energy^{ }conservation equation. ^{ }
The following example illustrates the guessing strategy for^{ }linear motion in a parabolic potential. This example also introduces^{ }the potential energy diagram, a central feature of our story^{ }line and an important tool in almost every undergraduate subject.^{ }^{ }
A. Harmonic oscillator
Our analysis begins with a qualitative prediction of^{ }the motion of a particle in a parabolic potential (or^{ }in any potential with motion bounded near a single potential^{ }energy minimum). If we consider the potential energy diagram for^{ }a fixed total energy, we can predict that the motion^{ }will be periodic. For a parabolic potential (Fig. 1) conservation^{ }of energy is expressed as
We rearrange Eq. (3) to^{ }read
Equation (4) reminds us of the trigonometric identity,
We^{ }set = t and equate the righthand sides of Eqs. (4)^{ }and (5) and obtain the solution
If we take the^{ }time derivative of x in Eq. (6) and use it^{ }to equate the lefthand sides of Eqs. (4) and (5),^{ }we find that^{18}
The fact that does not depend^{ }on the total energy E of the particle means that^{ }the period is independent of E and hence independent of^{ }the amplitude of oscillation described by Eq. (6). The simple^{ }harmonic oscillator is widely applied because many potential energy curves^{ }can be approximated as parabolas near their minima. ^{ }
Figure 1. At this^{ }point, it would be desirable to introduce the concept of^{ }the worldline, a position versus time plot that completely describes^{ }the motion of a particle. ^{ }
III. ACCELERATION AND FORCE
In^{ }the absence of dissipation, the force can be defined in^{ }terms of the energy. We start with conservation of energy:^{ }
We take the time derivative of both sides and use^{ }the chain rule:
By invoking the tendency of a ball^{ }to roll downhill, we can define force as the negative^{ }spatial derivative of the potential,
from which we see that^{ }F = –kx for the harmonic potential and F = –mg for the gravitational^{ }force near the earth's surface. ^{ }
IV. NONINTEGRABLE^{ }MOTIONS IN ONE DIMENSION
We guessed a solution for simple harmonic^{ }motion, but it is important for students to know that^{ }for most mechanical systems, analytical solutions do not exist, even^{ }when the potential can be expressed analytically. For these cases,^{ }our strategy begins by asking students to make a detailed^{ }qualitative prediction of the motion using the potential energy diagram,^{ }for example, describing the velocity, acceleration, and force at different^{ }particle positions, such as A through F in Fig. 1.^{ }^{ }
The next step might be to ask the student to^{ }plot by hand a few sequential points along the worldline^{ }using a difference equation derived from energy conservation corresponding to^{ }Eq. (3),
where U(x) describes an arbitrary potential energy. The^{ }process of plotting necessarily invokes the need to specify the^{ }initial conditions, raises the question of accuracy as a function^{ }of step size, and forces an examination of the behavior^{ }of the solution at the turning points. Drawing the resulting^{ }worldline can be automated using a spreadsheet with graphing capabilities,^{ }perhaps comparing the resulting approximate curve with the analytic solution^{ }for the simple harmonic oscillator. ^{ }
After the drudgery of these^{ }preliminaries, students will welcome a more polished interactive display that^{ }numerically integrates the particle motion in a given potential. On^{ }the potential energy diagram (see Fig. 2), the student sets^{ }up initial conditions by dragging the horizontal energy line up^{ }or down and the particle position left or right. The^{ }computer then moves the particle back and forth along the^{ }Eline at a rate proportional to that at which the^{ }particle will move, while simultaneously drawing the worldline (upper plot^{ }of Fig. 2). ^{ }
Figure 2. V. REDUCTION TO ONE COORDINATE
The analysis of^{ }onedimensional motion using conservation of energy is powerful but limited.^{ }In some important cases we can use constraints and constants^{ }of the motion to reduce the description to one coordinate.^{ }In these cases we apply our standard procedure: qualitative analysis^{ }using the (effective) potential energy diagram followed by interactive computer^{ }solutions. We illustrate this procedure by some examples. ^{ }
A. Projectile^{ }motion
For projectile motion in a vertical plane subject to a^{ }uniform vertical gravitational field in the ydirection, the total energy^{ }is
