American Journal of Physics, Vol. 73, No. 7, pp. 603–610, July 2005
©2005 American Association of Physics Teachers. All rights reserved.
Variational^{ }mechanics in one and two dimensions
Technical University, Vysokoskolska 4, 042 00 Kosice,^{ }Slovakia
Edwin F. Taylor^{b)}
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Slavomir Tuleja^{c)}
Gymnazium arm. gen. L.^{ }Svobodu, Komenskeho 4, 066 51 Humenne, Slovakia
(Received: 29 July 2004; accepted: 23 November 2004)We develop^{ }heuristic derivations of two alternative principles of least action. A^{ }particle moving in one dimension can reverse direction at will^{ }if energy conservation is the only criterion. Such arbitrary changes^{ }in the direction of motion are eliminated by demanding that^{ }the Maupertuis–Euler abbreviated action, equal to the area under the^{ }momentum versus position curve in phase space, has the smallest^{ }possible value consistent with conservation of energy. Minimizing the abbreviated^{ }action predicts particle trajectories in two and three dimensions and^{ }leads to the more powerful Hamilton principle of least action,^{ }which not only generates conservation of energy, but also predicts^{ }motion even when the potential energy changes with time. Introducing^{ }action early in the physics program requires modernizing the current^{ }obscure and confusing terminology of variational mechanics. © 2005 American^{ }Association of Physics Teachers. ^{ }
Contents
I.^{ }PREVIEW: LIST OF ACTORS IN ORDER OF APPEARANCE
An asteroid named^{ }Woolsthorpe, roaming between the orbits of Mars and Jupiter, appears^{ }to be heading in the general direction of Earth. Should^{ }we worry? Given the initial position and velocity of the^{ }asteroid and data on the shifting positions of nearby asteroids^{ }and distant planets as well as the Sun, Isaac Newton^{ }(1642–1727) tells us how the asteroid's velocity changes from instant^{ }to instant. By summing the resulting increments, we derive the^{ }reassuring prediction that Woolsthorpe will pass farther from Earth than^{ }our Moon. Newton answers the questions, "What happens next?" ^{ }
Newton's^{ }incremental construction of the path is not so useful in^{ }basketball. Given the initial position and speed (and therefore energy)^{ }of the ball, the shooter wants to know what direction^{ }of launch will place the center of the basketball at^{ }the center of the basketball hoop. Both launch and target^{ }points are defined in space. PierreLouis Moreau de Maupertuis (1698–1759)^{ }and Leonard Euler (1707–1783) offer us their abbreviated principle of^{ }least action^{1} to find the direction in which to launch^{ }the basketball.^{2} Maupertuis and Euler answer the question, "Starting from^{ }here, how do we get to there?" ^{ }
None of the^{ }three, Newton, Maupertuis, or Euler, could easily manage a Moon^{ }shot. The spaceship launches from its Earth orbit and coasts^{ }toward an orbit around the Moon. The Earth and Moon^{ }are in motion; the spaceship must arrive at the correct^{ }point in space when the Moon is nearby. Both launch^{ }and arrival specify events, indexed by time as well as^{ }by location. William Rowan Hamilton (1805–1865) provides the principle of^{ }least action, which can be used to find the required^{ }initial speed (and therefore energy) and direction of launch to^{ }reach the Moon orbit safely. Hamilton answers the question, "Starting^{ }from here now, how do we get to there then?"^{ }^{ }
As these three examples demonstrate, least action principles and Newton's^{ }laws form a powerful combination for the analysis of motion.^{ }Introductory mechanics courses currently employ Newton's vector laws of motion^{ }to engage questions similar to that posed by the hurtling^{ }asteroid. Least action principles, which are powerful tools in analyzing^{ }tasks similar to basketball and Moon shots, are introduced only^{ }in advanced mechanics courses which use difficult and abstract mathematics.^{ }^{ }
Earlier we have advocated starting the study of mechanics with^{ }conservation of energy, leading more or less directly to the^{ }principle of least action.^{3} We have since come to believe^{ }that momentum and Newton's laws deserve their present prominence in^{ }introductory physics, but that action principles can and should be^{ }introduced early, not only because they prepare the way for^{ }advanced mechanics courses, but also because they are fundamental tools^{ }in many fields of physics such as optics, electromagnetism, quantum^{ }mechanics, and relativity. ^{ }
We do not yet have a strategy^{ }for introducing least action; this paper presents a first step^{ }toward that goal, a story line that such an introduction^{ }might follow. Our purpose is to stimulate discussion about introducing^{ }action principles early in the physics curriculum. ^{ }
II. ABBREVIATED ACTION IN ONE DIMENSION
A stone moves with^{ }varying velocity in the y direction in a timeindependent potential^{ }such as that due to gravity near the Earth's surface.^{ }Conservation of energy is (almost) sufficient to predict its motion^{ }in one dimension. The total energy E is
where the^{ }kinetic energy is represented by K, the symbol used in^{ }most introductory physics texts, U(y) is the potential energy, and^{ }v is the speed of the stone. This speed is^{ }approximated by vs/t, where s is the incremental distance covered^{ }by the stone in time t. For motion in the^{ }ydirection s=y. Manipulation of Eq. (1) leads to a relation^{ }between s and the corresponding time increment t:
For the^{ }uniform gravitational field near the Earth's surface, the differential version^{ }of Eq. (2) easily integrates to an analytic solution. But^{ }analytic solutions are available for only a limited number of^{ }potential energy functions. In contrast, the onedimensional motion in all^{ }reasonable potentials is easily predicted using a simple numerical integration^{ }method based on Eq. (2) or improved numerical methods, which^{ }are straightforward, conceptually transparent, and already in the toolkits of^{ }many undergraduates. ^{ }
Interactive displays can encourage students to manipulate fundamental^{ }concepts in mechanics, as illustrated in Fig. 1:^{4} (bottom panel)^{ }the energy diagram; (central panel) the position versus time curve^{ }called the worldline, a term which should be introduced long^{ }before relativity; (top panel) the velocity versus position diagram, which^{ }becomes the phase diagram when the velocity is multiplied by^{ }the mass. ^{ }
Figure 1. The particle motion shown in Fig. 1 leads^{ }to a worldline AB that exhibits two "kinks," sharp changes^{ }in slope. The energy diagram shows that each kink occurs^{ }at the location of an abrupt change in the potential^{ }energy. The result is a sudden change in the velocity^{ }(the velocity is the inverse slope of the worldline), displayed^{ }in the velocity versus position graph. So the kinks in^{ }the worldline AB have a physical basis as idealizations. ^{ }
