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From Conservation of Energy to the Principle of Least Action: A Story Line
Jozef Hanca)
Technical University, Vysokoskolska 4, 042 00 Kosice, Slovakia
Edwin F. Taylorb)
Massachusetts Institute of Technology, Cambridge, MA 02139
ABSTRACT
A schematic story line outlines an introduction to Newtonian mechanics that starts by
employing the conservation of energy to predict the motion of a particle in a one-dimensional
potential. Incorporating constraints and constants of the motion into the energy expression
allows analysis of some more complicated systems. A heuristic transition embeds kinetic and
potential energy in the still more powerful principle of least action and Lagrange's equations.
NOTES TO EDITORS AND REFEREES:
1. Preprints of some papers referenced here but not yet published and some papers published
elsewhere are available at the website http://www.eftaylor.com/leastaction.html
2. Comments and suggestions can conveniently be indexed by page and line number in this
pdf file.
I. INTRODUCTION
It is sometimes claimed that nature has a "purpose", in that it seeks to take the path that produces the
minimum action. . . . In fact [according to Feynman's version of quantum mechanics] nature does
exactly the opposite. It takes every path, treating them all on [an] equal footing. We simply end up
seeing the path with the stationary action, due to the way the quantum mechanical phases add. It
would be a harsh requirement, indeed, to demand that nature make a "global" decision (that is, to
compare paths that are separated by large distances), and to choose the one with the smallest action.
Instead, we see that everything takes place on a "local” scale. Nearby phases simply add and
everything works out automatically.
— David Morin
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Here is a nine-sentence history of Newtonian mechanics: In the second half of the 1600s
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Newton proposed his three laws of motion, including what has become F = ma. In the mid1700s Leonhard Euler devised and applied one version of the principle of least action using
mostly geometrical methods. On 12 August 1755 the 19-year-old Ludovico de la Grange
Tournier of Turin (whom we call Joseph Louis Lagrange) sent Euler a letter with a
mathematical attachment that streamlined Euler's methods into algebraic form. "[A]fter
seeing Lagrange's work Euler dropped his own method, espoused that of Lagrange, and
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renamed the subject the calculus of variations." Lagrange, in his path-breaking 1788 Analytical
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Mechanics, introduced what we call the Lagrangian function and Lagrange's equations of
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motion. Almost half a century later (1834-35) William Rowan Hamilton developed
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Hamilton's principle, to which Landau and Lifshitz and Feynman more recently reassigned
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the name principle of least action. Between 1840 and 1860 the conservation of energy was
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established in all its generality.
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In 1918 Emma Noether
proved relations between
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symmetries and conserved quantities. Finally, in 1949 Richard Feynman devised the
formulation of quantum mechanics that not only explicitly underpins the principle of least
action but also demarcates the limits of validity of Newtonian mechanics.
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applications of energy and its environmental consequences is a matter of choice.
In
addition to expressions for kinetic and potential energy of a particle, this introduction should
include angular velocity, moment of inertia, and the kinetic energy of a rigid body rotating
about a fixed or parallel-moving axis.
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We have proposed that the principle of least action and Lagrange's equations become the
basis of introductory Newtonian mechanics. Three recent articles in this Journal demonstrate
how to use elementary calculus to derive from the principle of least action (1) Newton's laws
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of motion, (2) Lagrange's equations, and (3) examples of Noether's theorem.
How are these concepts and methods to be introduced to students? In the present paper we
suggest a reversal of the historical order: Begin with conservation of energy and graduate to
the principle of least action followed by Lagrange's equations.
We start by using the conservation of energy to analyze particle motion in a one-dimensional
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 the conservation of energy, we not only preview more advanced concepts and
procedures but also invoke some of their power. For example, expressions for energy
conforming to constraints automatically eliminate corresponding forces of constraint from
the analysis of motion. Applying a limited version of Noether's theorem expressed in terms
of energy identifies constants of the motion. Using constraints and constants of the motion
permits us to reduce to one coordinate the description of some important multi-dimensional
systems, whose motion can then be solved completely using the conservation of energy.
Equilibrium and statics also derive from the conservation of energy.
Prerequisites for the proposed introduction include elementary trigonometry, polar
coordinates, introductory differential calculus and partial derivatives, and the idea of the
integral as a sum of increments. In preparation for later formalism, we symbolize the time
derivative with a dot over the variable.