The potential energy is not a function of x;^{ }therefore, as shown in the following, momentum in the xdirection,^{ }p_{x}, is a constant of the motion. The energy equation^{ }becomes
In Newtonian mechanics the zero of the energy is^{ }arbitrary, so we can reduce the energy to a single^{ }dimension y by making the substitution:
^{ }
Our analysis of projectile^{ }motion already has applied a limited version of a powerful^{ }theorem due to Noether.^{19} The version of Noether's theorem used^{ }here says that when the total energy E is not^{ }an explicit function of an independent coordinate, x for example,^{ }then the function E/ is a constant of the motion.^{20}^{ }We have developed a simple, intuitive derivation of this version^{ }of Noether's theorem. The derivation is not included in this^{ }brief story line. ^{ }
The above strategy uses a conservation law^{ }to reduce the number of dimensions. The following example uses^{ }constraints to the same end. ^{ }
B. Object rolling without slipping
Twodimensional^{ }circular motion and the resulting kinetic energy are conveniently described^{ }using polar coordinates. The fact that the kinetic energy is^{ }an additive scalar leads quickly to its expression in terms^{ }of the moment of inertia of a rotating rigid body.^{ }When the rotating body is symmetric about an axis of^{ }rotation and moves perpendicularly to this axis, the total kinetic^{ }energy is equal to the sum of the energy of^{ }rotation plus the energy of translation of the center of^{ }mass. (This conclusion rests on the addition of vector components,^{ }but does not require the parallel axis theorem.) Rolling without^{ }slipping is a more realistic idealization than sliding without friction.^{ }^{ }
We use these results to reduce to one dimension the^{ }description of a marble rolling along a curved ramp that^{ }lies in the xy plane in a uniform gravitational field.^{ }Let the marble have mass m, radius r, and moment^{ }of inertia I_{marble}. The nonslip constraint tells us that v = r.^{ }Conservation of energy leads to the expression:
where
The motion^{ }is described by the single coordinate y. Constraints are used^{ }twice in this example: explicitly in rolling without slipping and^{ }implicitly in the relation between the height y and the^{ }displacement along the curve. ^{ }
C. Motion in a central gravitational^{ }field
We analyze satellite motion in a central inversesquare gravitational field^{ }by choosing the polar coordinates r and in the^{ }plane of the orbit. The expression for the total energy^{ }is
The angle does not appear in Eq. (17).^{ }Therefore we expect that a constant of the motion is^{ }given by E/, which represents the angular momentum J. (We^{ }reserve the standard symbol L for the Lagrangian, introduced later^{ }in this paper.)
We substitute the resulting expression for ^{ }into Eq. (17) and obtain
Students employ the plot of^{ }the effective potential energy U_{eff}(r) to do the usual qualitative^{ }analysis of radial motion followed by computer integration. We emphasize^{ }the distinctions between bound and unbound orbits. With additional use^{ }of Eq. (18), the computer can be programmed to plot^{ }a trajectory in the plane for each of these cases.^{ }^{ }
D. Marble, ramp, and turntable
This system is more complicated, but^{ }is easily analyzed by our energybased method and more difficult^{ }to treat using F = ma. A relatively massive marble of mass^{ }m and moment of inertia I_{marble} rolls without slipping along^{ }a slot on an inclined ramp fixed rigidly to a^{ }light turntable which rotates freely so that its angular velocity^{ }is not necessarily constant (see Fig. 3). The moment of^{ }inertia of the combined turntable and ramp is I_{rot}. We^{ }assume that the marble stays on the ramp and find^{ }its position as a function of time. ^{ }
Figure 3. We start with^{ }a qualitative analysis. Suppose that initially the turntable rotates and^{ }the marble starts at rest with respect to the ramp.^{ }If the marble then begins to roll up the ramp,^{ }the potential energy of the system increases, as does the^{ }kinetic energy of the marble due to its rotation around^{ }the center of the turntable. To conserve energy, the rotation^{ }of the turntable must decrease. If instead the marble begins^{ }to roll down the ramp, the potential energy decreases, the^{ }kinetic energy of the marble due to its rotation around^{ }the turntable axis decreases, and the rotation rate of the^{ }turntable will increase to compensate. For a given initial rotation^{ }rate of the turntable, there may be an equilibrium value^{ }at which the marble will remain at rest. If the^{ }marble starts out displaced from this value, it will oscillate^{ }back and forth along the ramp. ^{ }