In^{ }two and three dimensions, conservation of energy alone is not^{ }sufficient to determine the particle motion. The reason is that^{ }energy is a scalar which tells us only the magnitude^{ } s of the next step along the trajectory, not^{ }its direction. Predicting motion in two and three dimensions also^{ }requires the direction of the next step. Equation (2) gives^{ }a preview of this difficulty for onedimensional motion. Conservation of^{ }energy yields s, the magnitude of the incremental displacement y.^{ }For onedimensional motion the two possible directions are either –v^{ }for which y=–s or +v with y=+s. For both cases^{ }Eq. (2) gives only the magnitude s. These two possibilities^{ }are illustrated in Fig. 2. Both worldlines AB and AC^{ }satisfy conservation of energy. Worldline AB is the sensible one,^{ }the worldline that meets our expectations as shown in Fig.^{ }1. In contrast, along the worldline AC the particle reverses^{ }direction twice in the vicinity of the position y_{o}, keeping^{ }the same speed and therefore the same kinetic energy. In^{ }principle, a particle moving in one dimension can reverse direction^{ }at will if energy conservation were the only criterion. ^{ }
Figure 2. The^{ }worldline AC is unacceptable, but not because it has kinks^{ }in it; the worldline AB also has kinks, sudden changes^{ }in velocity at the positions of idealized jumps in the^{ }potential energy as shown in the bottom panel. Rather, the^{ }worldline AC is unacceptable because it contains spontaneous reversals of^{ }the direction unrelated to changes in the potential energy. Worse^{ }is that there is no logical or physical safeguard against^{ }an arbitrary number of such spontaneous reversals of direction. It^{ }is clear that our understanding of particle motion in terms^{ }of conservation of energy alone is incomplete. ^{ }
What other principles^{ }can be used to reject spontaneous reversals of direction? Conservation^{ }of momentum applies to an isolated system; the presence of^{ }a potential energy function due to an external source tells^{ }us that the system is not isolated. Newton's second law^{ }allows us to reject spontaneous reversals of direction. In Fig.^{ }2 the reversal occurs where the potential energy is uniform,^{ }so the application of Newton's second law says that^{ }v(F/m)t=[(–dU/dy)/m]t=0 and prohibits these reversals. However, we seek an alternative^{ }procedure for predicting motion which employs conservation of scalar energy^{ }instead of Newton's vector law. ^{ }
In Fig. 2, worldlines AB^{ }and AC have identical slopes (corresponding to identical velocities) everywhere^{ }except in the dotshaded region of the spacetime diagram. So^{ }the velocity versus position curves provide a useful tool for^{ }highlighting the difference between these two possible motions. Pay attention^{ }to the area under the velocity versus position graphs in^{ }Figs. 1 and 2. The worldline AB generates the diagonally^{ }shaded area in Fig. 2, identical to the diagonally shaded^{ }area in Fig. 1. The velocities along the worldline AC^{ }are identical to those along AB except at the double^{ }jog near y_{o}. The resulting multiplevalued velocity versus position graph^{ }for AC encloses the same diagonally shaded area, and in^{ }addition encloses the dotshaded areas labeled 1 and 2. (Area^{ }1 is enclosed twice during the motion described by worldline^{ }AC.) ^{ }
The area enclosed by the velocity versus position graph^{ }provides a handle by which to understand the difference between^{ }the realistic motion described by worldline AB and the unrealistic^{ }motion depicted by worldline AC. One way to eliminate spontaneous^{ }reversals of motion as a possible feature of the worldline^{ }is to demand that the area under the v versus^{ }y curve have the smallest possible value consistent with conservation^{ }of energy. ^{ }
The momentum versus position diagram is called the^{ }phase diagram, and the trajectory in this diagram is called^{ }the phase curve. The area under the phase curve is^{ }the abbreviated action^{5} and is given the symbol S_{o}:
The^{ }reason for "abbreviated" will become apparent. The condition that eliminates^{ }the velocityreversing jogs in the worldline now becomes: The particle^{ }moves so that the abbreviated action has the smallest possible^{ }value, subject to conservation of energy. ^{ }
The area under the^{ }phase curve can be expressed as:^{6}
Equation (4) uses the^{ }speed v and the distance ds instead of the velocity^{ }dy/dt and the displacement dy, because all incremental contributions to^{ }the integral are positive. For a particle moving in the^{ }positive ydirection in Fig. 2, its velocity has the same^{ }positive value as the speed v and the incremental displacement^{ }dy is positive and equals ds. Therefore the incremental contribution^{ }to the area is positive: mv ds. For a particle moving^{ }in the negative ydirection, its velocity (–v) is the negative^{ }of the speed v and the incremental displacement (dy=–ds) is^{ }the negative of the incremental distance ds. As a result,^{ }the incremental contribution to the area also is positive: m(–v)(–ds)=mvds.^{ }^{ }
The abbreviated principle of least action requires that the value^{ }of S_{o} be a minimum for the actual motion of^{ }a particle. In other words, from all possible nearby trajectories^{ }in space beginning at a fixed launch point and ending^{ }at a fixed target point, the particle traverses the trajectory^{ }with the smallest value of S_{o}. The construction of the^{ }phase curve, m times the velocity plotted in Fig. 2,^{ }requires conservation of energy, so conservation of energy is assumed^{ }in the abbreviated principle of least action. Note that the^{ }abbreviated principle of least action is a prescription for the^{ }trajectory as a whole, which is different in character from^{ }the prescription provided by Newton's second law. ^{ }
III. ABBREVIATED ACTION IN^{ }TWO DIMENSIONS
The action integral, Eq. (4), also applies in two^{ }and three dimensions, where v is the speed and ds^{ }is the incremental distance along the two or threedimensional trajectory.^{ }The minimization of the action S_{o} selects from possible nearby^{ }energyconserving trajectories the actual trajectory followed by the particle. For^{ }simplicity, we assume that the potential energy function varies with^{ }the ycoordinate only, for example, U(y)=mgy for a basketball. ^{ }
To^{ }find a trajectory y(x) that leads to a minimum value^{ }of S_{o} in Eq. (4) for fixed initial and final^{ }locations, we divide and conquer. If the entire action integral^{ }is a minimum, then the contribution to the integral from^{ }each infinitesimal portion of the trajectory also must be a^{ }minimum.^{7} Figure 3 shows the initial infinitesimal portion of a^{ }trajectory in the xy plane (the trajectory is not a^{ }worldline). The dotted curve represents the beginning portion of a^{ }trial trajectory approximated by line segments A, B, and C.^{ }This approximation can be made as accurate as desired by^{ }choosing points 1–4 sufficiently close together. ^{ }