The story line presented in this article omits most details and is offered for discussion,
correction, and elaboration. The authors have no present intent to write an introductory text
on Newtonian mechanics that follows the story line of this article.
II. ENERGY CONSERVATION
Begin with a discursive description of the forms of energy and the law of global conservation
of energy. Introductory treatments of the conservation of energy and conversion among its
17, 18, 19
How far one goes into engineering
various forms are available in the literature.
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III. ONE-DIMENSIONAL MOTION: ANALYTIC SOLUTIONS
Narrow the focus to conservation of mechanical energy of a particle moving in a onedimensional potential. The complete description of such one-dimensional motion follows
from the conservation of energy. Unfortunately, an explicit function of position vs time can be
derived for only a small fraction of such systems. In other cases one is encouraged to guess
the analytic solution, as illustrated in the following example. Any proposed analytic solution
is easily checked by substitution into the energy equation. Heuristic guesses are assisted by
the fact that the first time derivative of position, not the second, appears in the energy
conservation equation. The following example introduces the potential energy diagram, a
central tool in our story line and important in later study of advanced mechanics, quantum
mechanics, general relativity, and many other subjects.
Harmonic oscillator
Energy
Total energy of particle
A
B
C
D
E
F
Kinetic
energy
Potential
energy
Position
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Figure 1. The potential energy diagram is central to our treatment. Sample student exercise: For
the value of the energy shown, describe the motion of the particle qualitatively but in detail.
Compare the magnitudes and directions of each of the following quantities that describe the
particle motion at different positions marked A through F: velocity, force, acceleration. In order to
describe the motion, what initial condition(s) must be specified in addition to an initial starting
point, and are these conditions required for all starting points? How do answers to these
questions change for different values of the total energy?
Qualitative analysis of the parabolic potential energy diagram (or indeed any diagram with
motion bounded near a single potential energy minimum) tells us that the motion will be
periodic. For the parabolic potential the conservation of energy is written
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1
1
E = mx˙ 2 + kx 2
2
2
(1)
Rearrange Eq. (1) to read:
m 2
k 2
x
x˙ = 1−
2E
2E
(2)
Recalling the periodicity of motion, we are reminded of trigonometric identity
cos2 θ = 1− sin 2 θ
(3)
Set θ = ωt and equate corresponding terms in Eqs. (2) and (3), choosing the positive root,
which yields:
 2E 1/ 2
x =   sin ωt
(4)
 k 
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Take the time derivative of this expression and compare its square with the left side of Eq.
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(2). The result tells us that
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 k 1/ 2
ω = 
m
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(5)
The fact that ω does not depend on the amplitude of the displacement function (4) means that
the period is the same whatever the value of the total energy E. The simple harmonic
oscillator has wide application because a large number of potential energy curves can be
approximated as parabolas near their minima.
IV. ACCELERATION AND FORCE
In the absence of dissipation, force can be defined in terms of energy. In the simple harmonic
oscillator example, take the time derivative of both sides of (1) (for constant energy E and
mass m).
0 = mx˙x˙˙ + kxx˙
(6)
or
k
x
m
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˙x˙ = −
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Define force as m times the acceleration. Then the simple harmonic force is –kx. More
generally, take the time derivative of both sides of the conservation of energy equation for a
particle moving in a one-dimensional potential:
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0 = m˙xx˙˙ +
(7)
dU ( x )
dU ( x ) dx
dU ( x )
= m˙xx˙˙ +
= m˙xx˙˙ +
x˙
dt
dx dt
dx
4
(8)
1
2
3
from which
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m˙x˙ ≡ F = −
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dU ( x )
dx
(9)
Thus for motion without dissipation force is defined in terms of energy.
V. NON-INTEGRABLE MOTIONS IN ONE DIMENSION
We cannot solve most one-dimensional motions in closed form, as we did for the simple
harmonic oscillator. Our strategy will be to ask the student to begin every solution with a
qualitatively analysis using the potential energy diagram (as exemplified in the caption of
Fig. 1), then to complete the exact solution employing an interactive computer display (Fig.
2). Such a computer display, yet to be programmed but easily specified, numerically
integrates the particle motion in a given potential. On this display the student sets up initial
conditions by dragging the horizontal energy line up or down and the position dot left or
right, both in the classically permitted region (lower panel of Fig. 2). The computer then
moves the dot back and forth along the E-line at a rate proportional to that at which the
particle will move, while simultaneously drawing the worldline (upper panel of Fig. 2).