More quantitatively, the square^{ }of the velocity of the marble is
Conservation of energy^{ }yields the relation
where M indicates the marble's mass augmented^{ }by the energy effects of its rolling along the ramp,^{ }Eq. (16). The right side of Eq. (21) is not^{ }an explicit function of the angle of rotation . Therefore^{ }our version of Noether's theorem tells us that E/ is^{ }a constant of the motion, which we recognize as the^{ }total angular momentum J:
If we substitute into Eq. (21)^{ }the expression for from Eq. (22), we obtain
The^{ }values of E and J are determined by the initial^{ }conditions. The effective potential energy U_{eff}(x) may have a minimum^{ }as a function of x, which can result in oscillatory^{ }motion of the marble along the ramp. If the marble^{ }starts at rest with respect to the ramp at the^{ }position of minimum effective potential, it will not move along^{ }the ramp, but execute a circle around the center of^{ }the turntable. ^{ }
VI. EQUILIBRIUM AND STATICS: PRINCIPLE OF LEAST POTENTIAL^{ }ENERGY
Our truncated story line does not include frictional forces or^{ }an analysis of the tendency of systems toward increased entropy.^{ }Nevertheless, it is common experience that motion usually slows down^{ }and stops. Our use of potential energy diagrams makes straightforward^{ }the intuitive formulation of stopping as a tendency of a^{ }system to reach equilibrium at a local minimum of the^{ }potential energy. This result can be formulated as the principle^{ }of least potential energy for systems in equilibrium. Even in^{ }the absence of friction, a particle placed at rest at^{ }a point of zero slope in the potential energy curve^{ }will remain at rest. (Proof: An infinitesimal displacement results in^{ }zero change in the potential energy. Due to energy conservation,^{ }the change in the kinetic energy must also remain zero.^{ }Because the particle is initially at rest, the zero change^{ }in kinetic energy forbids displacement.) Equilibrium is a result of^{ }conservation of energy. ^{ }
Here, as usual, Feynman is ahead of^{ }us. Figure 4 shows an example from his treatment of^{ }statics.^{21} The problem is to find the value of the^{ }hanging weight W that keeps the structure at rest, assuming^{ }a beam of negligible weight. Feynman balances the decrease in^{ }the potential energy when the weight W drops 4^{} with^{ }increases in the potential energy for the corresponding 2^{} rise^{ }of the 60 lb weight and the 1^{} rise in^{ }the 100 lb weight. He requires that the net potential^{ }energy change of the system be zero, which yields
or^{ }W = 55 lb. Feynman calls this method the principle of virtual work,^{ }which in this case is equivalent to the principle of^{ }least potential energy, both of which express conservation of energy.^{ }^{ }
Figure 4. There are many examples of the principle of minimum potential^{ }energy, including a mass hanging on a spring, a lever,^{ }hydrostatic balance, a uniform chain suspended at both ends, a^{ }vertically hanging slinky,^{22} and a ball perched on top of^{ }a large sphere. ^{ }
VII.^{ }FREE PARTICLE. PRINCIPLE OF LEAST AVERAGE KINETIC ENERGY
For all its^{ }power, conservation of energy can predict the motion of only^{ }a fraction of mechanical systems. In this and Sec. VIII^{ }we seek a principle that is more fundamental than conservation^{ }of energy. One test of such a principle is that^{ }it leads to conservation of energy. Our investigations will employ^{ }trial worldlines that are not necessarily consistent with conservation of^{ }energy. ^{ }
We first think of a free particle initially at^{ }rest in a region of zero potential energy. Conservation of^{ }energy tells us that this particle will remain at rest^{ }with zero average kinetic energy. Any departure from rest, say^{ }by moving back and forth, will increase the average of^{ }its kinetic energy from the zero value. The actual motion^{ }of this free particle gives the least average kinetic energy.^{ }The result illustrates what we will call the principle of^{ }least average kinetic energy. ^{ }
Now we view the same particle^{ }from a reference frame moving in the negative xdirection with^{ }uniform speed. In this frame the particle moves along a^{ }straight worldline. Does this worldline also satisfy the principle of^{ }least average kinetic energy? Of course, but we can check^{ }this expectation and introduce a powerful graphical method established by^{ }Euler.^{16} ^{ }