Figure 3. Figure 4 shows an^{ }alternative initial portion of the trial trajectory derived by displacing^{ }point 2 in the ydirection. Conservation of energy and the^{ }local value of the potential energy fix the average speed^{ }of the particle along each segment in Figs. 3 and^{ }4. Finding the portion of the trajectory given by segments^{ }A and B reduces to using algebra and elementary calculus^{ }to find the yposition of the center point 2 that^{ }minimizes the abbreviated action for these segments. Equation (4) expresses^{ }the abbreviated action S_{oAB} along segments A and B:
The^{ }lower case s in s_{A} and s_{B} refers to the^{ }incremental length of each segment; v_{A} and v_{B} are the^{ }corresponding average speeds whose values are derived from conservation of^{ }energy. To find the minimum value of the abbreviated action^{ }S_{oAB}, we take the derivative of both sides of Eq.^{ }(5) with respect to y (all remaining coordinates are fixed)^{ }and set the result equal to zero:
The notation in^{ }Fig. 4 leads to expressions for each of the terms^{ }in Eq. (6). The Pythagorean theorem tells us how the^{ }length s_{A} varies with y, the independent vertical coordinate of^{ }point 2:
Equation (6) requires the derivative of s_{A} with^{ }respect to y:
Conservation of energy determines the value of^{ }the average speed along each segment:
Here U_{A} is the^{ }average potential energy along segment A, approximated as the average^{ }of the values at its two ends:
The derivative of^{ }v_{A} with respect to y comes from Eqs. (9) and^{ }(10):
The time t_{2}–t_{1} taken to traverse segment A equals^{ }the length of the segment s_{A} divided by the average^{ }speed v_{A} across the segment:
The expressions for segment A^{ }from Eqs. (7,8,9,10,11,12) plus the corresponding expressions for segment B^{ }allow us to recast Eq. (6) into the form
and^{ }lead to a powerful result which is hidden in the^{ }notation:
The numerator on the right side of Eq. (14)^{ }is the difference between the average ymomenta p_{yA} and p_{yB}^{ }on segments A and B. The denominator approximates the time^{ }of travel from the midpoint of segment A to the^{ }midpoint of segment B. The right side of Eq. (14)^{ }thus approximates the time derivative of the ycomponent of the^{ }particle momentum. The left side of Eq. (14) is the^{ }value of the quantity –dU/dy at the displaced point 2,^{ }the expression for the ycomponent of the force in a^{ }given potential. Thus Eq. (14) is an approximation to the^{ }ycomponent of Newton's second law of motion, an approximation that^{ }becomes exact for infinitesimal segments:
where the dot over the^{ }momentum p indicates the time derivative. ^{ }
Figure 4. Alternatively, minimizing the abbreviated^{ }action S_{oAB} along segments A and B by varying the^{ }xcoordinate of the center point 2 leads to
Equation (16)^{ }expresses the conservation of xmomentum between segments A and B^{ }and is in accord with Newton's second law for the^{ }case in which the potential energy U is not a^{ }function of x, namely F_{x}=–dU/dx=0=_{x} so that p_{x} does not^{ }change with time. ^{ }
If your allegiance is to Newton's second^{ }law, then you can treat Eqs. (15) and (16) as^{ }validating the abbreviated principle of least action when the energy^{ }is conserved. Alternatively, you can view Eqs. (15) and (16)^{ }as demonstrating the priority of the abbreviated principle of least^{ }action, because Newton's second law and conservation of momentum both^{ }grow out of it. We prefer to emphasize here the^{ }consistency among these different predictors of motion. ^{ }
IV. CONSTRUCTING THE^{ }TRAJECTORY
The procedure outlined in Sec. III easily adapts to constructing^{ }the trajectory of a particle with fixed endpoints and conserved^{ }total energy. Let the segments in the lefthand graph in^{ }Fig. 5 be infinitesimal portions of a trial trajectory drawn^{ }arbitrarily to connect the fixed launch point (point 1) with^{ }the fixed target point (not shown) for a particle of^{ }fixed total energy. Points 2–4 are a sequence of adjacent^{ }points along the beginning of the trial trajectory. As explained^{ }in the caption of Fig. 5, we first vary the^{ }ycoordinate and then the xcoordinate of each intermediate point to^{ }find the location that minimizes S_{o} along the two adjacent^{ }segments. Repeat the sweep along the entire modified trial trajectory^{ }until no further displacements of the intermediate points occur.^{8} The^{ }resulting path approximates the trajectory taken by the particle. Equations^{ }(14,15,16) tell us that at every intermediate point along the^{ }resulting trajectory Newton's second law holds. ^{ }
Figure 5. The total energy and^{ }the value of the potential energy at the launch point^{ }yields the launch speed and the completed trajectory determines the^{ }initial direction of motion at launch. Thus the abbreviated principle^{ }of least action tells us how to launch the particle^{ }(speed and direction) from the fixed initial point so that^{ }it will arrive at the target. ^{ }
The trajectory alone does^{ }not fully describe the motion. A complete description includes not^{ }only the trajectory, but also the time at which the^{ }particle passes each point along the trajectory, plotted as the^{ }worldline. The basketball player does not care when the center^{ }of the ball reaches the center of the hoop.^{9} Nevertheless,^{ }our procedure can easily be extended to find the time^{ }at which the particle passes each point along the trajectory.^{ }Equation (12) yields the time t_{2}–t_{1} along segment A and^{ }similar equations give the time along later segments, resulting in^{ }the total time from the launch to any point on^{ }the trajectory. The combination of the trajectory plus the time^{ }at each point on the trajectory, embodied in the worldline,^{ }provides a complete description of the flight of the basketball.^{ }^{ }
A standard result of projectile motion for a value of^{ }energy greater than the minimum is the possibility of two^{ }trajectories between the launch and target, one path higher and^{ }of longer duration, the other flatter and of shorter duration.^{ }The technique for constructing trajectories described here will discover only^{ }one of these, depending on the initial arbitrarily chosen trial^{ }path. Finding the second trajectory requires a different initial trial^{ }function. This case, with its alternative analytic solutions, is a^{ }good one for introducing students to the qualitative skills required^{ }to guess trial trajectories when multiple paths are possible. ^{ }
V. JUST PLAIN ACTION
A^{ }small additional step reveals a truly remarkable and general expression^{ }due to Hamilton,^{10} for which the name action stands powerfully^{ }alone, with no modifier. This action earns the symbol S,^{ }without a subscript, and is defined as:
The following simple^{ }expression relates the action S to the abbreviated action S_{o}:^{11}^{ }
Equation (18) can be derived from an extension of Eq.^{ }(4):