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time
Position
E
Energy
U
Position
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2
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Figure 2. Interactive computer display of the worldline derived numerically from the initial
energy and the potential energy function.
VI. REDUCTION TO ONE COORDINATE
The analysis of one-dimensional motion using the conservation of energy is powerful but
limited. In some important cases we can use constraints and constants of the motion to
reduce the description of a wider class of motions to one coordinate. To these we apply our
standard procedure: qualitative analysis using the diagram of the (effective) potential energy
followed by interactive computer solution. Here are some examples.
(a) Object rolling without slipping
A marble with mass m, radius r, and moment of inertia Imarble rolls without slipping along a
curved ramp that lies in the xy plane. A uniform gravitational acceleration g acts in the
negative y-direction. The non-slip constraint tells us that v = rω . Conservation of energy leads
to the expression:
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2
1
v
1
1
1
1
E = mv 2 + Imarbleω 2 + mgy = mv 2 + Imarble  + mgy
 r
2
2
2
2
 2
I
1
1
=  m + marble
v + mgy = Mv 2 + mgy
2 
r 
2
2
2
3
where
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M =m+
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Imarble
r2
(10)
(11)
Constraints are used twice in this example: explicitly in rolling without slipping and
implicitly in the relation between height y and displacement along the curve.
(b) Projectile motion
For projectile motion in a vertical plane subject to a vertical uniform vertical gravitational
field, the total energy is
1
1
(12)
E = mx˙ 2 + my˙ 2 + mgy
2
2
The potential energy is not a function of x; therefore, as shown below, x-momentum px is a
constant of the motion. The energy equation becomes:
1
p2
E = m˙y 2 + x + mgy
2m
2
(13)
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In Newtonian mechanics the zero-point of energy is arbitrary, so we can reduce the analysis
to one dimension by making the substitution:
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E'= E −
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In analyzing projectile motion, we have already applied a limited version of the powerful
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Noether's theorem. The special version of Noether's theorem used here, expressed in terms
of energy, says that when the total energy E is not an explicit function of an independent
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coordinate, x for example, then the function ∂E ∂x˙ is a constant of the motion. In some
important cases we can use the resulting constants of the motion, together with constraints,
to describe some two- and even three-dimensional motions with just one coordinate.
Projectile motion, above, is one example. Here is another.
px2 1 2
= m˙y + mgy
2m 2
(14)
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(c) Marble, ramp, and turntable
m
x
θ
φ
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Figure 3. System of marble, ramp, and turntable.
[NOTE: This figure will be redrawn professionally after any referee suggestions.]
A marble of mass m and moment of inertia Imarble rolls without slipping along a slot on an
inclined ramp fixed rigidly to a turntable which rotates freely so that its angular velocity is
not necessarily constant (Fig. 3). The moment of inertia of the combined turntable and ramp
is Irot. Assuming that the marble stays on the ramp, find its position x as a function of time.
Start with a qualitative analysis, assuming the marble initially moves outward along the
ramp. This outward rolling will slow down the rotation of the turntable, perhaps leading the
marble to reverse direction. One possibility is oscillatory motion of the marble along the
ramp; another possibility is an equilibrium location of the marble on the ramp with simple
circular motion of the marble around the axis of the turntable. Now the analysis. The square
of the velocity of the marble is:
v 2 = x˙ 2 + φ˙ 2 x 2 cos2 θ
(15)
Conservation of energy yields the equation
1
1
1
Mx˙ 2 + Irot φ˙ 2 + Mx 2φ˙ 2 cos2 θ + mgx sin θ
2
2
2
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E=
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Here M indicates the marble's mass augmented by the energy effects of its rotation, Eq. (11).
The expression (16) is not an explicit function of the angle of rotation φ. Therefore our
version of Noether's theorem tells us that ∂E ∂φ˙ is a constant of the motion, which we
recognize as the total angular momentum J:
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(16)
∂E
= ( Irot + Mx 2 cos2 θ )φ˙ = J
˙
∂φ
(17)
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1
2
Substitute for φ˙ from Eq. (17) into (16):
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E=
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1
J2
1
M˙x 2 +
+ mgx sin θ = M˙x 2 + U eff ( x )
2
2
2
2
2( Irot + Mx cos θ )
(18)
Values of E and J are determined by initial conditions. The effective potential energy U eff ( x )
can have a minimum as a function of x. If the marble starts at rest at the position of this
minimum, it will move in a circle. The general motion of this system is difficult to analyze
using F = ma.