We fix two events A and C at the^{ }ends of the worldline (see Fig. 5) and vary the^{ }time of the central event B so that the kinetic^{ }energy is not the same on the legs labeled 1^{ }and 2. Then the time average of the kinetic energy^{ }K along the two segments 1 and 2 of the^{ }worldline is given by
We multiply both sides of Eq.^{ }(25) by the fixed total time and recast the velocity^{ }expressions using the notation in Fig. 5:
We find the^{ }minimum value of the average kinetic energy by taking the^{ }derivative with respect to the time t of the central^{ }event:
From Eq. (27) we obtain the equality
As expected,^{ }when the timeaveraged kinetic energy has a minimum value, the^{ }kinetic energy for a free particle is the same on^{ }both segments; the worldline is straight between the fixed initial^{ }and final events A and C and satisfies the principle^{ }of minimum average kinetic energy. The same result follows if^{ }the potential energy is uniform in the region under consideration,^{ }because the uniform potential energy cannot affect the kinetic energy^{ }as the location of point B changes on the spacetime^{ }diagram. ^{ }
Figure 5. In summary, we have illustrated the fact that for^{ }the special case of a particle moving in a region^{ }of zero (or uniform) potential energy, the kinetic energy is^{ }conserved if we require that the time average of the^{ }kinetic energy has a minimum value. The general expression for^{ }this average is
^{ }
In an introductory text we might introduce^{ }at this point a sidebar on Fermat's principle of least^{ }time for the propagation of light rays. ^{ }
Now we are^{ }ready to develop a similar but more general law that^{ }predicts every central feature of mechanics. ^{ }
VIII. PRINCIPLE OF^{ }LEAST ACTION
Section VI discussed the principle of least potential energy^{ }and Sec. VII examined the principle of least average kinetic^{ }energy. Along the way we mentioned Fermat's principle of least^{ }time for ray optics. We now move on to the^{ }principle of least action, which combines and generalizes the principles^{ }of least potential energy and least average kinetic energy. ^{ }
A.^{ }Qualitative demonstration
We might guess (incorrectly) that the time average of^{ }the total energy, the sum of the kinetic and potential^{ }energy, has a minimum value between fixed initial and final^{ }events. To examine the consequences of this guess, let us^{ }think of a ball thrown in a uniform gravitational field,^{ }with the two events, pitch and catch, fixed in location^{ }and time (see Fig. 6). How will the ball move^{ }between these two fixed events? We start by asking why^{ }the baseball does not simply move at constant speed along^{ }the straight horizontal trajectory B in Fig. 6. Moving from^{ }pitch to catch with constant kinetic energy and constant potential^{ }energy certainly satisfies conservation of energy. But the straight horizontal^{ }trajectory is excluded because of the importance of the averaged^{ }potential energy. ^{ }
Figure 6. We need to know how the average kinetic^{ }energy and average potential energy vary with the trajectory. We^{ }begin with the idealized triangular path T shown in Fig.^{ }6. If we assume a fixed time between pitch and^{ }catch and that the speed does not vary wildly along^{ }the path, the kinetic energy of the particle is approximately^{ }proportional to the square of the distance covered, that is,^{ }proportional to the quantity x + y using the notation in Fig.^{ }6. The increase in the kinetic energy over that of^{ }the straight path is proportional to the square of the^{ }deviation y_{0}, whether that deviation is below or above the^{ }horizontal path. But any incremental deviation from the straightline path^{ }can be approximated by a superposition of such triangular increments^{ }along the path. As a result, small deviations from the^{ }horizontal path result in an increase in the average kinetic^{ }energy approximately proportional to the average of the square of^{ }the vertical deviation y_{0}. ^{ }
The average potential energy increases or^{ }decreases for trajectories above or below the horizontal, respectively. The^{ }magnitude of the change in this average is approximately proportional^{ }to the average deviation y_{0}, whether this deviation is small^{ }or large. ^{ }