The first integral on the left in Eq. (19)^{ }is a summation along the trajectory in space, from the^{ }launch to the target. In contrast, the last integral on^{ }the right in Eq. (19) multiplies twice the kinetic energy^{ }along each segment of the path by the incremental time^{ }required to traverse that segment and sums the result. The^{ }last integral can be regarded as a summation along the^{ }worldline. Adding and subtracting the potential energy function U on^{ }the right side of Eq. (19) gives the result:
The^{ }integral containing K–U on the righthand side of Eq. (20)^{ }is the action S, as defined in Eq. (17). In^{ }the last integral on the righthand side of Eq. (20)^{ }the total energy E=K+U does not change with time,^{ }and therefore the integral equals (t_{final}–t_{initial})E; Eq. (18) follows immediately.^{ }^{ }
Equation (18) deserves close scrutiny. Suppose that the procedure outlined^{ }in Sec. IV leads to the actual trajectory in space^{ }that minimizes S_{o} on the lefthand side of Eq. (18).^{ }Section IV tells us how to complete the worldline, which^{ }gives the time of arrival at the target, t_{final}. Hence^{ }minimizing S_{o} on the left side of Eq. (18) gives^{ }us everything about the right side, including the worldline along^{ }which the integral S is taken and the time limits^{ }of the integration. ^{ }
Instead of employing the abbreviated action S_{o}^{ }on the left side of Eq. (18), we try using^{ }the action S on the right side to predict the^{ }motion. The action integral S is taken along the worldline^{ }between the fixed initial and final events. We boldly postulate^{ }the principle of least action, which requires that the integral^{ }S in Eq. (17) have a minimum value for the^{ }worldline taken by the particle between fixed events, that is,^{ }with known elapsed time t_{final}–t_{initial} on the right side of^{ }Eq. (18). The new least action principle implies a change^{ }from the original constraints—fixed endpoints in space and fixed energy—to^{ }physically less restrictive constraints—fixed initial and final events in spacetime.^{ }For all nearby worldlines anchored on the same initial and^{ }final events, we predict that the particle moves along that^{ }worldline for which the value of S is a minimum.^{ }^{ }
Specifying the arrival event means specifying in advance the arrival^{ }time t_{final}. But this final time affects the kinetic energy^{ }K along different parts of the worldline required to meet^{ }the specified deadline and therefore affects the value of the^{ }total energy E. It turns out that minimizing the value^{ }of the action S not only validates the conservation of^{ }energy along the actual worldline, but also yields the value^{ }of the conserved energy E. ^{ }
To simplify our study of^{ }the action integral S, we return to onedimensional motion and^{ }seek a worldline y(t) that minimizes the value of S^{ }between fixed initial and final events. The trial worldline, a^{ }portion of which is shown in Fig. 6, need not^{ }have a directionreversing jog in it, as was required for^{ }alternative worldlines when the value of the energy was fixed^{ }in advance (see Fig. 2). Our expectation that the derived^{ }energyconserving worldline be smooth for a smooth potential energy function^{ }will turn out to be justified. ^{ }
Figure 6. Figure 6 approximates a^{ }portion of the trial worldline with incremental straight segments, with^{ }the independent coordinate t along the horizontal axis, just as^{ }Fig. 4 plots the independent coordinate x along the horizontal^{ }axis. Figure 6 focuses on two adjacent segments, A and^{ }B, that lie somewhere along this trial worldline. We temporarily^{ }fix events 1 and 3 while moving event 2 in^{ }the ydirection to study the effect of this displacement on^{ }the value of the action S_{AB} summed along segments A^{ }and B. A student exercise shows that the outcome is^{ }the same: the ycomponent of Newton's law F_{y}=_{y}, Eq. (15).^{ }Therefore minimizing the action S is equivalent to invoking Newton's^{ }second law. ^{ }
In finding the worldline using the principle of^{ }least action, we did not assume in advance that energy^{ }is conserved for a potential energy U(y). Nevertheless, if the^{ }expression U for potential energy does not contain the time^{ }explicitly, the principle of least action includes the conservation of^{ }energy. The actual worldline of least action must satisfy a^{ }minimal condition not only with respect to the space coordinates,^{ }but also with respect to time. The equation expressing a^{ }zero derivative of S with respect to time is nothing^{ }but conservation of energy. This result follows from varying the^{ }tlocation of event 2 in Fig. 6 while keeping its^{ }ycoordinate constant. The approximate expression for the action S_{AB} along^{ }worldline segments A and B is
where t is the^{ }time of event 2. The minimum value of Eq. (21)^{ }follows from setting its derivative with respect to t equal^{ }to zero. The average kinetic energy along segment A is^{ }
The time derivative of (t_{3}–t)K_{B} in Eq. (21) yields the^{ }result K_{B}. ^{ }
Because the potential energy is a function of^{ }space coordinate y only, the average potential energy U_{A} along^{ }segment A does not change its value with the time^{ }displacement of event 2. Therefore the time derivative of the^{ }term in Eq. (21) that includes U_{A} becomes
The time^{ }derivative of (t_{3}–t)U_{B} in Eq. (21) yields the result –U_{B}.^{ }^{ }
If we substitute Eqs. (22) and (23) for segment A^{ }plus the corresponding expressions for segment B into the expression^{ }for the derivative of Eq. (21) with respect to the^{ }time t of event 2 and set the result equal^{ }to zero, we obtain
Equation (24) can be written as^{ }
which says that energy is conserved along segments A and^{ }B. ^{ }
VI. CONSTRUCTING^{ }THE WORLDLINE
The worldline of a particle derives from the principle^{ }of least action using an iterative process similar to that^{ }used to find the trajectory in Sec. IV and illustrated^{ }in Fig. 5. The sequence of steps is essentially identical^{ }to that used for the abbreviated action when the original^{ }x, y coordinates for a trajectory become the t, y^{ }coordinates for a worldline, as shown by the alternative t,^{ }y coordinate axes in Fig. 5. We first move the^{ }ycoordinate and then the tcoordinate of each intermediate event to^{ }minimize the action along the adjacent pair of segments, sweeping^{ }repeatedly along the worldline between fixed initial and final events^{ }until the intermediate events no longer move on the spacetime^{ }diagram. The resulting segmented worldline approximates the worldline followed by^{ }the particle. ^{ }