(d) Motion in a central potential
Analyze satellite motion in a central inverse-square gravitational field. Choose coordinates r
and φ in the plane of the orbit. Then the expression for total energy is:
1
GMm 1
GMm
E = mv 2 −
= m( r˙ 2 + r 2φ˙ 2 ) −
2
r
2
r
(19)
The angle φ does not appear in this equation. Therefore we expect that a constant of the
motion is given by ∂E ∂φ˙ , which is the angular momentum J:
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∂E
= mr 2φ˙ = J
˙
∂φ
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21
22
Substitute the resulting expression for φ˙ in Eq. (20) into the energy equation (19).
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1
J2
GMm 1 2
E = m˙r 2 +
−
= m˙r + U eff ( r)
2
2mr
2
r
2
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(20)
(21)
The student employs the plot of effective potential energy to carry out the usual qualitative
analysis of radial motion followed by computer integration. Emphasize the distinction
between bound orbits, unbound orbits, and the intermediate limiting case. With additional
use of Eq. (20), the computer can plot the trajectory in the plane.
VII. EQUILIBRIUM AND STATICS
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Friction is an important feature of motion in everyday life, but it is complicated to analyze.
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Frictional forces are often ignored in advanced mechanics texts. The tendency of a system
toward increased entropy and the statistical mechanics and thermodynamics that account for
this tendency are not part of our story line here. Nevertheless, in common experience motion
usually slows down and stops. Our use of potential energy diagrams makes straightforward
the formulation of "stopping" as a tendency of systems to reach equilibrium at a local
minimum of the potential energy. (In principle the system could come to equilibrium
perched at a maximum of the potential energy, or even at a zero-slope inflection point.) In the
absence of dissipative processes, equilibrium is really an application of conservation of
energy; a system released from rest in a configuration at a minimal (or stationary) value of
the potential energy will remain at rest. (Proof: An infinitesimal displacement results in zero
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change in potential energy. Because of energy conservation, the change in kinetic energy
must also remain zero. Since the system is initially at rest, the zero change in kinetic energy
forbids displacement.)
Here, as usual, Feynman is ahead of us. Figure 4 shows an example from his treatment of
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statics. The problem is to find the value of the weight W that keeps the structure at rest.
Using units of feet, inches, and pounds, Feynman balances the decrease in potential energy
when the weight W drops 4 inches with increases in potential energy for the corresponding
2-inch rise of the 60 pound weight plus the 1-inch rise in the 100 pound weight. He demands
that the net potential energy change of the system be zero, yielding
Figure 4. Feynman's example of
the principle of virtual work
(−4 inch)W + (2 inch)(60 pounds) + (1 inch)(100 pounds) = 0
(22)
or W = 55 pounds. This method, known formally as the principle of virtual work, is simply an
application of the conservation of energy.
There are many examples of the principle of virtual work, including a mass hanging on a
spring, the lever, hydrostatic balance, uniform chain suspended at both ends, vertically
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hanging "slinky," and a ball perched on top of a large sphere, not to mention every static
engineering and natural structure in the universe.
VIII. FREE PARTICLE. PRINCIPLE OF LEAST AVERAGE KINETIC ENERGY
The conservation of energy is limited in predicting the motion of mechanical systems. Now
we begin the transition to the more powerful principle of least action by looking at the time
average of the kinetic energy of a free particle, a particle that moves in a region of zero
potential energy or potential energy that has the same value everywhere in the region. In this
and the following sections we seek a principle that is more fundamental than the
conservation of energy. We will test any fundamental principle we uncover by demanding
that, at least, it lead back to the conservation of energy. For this reason the trial worldlines we
explore hereafter do not necessarily satisfy conservation of energy.
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C fixed
ttotal
ttotal – t
2
t
B
1
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A fixed
d
2d
x
Figure 5. Varying the time of the middle event to determine the path
of minimum time-averaged kinetic energy.
The solid line in Fig. 5 shows the trial worldline of a particle that moves freely between fixed
events A and C. We know that a free particle travels with constant velocity, that is, along a
straight worldline. Verify that this is the actual motion by examining a trial worldline for
which the central event B in the worldline is displaced in time (and therefore kinetic energy is
not the same on legs 1 and 2). Whether the potential energy is zero or uniform in this region,
its contribution to the time average energy does not change with the time-displacement of
event B, so we ignore the potential energy. Let K represent the kinetic energy. Then the time
average of the kinetic energy along the two segments 1 and 2 of the worldline is given by the
expression.