We can apply these conclusions to the average^{ }of E = K + U as the path departs from the horizontal. For^{ }paths above the horizontal, such as C, D, and E^{ }in Fig. 6, the averages of both K and U^{ }increase with deviation from the horizontal; these increases have no^{ }limit for higher and higher paths. Therefore, no upward trajectory^{ }minimizes the average of K + U. In contrast, for paths that^{ }deviate downward slightly from the horizontal, the average K initially^{ }increases slower than the average U decrease, leading to a^{ }reduction in their sum. For paths that dip further, however,^{ }the increase in the average kinetic energy (related to the^{ }square of the path length) overwhelms the decrease in the^{ }average potential energy (which decreases only as y_{0}). So there^{ }is a minimum of the average of K + U for some^{ }path below the horizontal. The path below the horizontal that^{ }satisfies this minimum is clearly not the trajectory observed for^{ }a pitched ball. ^{ }
Suppose instead we ask how the average^{ }of the difference K–U behaves for paths that deviate from^{ }the horizontal. By an argument similar to that in the^{ }preceding paragraph, we see that no path below the horizontal^{ }can have a minimum average of the difference. But there^{ }exists a path, such as D, above the horizontal for^{ }which the average of K–U is a minimum. For the^{ }special case of vertical launch, students can explore this conclusion^{ }interactively using tutorial software.^{23} ^{ }
B. Analytic demonstration
We can check the^{ }preceding result analytically in the simplest case we can imagine^{ }(see Fig. 7). In a uniform vertical gravitational field a^{ }marble of mass m rolls from one horizontal surface to^{ }another via a smooth ramp so narrow that we may^{ }neglect its width. In this system the potential energy changes^{ }just once, halfway between the initial and final positions. ^{ }
Figure 7. The^{ }worldline of the particle will be bent, corresponding to the^{ }reduced speed after the marble mounts the ramp, as shown^{ }in Fig. 8. We require that the total travel time^{ }from position A to position C have a fixed value^{ }t_{total} and check whether minimizing the time average of the^{ }difference K–U leads to conservation of energy:
We define a^{ }new symbol S, called the action:
We can use the^{ }notation in Fig. 8 to write Eq. (31) in the^{ }form
where M is given by Eq. (16). We require^{ }that the value of the action be a minimum with^{ }respect to the choice of the intermediate time t:
This^{ }result can be written as
A simple rearrangement shows that^{ }Eq. (34) represents conservation of energy. We see that energy^{ }conservation has been derived from the more fundamental principle of^{ }least action. Equally important, the analysis has completely determined the^{ }worldline of the marble. ^{ }
Figure 8. The action Eq. (31) can be^{ }generalized for a potential energy curve consisting of multiple steps^{ }connected by smooth, narrow transitions, such as the one shown^{ }in Fig. 9:
The argument leading to conservation of energy,^{ }Eq. (34), applies to every adjacent pair of steps in^{ }the potential energy diagram. The computer can hunt for and^{ }find the minimum value of S directly by varying the^{ }values of the intermediate times, as shown schematically in Fig.^{ }10. ^{ }
Figure 9. Figure 10. A continuous potential energy curve can be regarded as^{ }the limiting case of that shown in Fig. 9 as^{ }the number of steps increases without limit while the time^{ }along each step becomes an increment t. For the resulting^{ }potential energy curve the general expression for the action S^{ }is
Here L (= K–U for the cases we treat) is^{ }called the Lagrangian. The principle of least action says that^{ }the value of the action S is a minimum for^{ }the actual motion of the particle, a condition that leads^{ }not only to conservation of energy, but also to a^{ }unique specification of the entire worldline. ^{ }
More general forms of^{ }the principle of least action predict the motion of a^{ }particle in more than one spatial dimension as well as^{ }the time development of systems containing many particles. The principle^{ }of least action can even predict the motion of some^{ }systems in which energy is not conserved.^{24} The Lagrangian L^{ }can be generalized so that the principle of least action^{ }can describe relativistic motion^{7} and can be used to derive^{ }Maxwell's equations, Schroedinger's wave equation, the diffusion equation, geodesic worldlines^{ }in general relativity, and steady electric currents in circuits, among^{ }many other applications. ^{ }