The limittaking process of applying the principle of^{ }least action to each pair of adjacent segments in sequence,^{ }when multiple sweeps along the worldline are completed, satisfies Newton's^{ }second law and matches the value of the energy between^{ }every adjacent pair of segments, Eq. (25), and therefore determines^{ }the value of the energy on every segment of the^{ }worldline. The value of E derived from the principle of^{ }least action multiplied by the total elapsed time (fixed at^{ }the beginning of the analysis) completes the right side of^{ }Eq. (18). Knowledge of the worldline yields the trajectory, which^{ }allows us to evaluate the integral S_{o} on the left^{ }side of Eq. (18) for the nowdetermined value of the^{ }energy E. In brief, either side of Eq. (18) predicts^{ }the motion of a particle in a timeindependent potential U:^{ }Minimizing S_{o} on the left side for a fixed energy^{ }determines the trajectory; minimizing S on the right side for^{ }a fixed time lapse determines the worldline. ^{ }
One payoff of^{ }describing motion using a scalar energy and a scalar action^{ }is the straightforward generalization of the analysis to motion in^{ }three dimensions, in which the potential energy function has the^{ }general form U(x,y,z). This generalization requires only a simple extension^{ }of the analysis in Secs. III–VI. The partial derivative with^{ }respect to any spatial coordinate that minimizes the action or^{ }the abbreviated action leads to the corresponding component of Newton's^{ }second law; the partial derivative with respect to time that^{ }minimizes action leads to the conservation of energy. ^{ }
VII. TIMEDEPENDENT POTENTIAL ENERGY
In many cases the potential energy changes^{ }with time. For example, the gravitational potential at a point^{ }in space between the Earth and Moon changes as the^{ }Moon moves. This timevarying potential can change the energy E^{ }of a spaceship moving through that point. Newton's second law^{ }relates the acceleration to the instantaneous force described by the^{ }spatial derivative of the potential energy, Eq. (14). Therefore Newton's^{ }law applies from instant to instant, even if the potential^{ }changes with time. ^{ }
Constructing the worldline using the principle of^{ }least action involves a variation of the ycoordinate of each^{ }intermediate event on the worldline, while holding constant the time^{ }of that event, as illustrated in Fig. 6. Our approximation^{ }takes the value of the potential energy along each segment^{ }to be the average of its two endpoints. The endpoints^{ }in Fig. 6 are events, and the approximation corresponding to^{ }Eq. (10) is
The variation of the ycoordinate of event^{ }2 does not change the time t_{2} of that event,^{ }so the minimum action again leads to Newton's second law.^{ }Indeed, it can be shown that the principle of least^{ }action is equivalent to Newton's second law for nondissipative systems.^{12}^{ }Therefore the principle of least action is also valid for^{ }timevarying potentials. In contrast, the principle of abbreviated action assumes^{ }conservation of energy, so it cannot predict motion when the^{ }potential energy varies with time. ^{ }
Does the principle of least^{ }action also lead to a correct accounting of the changing^{ }particle energy when the potential energy changes with time? The^{ }analysis is a simple extension of the one that leads^{ }to Eq. (25). Minimizing the action along segments A and^{ }B in Fig. 6 requires varying the tcoordinate of point^{ }2, while keeping its ycoordinate constant. Equations (21) and (22)^{ }remain valid for the timedependent potential energy, but Eq. (23)^{ }is altered using Eq. (26). Setting the time derivative of^{ }the action equal to zero leads to
where t is^{ }the elapsed time as the particle moves between the midpoints^{ }of the adjacent segments. Equation (27) correctly approximates the increase^{ }in the total energy of the particle as it passes^{ }across segments A and B. ^{ }
Another important type of motion^{ }takes place under a constraint, for example, a bead sliding^{ }without friction along a rod that rotates at a constant^{ }rate.^{13} In such motion the forces of constraint typically change^{ }the energy of the particle. Newton's laws are awkward for^{ }describing motion under such a constraint because of its vector^{ }nature (and for additional reasons). In contrast, the scalar principle^{ }of least action treats such constrained motion with simplicity and^{ }power. ^{ }
VIII. SELFDESCRIPTIVE TERMINOLOGY
Before we can move the^{ }two principles of least action to a position earlier in^{ }the physics curriculum, we need to update the language of^{ }variational mechanics, making its terminology selfconsistent, transparent, and easy to^{ }understand. The present terminology of variational mechanics is clogged with^{ }the accumulated sludge of ancient trial and error and the^{ }detritus of genius. Our Murky Terminology Award goes to adjectives^{ }describing constraints of motion: holonomic, semiholonomic, rheonomous, and scleronomous. Terms^{ }encrusted with the barnacles of eminent contributors' names obstruct the^{ }flow of understanding. Who could know from the names Hamilton's^{ }principal function or Lagrange's equations what each is about, how^{ }they are used, or why they might be important?^{14} Students^{ }should not be forced to master a polyglot language before^{ }they can revel in the simplicity and power of action.^{ }^{ }
The terms for our mechanics tools should be selfdescriptive. The^{ }selfdescriptive name of an object, principle, or application recalls for^{ }us and drives home its key feature every time we^{ }read, speak, hear, or apply it. First prize for a^{ }selfdescriptive name goes to black hole, which summarizes the properties^{ }of a mighty astronomical object. The term black hole is^{ }not only descriptive but also exciting, firing the public imagination.^{ }The original study of collapsed gravitational structures did not lead^{ }to this term. The name was long sought, stumbled upon,^{ }recognized, and promoted by John Archibald Wheeler.^{15} ^{ }
Why not replace^{ }abbreviated action with the term fixedenergy action or trajectory action?^{ }Could Lagrange's equations become local equations of motion? It may^{ }be difficult to find selfdescriptive names for fundamental concepts of^{ }a field, but these names should at least be snappy^{ }and motivating. We would not be writing a paper promoting^{ }the use of Hamilton's principal function, but Landau and Lifschitz,^{ }and later Feynman, renamed it action, cutting the number of^{ }syllables by threequarters and invigorating the field.^{16} ^{ }
Even the field^{ }of mechanics needs new names. Classical mechanics currently means nonquantum^{ }mechanics; special and general relativity are classical subjects. This paper^{ }focuses on the more restricted field of Newtonian mechanics, which^{ }predicts nonrelativistic, nonquantum motion. Newton's greatness is not enhanced by^{ }using his name to smother magisterial contributions by Euler, Lagrange,^{ }Jacobi, Hamilton, and others. ^{ }
ACKNOWLEDGMENTS
The authors wish to thank Kenneth Ford, Don^{ }S. Lemons, and Jon Ogborn for useful discussions and helpful^{ }suggestions. ^{ }
REFERENCES
Citation links [e.g., Phys. Rev. D 40, 2172 (1989)] go to online journal abstracts. Other links (see Reference Information) are available with your current login. Navigation of links may be more efficient using a second browser window.