K1t1 + K 2 t 2
1 1 2
1
=
mv1 t1 + mv 22 t 2


t total  2
2
t total
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K average =
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19
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Multiply through by the fixed total time and expand the velocity expressions using the
notation displayed in Fig. 5:
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(23)
1 d2
1
d2
K averaget total = m 2 t + m
( ttotal − t )
2 t
2 ( t total − t ) 2
(24)
1 d2 1
d2
= m + m
2 t 2 (t total − t )
22
11
1
2
3
Find the minimum value of the average kinetic energy by taking the derivative with respect
to the time t of the central event and setting the result equal to zero:
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 dK average 
1 d2 1
d2
1 2 1 2
t
=
−
m
+
m
mv 2 = 0

 total
2 = − mv1 +
2
2 t
2 ( t total − t )
2
2
 dt 
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6
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The right-hand equation leads to the equality:
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1 2 1 2
mv 2 = mv1
2
2
(25)
(26)
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As expected, when the time-average 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. 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, kinetic energy is
conserved if we demand that the time average of the kinetic energy has a minimum value.
The general expression for this average is:
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K average =
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1
t total
∫ Kdt
(27)
In an introductory text one might insert at this point a sidebar on Fermat's principle of least
time for the propagation of light rays, which predicts every central feature of ray optics. Now
we are ready to develop a similar law that predicts every central feature of mechanics.
IX. PRINCIPLE OF LEAST ACTION
Section VIII examined the principle of least average kinetic energy, which predicts the motion
of a particle moving in a region of zero (or uniform) potential energy. Along the way we
mentioned Fermat's principle of least time that describes ray optics. Now we move on to the
more general principle which includes the principle of least average kinetic energy as a
special case. Can we find some quantity whose time average value is a minimum when both
kinetic energy and potential energy are functions of position?
Let K represent kinetic energy and U represent potential energy. One might guess
(incorrectly) that the time averaged total K + U would have a minimum value between fixed
initial and final events. (Remember: In calculating the value of this average we do not assume
the conservation of total energy, but rather demand that energy conservation emerge as one
result of our analysis.)
It turns out that the time averaged difference K – U between kinetic and potential energy has a
minimum value for the path taken by the particle. How can this be? To sort this out, think of
a baseball pitched in a uniform gravitational field, with the two events of pitch and catch
fixed in both location and time (Fig. 6). How will the ball move between these two fixed
events?
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y
E
(K − U) avg minimum
D
C
pitch
catch
B
x
A
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(K + U )avg
minimum?
Figure 6. Alternative trial trajectories of a pitched baseball.
To begin with, forget about averages and ask why the baseball does not simply move steadily
along the straight horizontal trajectory B in the figure. 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.
To see why the horizontal trajectory will not occur, go back to time averages and try the
(incorrect) assumption that the trajectory taken by the baseball is the one that has a minimum
value of the time average of the sum K + U. That trajectory would fall and then rise, like curve
A in Fig. 6. Here is why: By moving downward the ball decreases the average value of the
potential energy U (while increasing K a little because of the longer path). This is clearly not
what occurs when a baseball is pitched!
As an alternative hypothesis, try the assumption that the trajectory taken by the baseball is
the one that leads to a minimum value of the time average of the difference K – U. Which of
the trajectories C, D, or E can satisfy this criterion? By following a trajectory such as C that
rises and then falls, the baseball increases the time average value of its potential energy U.
Since U enters the average of K – U with a minus sign, a higher trajectory decreases the overall
contribution of the potential energy to the average (while increasing kinetic energy
somewhat). A minimum value of the time average of K – U is achieved along some trajectory
such as D in Fig. 6. For a still higher trajectory E the average value of the potential energy U
increases further, but the kinetic energy (proportional to the square of the speed) increases
even faster, because the baseball must complete in the same time the longer path between the
fixed events of pitch and catch. For higher and higher trajectories the average kinetic energy
increases without limit, leading to no maximum value. The actual path, indicated
schematically by D in Fig. 6, represents a compromise that makes the time-average value of K
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2
3
4
5
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– U along the path a minimum. We will see that total energy is automatically conserved
along this path of minimum time average of the quantity (K – U) .