We have not provided a proof of^{ }the principle of least action in Newtonian mechanics. A fundamental^{ }proof rests on nonrelativistic quantum mechanics, for instance that outlined^{ }by Tyc^{25} which uses the deBroglie relation to show that^{ }the phase change of a quantum wave along any worldline^{ }is equal to S/, where S is the classical action^{ }along that worldline. Starting with this result, Feynman and Hibbs^{ }have shown^{26} that the sumoverallpaths description of quantum motion reduces^{ }seamlessly to the classical principle of least action as the^{ }masses of particles increase. ^{ }
Once students have mastered the principle^{ }of least action, it is easy to motivate the introduction^{ }of nonrelativistic quantum mechanics. Quantum mechanics simply assumes that the^{ }electron explores all the possible worldlines considered in finding the^{ }Newtonian worldline of least action. ^{ }
IX. LAGRANGE'S^{ }EQUATIONS
Lagrange's equations are conventionally derived from the principle of least^{ }action using the calculus of variations.^{27} The derivation analyzes the^{ }worldline as a whole. However, the expression for the action^{ }is a scalar; if the value of the sum is^{ }minimum along the entire worldline, then the contribution along each^{ }incremental segment of the worldline also must be a minimum.^{ }This simplifying insight, due originally to Euler, allows the derivation^{ }of Lagrange's equations using elementary calculus.^{14} The appendix shows an^{ }alternative derivation of Lagrange's equation directly from Newton's equations. ^{ }
ACKNOWLEDGMENTS
The authors acknowledge useful comments by^{ }Don S. Lemons, Jon Ogborn, and an anonymous reviewer. ^{ }
APPENDIX: DERIVATION OF LAGRANGE'S EQUATIONS FROM NEWTON'S^{ }SECOND LAW
Both Newton's second law and Lagrange's equations can be^{ }derived from the more fundamental principle of least action.^{13}^{,}^{14} Here^{ }we move from one derived formulation to the other, showing^{ }that Lagrange's equation leads to F = ma (and vice versa) for^{ }particle motion in one dimension.^{28} ^{ }
If we use the definition^{ }of force in Eq. (10), we can write Newton's second^{ }law as:
We assume that m is constant and rearrange^{ }and recast Eq. (A1) to read
On the righthand side^{ }Eq. (A2) we add terms whose partial derivatives are equal^{ }to zero:
We then set L = K–U, which leads to the^{ }Lagrange equation in one dimension:
This sequence of steps can^{ }be reversed to show that Lagrange's equation leads to Newton's^{ }second law of motion for conservative potentials. ^{ }
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Citation links [e.g., Phys. Rev. D 40, 2172 (1989)] go to online journal abstracts. Other links (see Reference Information) are available with your current login. Navigation of links may be more efficient using a second browser window.

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Jozef Hanc, Edwin F. Taylor, and Slavomir Tuleja, "Deriving Lagrange's
equations using elementary calculus," accepted for publication in Am.
J. Phys. Preprint available at http://www.eftaylor.com.
first citation in article

Jozef Hanc, Slavomir Tuleja, and Martina Hancova, "Symmetries and
conservations laws: Consequences of Noether's theorem," accepted for
publication in Am. J. Phys. Preprint available at http://www.eftaylor.com.
first citation in article

Jozef Hanc, "The original Euler's calculusofvariations method: Key to
Lagrangian mechanics for beginners," submitted to Am. J. Phys. Preprint
available at http://www.eftaylor.com.
first citation in article

Thomas A. Moore, "Getting the most action out of least action," submitted to Am. J. Phys. Preprint available at http://www.eftaylor.com.
first citation in article

Student exercise: Show that B cos t and C cos t + D sin t are also solutions. Use the occasion to discuss the importance of relative phase.
first citation in article

Benjamin Crowell bases an entire introductory treatment on Noether's theorem: Discover Physics (Light and Matter, Fullerton, CA, 1998–2002). Available at http://www.lightandmatter.com.
first citation in article

Noether's theorem^{10} implies that when the Lagrangian of a system (L = K–U for our simple cases) is not a function of an independent coordinate, x for example, then the function L/
is a constant of the motion. This statement also can be expressed in
terms of Hamiltonian dynamics. See, for example, Cornelius Lanczos, The Variational Principles of Mechanics
(Dover, New York, 1970), 4th ed., Chap. VI, Sec. 9, statement above Eq.