We use the name least action instead of the technically correct stationary action
for several reasons: (a) Many cases involve a local minimum of the
action. (b) The value of the action is always a minimum for a
sufficiently small segment of the curve. (c) The word least is selfdescriptive, but stationary
requires additional explanation. (d) The word least does not lead to
the error that the value of either form of action, Eqs. (4) and (17),
can be a maximum for an actual path, which it cannot. (e) Least action
is the name most often used in the historical literature on the
subject. We recommend that the term stationary action be introduced,
with careful explanation, not long after the term least action itself. first citation in article

For linear gravitational potential energy near the Earth's surface, we
can integrate Newton's second law to derive an analytic expression for
the basketball trajectory and hence the required direction of launch.
However, in more complicated potentials we are reduced to trial and
error to find a path that passes through the basket. Minimizing the
Maupertuis–Euler abbreviated action finds the trajectory in one stroke.
A similar comment applies to the Moon shot described in the following
paragraph: Minimizing Hamilton's action gives us the worldline
directly. first citation in article

Jozef Hanc and Edwin F. Taylor, "From conservation of energy to the principle of least action: A story line," Am. J. Phys. 72, 514–521 (2004). [ISI]
first citation in article

Derivations of the action outlined in this paper were stimulated by an
interactive Java program developed by one of the authors (ST). This
display numerically integrates Eq. (2), solving for the onedimensional
motion of a particle in a timeindependent potential. In its extended
form, the program shows all three panels in Fig. 1. The program is
available at http://vscience.euweb.cz/worldlines/Worldlines.html.
first citation in article