Check this result analytically in the simplest case we can imagine (Fig. 7). In a uniform
vertical gravitational field a marble of mass m and momentum of inertia Imarble rolls from
one horizontal surface to another via a smooth ramp so narrow that we neglect its width. In
this system the potential energy changes just once, halfway between initial and final
positions.
1
2
v1
A
z1
z2
B
x
d
9
10
11
12
13
C
d
Figure 7. Motion across a potential energy step.
The worldline of the particle will be bent, corresponding to the reduced speed after the
marble mounts the ramp, as shown in Fig. 8.
14
1
2
C fixed
ttotal
ttotal – t
2
B
t
1
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
d
A fixed
2d
x
Figure 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
We demand that the total travel time from position A to position C have a fixed value ttotal
and check whether minimizing the time average of the difference (K – U) leads to the
conservation of energy.
time average =
1
t total
[(K − U ) t + (K − U ) t ]
1 1
(28)
2 2
Define a new symbol S, called the action:
Action ≡ S ≡ time average × t total = (K − U )1 t1 + (K − U ) 2 t 2
(29)
Use the symbols in Fig. 8 to write Eq. (29) in the form
1
1
S = mv12 t1 − mgz1t1 + mv 22 t 2 − mgz2 t 2
2
2
2
md
md 2
=
− mgz1t +
− mgz 2 (t total − t )
2t
2(t total − t )
(30)
15
1
2
3
4
(For simplicity we neglect the marble's kinetic energy of rotation.) Demand that the value of
the action be a minimum with respect to the choice of the intermediate time t:
5
dS
md 2
md 2
= − 2 − mgz1 +
2 + mgz 2 = 0
dt
2t
2(t total − t )
6
7
8
This result can be written:
9
10
11
12
13
(31)
 1

1
dS
= −  mv12 + mgz1 +  mv 22 + mgz2  = 0
 2

2
dt
(32)
A simple rearrangement shows that Eq. (32) represents the conservation of energy. Here
energy conservation has been derived from the more fundamental principle of least action.
3
2
z1
1
d
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
z3
z2
4
z4
d
d
d
Figure 9. A more complicated potential energy diagram, leading in the
limit to a potential energy curve that varies smoothly with position
The action Eq. (29) can be generalized for multiple steps connected by smooth, narrow
transitions, such as the one shown in Fig. 9. The argument leading to conservation of Eq. (32)
then applies to every adjacent pair of steps in the potential energy diagram. The
corresponding generalization of the action is:
S = (K − U )1 t1 + (K − U ) 2 t 2 + (K − U ) 3 t 3 + (K − U ) 4 t 4
(33)
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.
16
t
E
Fixed
D
C
B
A
Fixed
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
26
27
28
29
30
1
4
x
Figure 10. Varying the times at the potential energy steps to find a worldline of least
action. The computer program temporarily fixes the end events of a two-segment
section, say events B and D, then varies the middle event C to find the minimum value
of the total S, then varies D while C is kept fixed, and so on. (End-events A and E
always 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.
A continuous potential energy curve is 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 dt. For
the resulting potential energy curve the general expression for the action S is:
Action ≡ S ≡
∫ L dt = ∫ (K − U ) dt
entire
path
16
17
18
19
20
21
22
23
24
25
3
2
(34)
entire
path
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 worldline.
A more general form of the principle of least action also predicts 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
30
which energy is not conserved. With generalizations of the Lagrangian L, the principle of
31
least action describes relativistic motion, governs many non-mechanical systems, and can
be used in the derivations of Maxwell's equations, Schroedinger's wave equation, the
diffusion equation, geodesic worldlines in general relativity, and electric currents in steadystate circuits, among many other applications.
17
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2
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4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
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22
23
24
25
26
27
28
29
30
31
32
The present section has not provided a proof of the principle of least action in Newtonian
mechanics. A fundamental proof rests on nonrelativistic quantum mechanics, for instance
32
that outlined by Tyc which uses the deBroglie relation to show that the phase change of a
quantum wave along any worldline is equal to S h where S is the classical action along that
33
worldline. Starting with this result, Feynman and Hibbs show that his "sum over all paths"
description of quantum motion reduces seamlessly to the Newtonian principle of least action
as the mass of particles increases.
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 actually explores all the possible worldlines considered in finding the Newtonian
worldline of least action.