(69.1). In all the mechanical systems that we consider here, the total
energy is conserved and thus automatically does not contain time
explicitly. Also the expression for the kinetic energy K is quadratic in the velocities, and the potential energy U
is independent of velocities. These conditions are sufficient for the
Hamiltonian of the system to be the total energy. Then our limited
version of Noether's theorem is identical to the statement in Lanczos. first citation in article

Reference 7, Vol. 1, p. 45.
first citation in article

Don S. Lemons, Perfect Form (Princeton U.P., Princeton, 1997), Chap. 4.
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Slavomir Tuleja and Edwin F. Taylor, Principle of Least Action Interactive, available at http://www.eftaylor.com.
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Herbert Goldstein, Charles Poole, and John Safko, Classical Mechanics (AddisonWesley, San Francisco, 2002), 3rd ed., Sec. 2.7.
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Tomas Tyc, "The de Broglie hypothesis leading to path integrals," Eur. J. Phys. 17 (5), 156–157 (1996). [Inspec]
An extended version of this derivation is available at http://www.eftaylor.com.
first citation in article

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw–Hill, New York, 1965), p. 29.
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Reference 6, Chap. 1; Ref. 24, Sec. 2.3; D. Gerald Jay Sussman and Jack Wisdom, Structure and Interpretation of Classical Mechanics (MIT, Cambridge, 2001), Chap. 1.
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Ya. B. Zeldovich and A. D. Myskis, Elements of Applied Mathematics, translated by George Yankovsky (MIR, Moscow, 1976), pp. 499–500.
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CITING ARTICLES
This list contains links to other online articles that cite the article currently being viewed.

Hamilton's principle: Why is the integrated difference of the kinetic and potential energy minimized?
Alberto G. Rojo, Am. J. Phys. 73, 831 (2005)

Variational mechanics in one and two dimensions
Jozef Hanc et al., Am. J. Phys. 73, 603 (2005)
FIGURES
Full figure (6 kB)Fig. 1. The potential energy diagram is central to our treatment and^{ }requires a difficult conceptual progression from a ball rolling down^{ }a hill pictured in an xy diagram to a graphical^{ }point moving along a horizontal line of constant energy in^{ }an energyposition diagram. Making this progression allows the student to^{ }describe qualitatively, but in detail, the motion of a particle^{ }in a onedimensional potential at arbitrary positions such as A^{ }through F. First citation in article
Full figure (7 kB)Fig. 2. Mockup of an interactive computer display of the worldline^{ }derived numerically from the energy and the potential energy function. First citation in article
Full figure (8 kB)Fig. 3. System^{ }of marble, ramp, and turntable. First citation in article
Full figure (6 kB)Fig. 4. Feynman's example of the principle of^{ }virtual work. First citation in article
Full figure (10 kB)Fig. 5. The time of the middle event is varied to^{ }determine the path of the minimum timeaveraged kinetic energy. First citation in article
Full figure (10 kB)Fig. 6. Alternative trial^{ }trajectories of a pitched ball. For path D the average^{ }of the difference K–U is a minimum. First citation in article
Full figure (4 kB)Fig. 7. Motion across a potential^{ }energy step. First citation in article
Full figure (8 kB)Fig. 8. Broken worldline of a marble rolling across steps connected^{ }by a narrow ramp. The region on the right is^{ }shaded to represent the higher potential energy of the marble^{ }on the second step. First citation in article
Full figure (5 kB)Fig. 9. A more complicated potential energy diagram, leading^{ }in the limit to a potential energy curve that varies^{ }smoothly with position. First citation in article
Full figure (6 kB)Fig. 10. The computer program temporarily fixes the end events^{ }of a twosegment section, say events B and D, then^{ }varies the time coordinate of the middle event C to^{ }find the minimum value of the total S, then varies^{ }D while C is kept fixed, and so on. (The^{ }endevents A and E remain fixed.) The computer cycles through^{ }this process repeatedly until the value of S does not^{ }change further because this value has reached a minimum for^{ }the worldline as a whole. The resulting worldline approximates the^{ }one taken by the particle. First citation in article
FOOTNOTES
^{a}Electronic mail: jozef.hanc@tuke.sk
^{b}Author to whom correspondence should^{ }be addressed. Electronic mail: eftaylor@mit.edu
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