The name abbreviated action and the symbol S_{o} are used by L. D. Landau and E. M. Lifschitz, Mechanics (ButterworthHeinemann, London, 1999), Vol. 1, 3rd ed., p. 141, and Herbert Goldstein, Charles Poole, and John Safko, Classical Mechanics (Addison–Wesley, San Francisco, 2002), 3rd ed., pp. 359, 434. We have named the corresponding variational principle the abbreviated principle of least action, rather than the more technically correct principle of least abbreviated action, believing that "least abbreviated" might be incorrectly interpreted as "augmented."
first citation in article

Wolfgang Yourgrau and Stanley Mandelstam, Variational Principles in Dynamics and Quantum Theory (Dover, New York, 1979), pp. 24–29.
first citation in article

Richard P. Feynman, Robert B. Leighton, and Matthew Sands, The Feynman Lectures on Physics (Addison–Wesley, San Francisco, 1964), Vol. II, p. 198.
first citation in article

The minimization procedure for constructing a trajectory or worldline
described in the caption to Fig. 5 is conceptually simple but not the
most effective in practice. In both cases it is more efficient to start
with a trial segmented curve with equal increments along the horizontal
axis. Then we vary only the ycoordinates
of intermediate points to minimize the action, obtaining the actual
path. The minimization of action with respect to coordinates along the
horizontal axis is not necessary because the result is just points
uniformly distributed on the horizontal axis, which was our initial
assumption. We do not discuss here the proof of this statement or the
convergence of the algorithm, because it goes beyond the scope of the
present paper. first citation in article