X. LAGRANGE'S EQUATIONS
Lagrange's equations are conventionally derived from the principle of least action using the
34
calculus of variations. These derivations analyze the worldline as a whole. However, the
action is a summation; if the value of the sum is minimum along the entire worldline, then
the contribution along each incremental segment of the worldline must also be a minimum.
This simplifying insight, due originally to Euler, allows a derivation of Lagrange's equations
35
using elementary calculus.
directly from F = ma.
The appendix shows a derivation of Lagrange's equation
ACKNOWLEDGMENT
The authors would like to acknowledge useful comments by Don S. Lemons.
APPENDIX: DERIVATION OF LAGRANGE'S EQUATIONS DIRECTLY FROM F=ma
Both F = ma and Lagrange's equations can be derived from the more fundamental principle of
36
least action. Here we move laterally from one derived formulation to the other, showing
that Lagrange's equation leads to F = ma for particle motion in one dimension, as adapted
37
from a similar derivation by Zeldovich and Myskis. Using the definition of force in Eq. (9),
we write Newton's second law as:
dU
= mx˙˙
dx
33
F =−
34
35
36
Assume that m is constant. Rearrange and recast Eq. (A1) to read:
(A1)
37
d(−U ) d  d  m˙x 2 
d(−U ) d
− (m˙x ) =
−  
 = 0
dt
dt  d˙x  2 
dx
dx
38
39
40
41
In the right-hand Eq. (A2) add, under the derivative signs, specifically chosen terms whose
partial derivatives are equal to zero:
(A2)
18
1
 d  ∂  m˙x 2

∂  m˙x 2
− U ( x ) −  
− U ( x ) = 0

∂x  2
 dt ∂x˙  2

2
3
4
Set L = K – U, leading to the Lagrange equation in one dimension:
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
(A3)
∂L d ∂L 
=0
−
∂x dt  ∂x˙ 
(A4)
This sequence of steps can be reversed to show that Lagrange's equation leads to Newton's
second law of motion F = ma.
FOOTNOTES AND REFERENCES
a)
email: [email protected]
b)
email: [email protected]
1
David Morin, "The Lagrangian Method," Chapter 5 of a draft honors introductory mechanics
text. Available at
http://www.courses.fas.harvard.edu/~phys16/handouts/textbook/ch5.pdf (no final slash).
2
We use the phrase Newtonian mechanics to mean non-relativistic, non-quantum mechanics.
The alternative phrase classical mechanics also applies to relativity, both special and general,
and is thus inappropriate in the context of the present paper.
3
Sir Isaac Newton's Mathematical Principles of Natural Philosophy and his System of the World, Tr.
Andrew Motte and Florian Cajori, University of California Press, 1946, page 13.
4
Herman H. Goldstine, A History of the Calculus of Variations from the 17th through the 19th
Century, Springer-Verlag, 1980, page 110.
5
J. L. Lagrange, Analytic Mechanics, Victor N. Vagliente and Auguste Boissonnade Tr.
(Kluwer Academic Publishers, 2001).
6
William Rowan Hamilton, "On a general method in dynamics, by which the study of the
motions of all free systems of attracting or repelling points is reduced to the search and
differentiation of one central Relation or characteristic Function," Philosophical Transactions of
the Royal Society, part II for 1834, pp. 247-308; "Second essay on a general method in
dynamics," Philosophical Transactions of the Royal Society, part I for 1835, pp. 95-144. Both
papers are available at http://www.emis.de/classics/Hamilton/
19
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2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
7
Landau and E. M. Lifshitz, Mechanics, Course of Theoretical Physics (Butterworth-Heinemann,
London, 1976), 3rd ed., Vol. 1, Chap. 1. Their renaming first occurred in the original 1957
Russian edition.
8
Richard P. Feynman, Robert B. Leighton and Matthew Sands, The Feynman Lectures on
Physics (Addison-Wesley, 1964), Vol. II, p. 19-8.
9
The technically correct term is principle of stationary action. See I. M. Gelfand and S. V. Fomin,
Calculus of Variations, Richard A. Silverman, Tr., (1963, Dover Publications 2000) Chapter 7,
Sec. 36.2. However, we have a strong preference for the conventional term principle of least
action for reasons not central to the argument of the present paper.
10
Thomas H. Kuhn, "Energy Conservation as an Example of Simultaneous Discovery?" in The
Conservation of Energy and the Principle of Least Action, I. Bernard Cohen, ed, Arno Press, 1981.