According to the official rules of the National Basketball Association,
a basket is scored after the final buzzer provided the ball is launched
before the buzzer sounds. first citation in article

For Hamilton's development of the principle of least action, see two of his papers at http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics/.
first citation in article

Landau and Lifschitz, Ref. 5, p. 141, Eq. (44.3); Goldstein et al., Ref. 5, p. 359.
first citation in article

Our derivation can be reversed to show the equivalence of Newton's
second law and the principle of least action. See also Goldstein et
al., Ref. 5, p. 35. first citation in article

The example of a bead sliding along a uniformly rotating rod is in
Goldstein et al., Ref. 5, pp. 28–29. Additional example is a pendulum
whose string support is slowly pulled up through a small hole. See
Cornelius Lanczos, The Variational Principle of Mechanics (Dover, New York, 1986), 4th ed., p. 124.
first citation in article

Further examples of nameencrusted terminology: d'Alembert's principle,
Hamiltonian, Hamilton's principle, Hamilton's equations,
Hamilton–Jacobi equation, Jacobi identity, Jacobi principle, Jacobi
condition, Jacobi's theorem, Lagrangian, Poisson bracket, Poisson's
equations, Hilbert integral, Legendre condition, Poincare invariants,
Cartheodory's method, Bernoulli's method, Clebsch condition, Clebsch
relation, Clebsch transformation, Descartes–Snell rule, Noether's
theorem, Rayleigh's dissipation function, Routh's procedure, Staeckel
conditions, Weierstrass condition, Weierstrass–Erdmann corner
condition. first citation in article

Kip S. Thorne, Black Holes and Time Warps: Einstein's Outrageous Legacy (Norton, New York, 1994), pp. 256–257; John Archibald Wheeler with Kenneth Ford, Geons, Black Holes, and Quantum Foam: A Life in Physics (Norton, New York, 1998), pp. 296–297.
first citation in article

Landau and Lifschitz, Ref. 5, Chap. 1; Landau and Lifschitz rechristened Hamilton's principal function as the action in the first Russian edition in 1958, and in a 1940 textbook, a precursor of Ref. 5; See also Feynman, Ref. 7.
first citation in article
FIGURES
Full figure (15 kB)Fig. 1. Screen shot of the interactive program showing the energy^{ }and potential energy diagram in the bottom panel, the worldline^{ }(the plot relating position and time) in the middle panel,^{ }and a plot of velocity versus position y in the^{ }top panel. The diagonal shading of the area under the^{ }velocity versus position graph has been added for use in^{ }comparing alternative motions depicted in Fig. 2. First citation in article
Full figure (18 kB)Fig. 2. Two sequential spontaneous reversals^{ }of direction with the same speed near position y_{o} satisfy^{ }conservation of energy but are unphysical. The original worldline AB^{ }shown in Fig. 1 generates the diagonally shaded area under^{ }the velocity curve at the top of Fig. 1. The^{ }worldline AC not only generates the same enclosed area, but^{ }also adds the superposed dotshaded areas labeled 1 and 2.^{ }Area 1 is enclosed twice in this process. First citation in article
Full figure (4 kB)Fig. 3. The infinitesimal initial^{ }portion of the curved trajectory of a particle in the^{ }xy plane (dotted line) might represent the first millisecond of^{ }motion of a particle that starts at fixed point 1.^{ }The dotted trajectory is approximated by three connected straight segments^{ }A, B, C. First citation in article
Full figure (4 kB)Fig. 4. We vary the yposition of point 2 to^{ }minimize the abbreviated action along segments A and B. First citation in article
Full figure (10 kB)Fig. 5. Construction of^{ }the trajectory y(x) in two dimensions. The initial segments are^{ }shown. Point 1 is the fixed launch point. (a) We^{ }first vary the ycoordinate and then the xcoordinate of point^{ }2 to find the location that minimizes the value of^{ }the abbreviated action S_{oAB} along segments A and B. (b)^{ }We move point 2 to that new location, then vary^{ }the y coordinate, then the xcoordinate of point 3 to^{ }find the location that minimizes the value of S_{oBC} along^{ }segments B and C. (c and continuation) We move point^{ }3 to that new location and continue moving later points^{ }on the trial trajectory all the way to the fixed^{ }final point (the target, not shown). We repeatedly sweep the^{ }entire trial trajectory until the intermediate points no longer change.^{ }The resulting trajectory approximates that taken by the particle. As^{ }described in Sec. VII, a similar construction with y and^{ }t coordinates approximates the worldline which minimizes Hamilton's action S. First citation in article
Full figure (5 kB)Fig. 6. Two^{ }adjacent infinitesimal segments A and B are chosen arbitrarily along^{ }the trial worldline of a particle moving in one dimension^{ }in a timeindependent potential. Events 1 and 3 are temporarily^{ }fixed, while the ycoordinate of event 2 is displaced in^{ }order to find its position 2 for the minimum value^{ }of the action along segments A and B. 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
^{c}Electronic mail:^{ }tuleja@stonline.sk
Up: Issue Table of Contents
Go to: Previous Article  Next Article
Other formats: HTML (smaller files)  PDF (107 kB)