11
E. Noether, "Invariante Variationsprobleme," Goett. Nachr. 1918, pp. 235-257.
12
Richard P. Feynman, Nobel Lecture, December 11, 1965, from Nobel Lectures, Physics 19631970 (Elsevier).
13
Edwin F. Taylor, "A Call to Action," Am. J. Phys., 71 (5), May 2003, 423-425.
14
Jozef Hanc, Slavomir Tuleja, Martina Hancova, "Simple derivation of Newtonian
mechanics from the principle of least action," Am. J. Phys., 71 (4) April 2003, 385-391.
15
Jozef Hanc, Edwin F. Taylor, and Slavomir Tuleja, "Deriving Lagrange's Equations Using
Elementary Calculus," accepted for publication in Am. J. Phys..
16
Jozef Hanc, Slavomir Tuleja, Martina Hancova, "Symmetries and Conservations Laws:
Consequences of Noether's Theorem," accepted for publication in Am. J. Phys.
17
Reference 8, Vol. 1, Chapter 4. Feynman's parable about Dennis the Menace and his
indestructible blocks is classic. The final statement "There are no blocks" emphasizes that
energy is not a thing, cannot be pointed at, and typically cannot even be localized; it is the
property of a system.
18
Benjamin Crowell: Conservation Laws (Light and Matter, Fullerton, CA, 1998-2002).
Available at http://www.lightandmatter.com
19
Alan Van Heuvelen and Xueli Zou, "Multiple representations of work-energy processes,"
Am. J. Phys. 69 (2), 2001, pp 184-194
20
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2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
20
Roger A. Hinrich and Merlin Kleinbach, Energy: Its Use and the Environment (Harcourt, Inc.
2002).
21
Gordon J. Aubrecht, Energy, 2nd ed. Edition (Prentice-Hall, 1995).
22
Student exercise: Show that B cos ωt and C cos ωt + D sin ωt are also solutions. Use the
occasion to discuss the importance of relative phase.
23
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
24
Noether's theorem says 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 d˙x
is a constant of the motion. This statement can also be expressed in terms of Hamiltonian
dynamics. See for example Cornelius Lanczos, The Variational Principles of Mechanics, 4th ed.
(Dover Publications, 1970), Chap VI, Sec. 9, statement above Eq. (69.1). In all the mechanical
systems that we consider here, total energy is conserved and thus automatically does not
contain time explicitly. The expression for kinetic energy K is always quadratic in the
velocities, while 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 referenced above. However, we
have prepared for students a simpler, deeper, and more intuitive argument that justifies our
modified Noether theorem.
25
Ruth Chabay and Bruce Sherwood, Matter & Interactions (John Wiley, 2002) Vol. 1, pp 188194 and 244-248. Page 248 lists background references in Am. J. Phys.
26
Of course on the molecular level there is no friction. Of the current advanced mechanics
texts listed in our references, only Landau and Lifshitz, Ref. 7, mentions sliding friction even
in passing (p. 122); concerning motion in a dissipative medium they say (p. 74) " . . . there are
no equations of motion in the mechanical sense. Thus the problem of the motion of a body in
a medium is not one of mechanics." Their conclusion would surprise Isaac Newton.
27
Reference 8,Vol. 1, page 4-5.
28
Don S. Lemons, Perfect Form (Princeton University Press, 1997), Chapter 4.
29
Slavomir Tuleja, Edwin F. Taylor, Java software: Principle of Least Action Interactive, 2003,
available at the website http://www.eftaylor.com/leastaction.html.
30
Herbert Goldstein, Charles Poole, and John Safko, Classical Mechanics, 3rd ed.(AddisonWesley, Reading, MA 2002), Section2.7.
21
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2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
31
Reference 8, Vol. II, p. 19-8
32
Tomas Tyc, "The de Broglie hypothesis leading to path integrals," Eur. J. Phys. 17 (May
1996), 156-157. Version available at the URL given in reference 29.
33
R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965),
p. 29.
34
Reference 7, Chap. 1; Reference 30, Sect. 2.3; D. Gerald Jay Sussman and Jack Wisdom,
Structure and Interpretation of Classical Mechanics (MIT, Cambridge, 2001), Chap. 1.
35
Reference 15
36
References 14 and 15.
37
Ya. B. Zeldovich and A. D. Myskis, Elements of Applied Mathematics, George Yankovsky Tr.
(MIR Publishers Moscow,1976) pages 499-500
22