Table Of ContentFeynman Simplified
2D: Magnetic Matter,
Elasticity, Fluids, &
Curved Spacetime
Everyone’s Guide
to the
Feynman Lectures on Physics
by
Robert L. Piccioni, Ph.D.
Copyright © 2015
by
Robert L. Piccioni
Published by
Real Science Publishing
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Edited by Joan Piccioni
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Everyone’s Guide to the
Feynman Lectures on Physics
Feynman Simplified gives mere mortals access to the fabled Feynman Lectures on Physics.
This Book
Feynman Simplified: 2D covers the final quarter of Volume 2 of The Feynman Lectures on
Physics. The topics we explore include:
Principle of Least Action
Tensors in 3-D and 4-D Spacetime
Magnetic Materials
Diamagnetism & Paramagnetism
Ferromagnetism
Elasticity & Elastic Matter
Viscosity & Liquid Flow
Gravity and Curved Spacetime
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Table of Contents
Chapter 36: Principle of Least Action
Chapter 37: Tensors
Chapter 38: Magnetic Matter
Chapter 39: Paramagnetism & Resonance
Chapter 40: Theories of Ferromagnetism
Chapter 41: Practical Ferromagnetism
Chapter 42: Elasticity
Chapter 43: Elastic Materials
Chapter 44: Non-Viscous Fluid Flow
Chapter 45: Viscous Fluid Flow
Chapter 46: Curved Spacetime
Chapter 36
Principle
of Least Action
This is a special lecture on a general principle that applies to all of physics, not just
electromagnetism.
Feynman is famous for his profound understanding of the Principle of Least Action.
The Feynman Lectures state this lecture: “is intended to be for ‘entertainment’.” That is code for:
“this won’t be on the exam.” But, that does not mean this is unimportant.
In fact, the Principle of Least Action is one of the most important principles of physics — a principle
every serious physicist should understand.
On V2p19-1, Feynman says:
“When I was in high school, my physics teacher—whose name was Mr. Bader—called me down
one day after physics class and said, ‘You look bored; I want to tell you something interesting.’
Then he told me something which I found absolutely fascinating, and have, since then, always
found fascinating. Every time the subject comes up, I work on it. In fact, when I began to prepare
this lecture I found myself making more analyses on the thing. Instead of worrying about the
lecture, I got involved in a new problem. The subject is this—the principle of least action.
“Mr. Bader told me the following: Suppose you have a particle (in a gravitational field, for
instance) which starts somewhere and moves to some other point by free motion—you throw it,
and it goes up and comes down. It goes from the original place to the final place in a certain
amount of time. Now, you try a different motion. Suppose that to get from here to there, it went
[along a very different path] but got there in just the same amount of time. Then he said this: If
you calculate the kinetic energy at every moment on the path, take away the potential energy, and
integrate it over the time during the whole path, you’ll find that the number you’ll get is bigger
than that for the actual motion.
“In other words, the laws of Newton could be stated not in the form F = ma but in the form: the
average kinetic energy less the average potential energy is as little as possible for the path of an
object going from one point to another.”
We define an object’s action S to be its kinetic energy minus potential energy. In a gravitational field,
this is:
S = mv2/2 – mgx
Here, g is the acceleration of gravity, m is the object’s mass, v is its velocity, and x is its height above
any convenient base elevation, such as sea level.
If the potential energy represents all active forces, the principle of least action says:
Objects follow the path of least action.
Let’s consider Mr. Bader’s simple example: a ball thrown upward in a uniform gravitational field.
The total action from time t=A to time t=B is:
S = ∫ B { mv2/2 – mgx } dt
AB A
Figure 36-1 shows two possible paths, with x plotted vertically and time t plotted horizontally.
Figure 36-1 Two Possible Paths
The actual path taken by a real ball is a parabola, shown as the solid curve in Figure 36-1. An
alternative that we might imagine is shown as the dashed curve. Both curves start at the same x and t,
and both end at the same x and t. We see that the alternative path is more “interesting”, with more
structure and sharper turns.
We might imagine many alternatives to nature’s actual path, but as Feynman says in V2p19-2: “The
miracle is that the true path is the one for which [S ] is least.”
AB
Let’s make the problem even simpler. Let’s suppose no forces act on the ball, and therefore there is no
potential energy term in our integral. The action then reduces to:
S = ∫ B { mv2/2 } dt
AB A
Now, we know what the average velocity <v> must be: total distance traveled Δx divided by total
travel time Δt. We then write:
Δx = ∫ B v dt = ∫ B (dx/dt) dt = ∫ B dx
A A A
Δt = ∫ B dt
A
<v> = Δx / Δt
Feynman uses the following argument to show that the action integral is minimized if v is always
equal to <v>:
“As an example, say your job is to start from home and get to school in a given length of time
with the car. You can do it several ways: You can accelerate like mad at the beginning and slow
down with the brakes near the end, or you can go at a uniform speed, or you can go backwards
for a while and then go forward, and so on. The thing is that the average speed has got to be, of
course, the total distance that you have gone over the time. But if you do anything but go at a
uniform speed, then sometimes you are going too fast and sometimes you are going too slow.
Now the mean square of something that deviates around an average, as you know, is always
greater than the square of the mean; so the kinetic energy integral would always be higher if you
wobbled your velocity than if you went at a uniform velocity.”
Personally, I am more comfortable with mathematical proofs than with verbal arguments and
analogies. Analogies are almost never perfect, and arguments are often won by the loudest and most
forceful, whether or not they are right. Despite some glaring errors in his theories, no one successfully
argued against Aristotle for 2000 years.
So, let’s do the math, starting with this little trick:
(v – <v>)2 = v2 – 2v<v> + <v>2
v2 = (v – <v>)2 + 2v<v> – <v>2
S = (m/2) ∫ B v2 dt
AB A
2S /m = ∫ B {(v–<v>)2 + 2v<v> – <v>2} dt
AB A
The last term is easy:
– <v>2 ∫ B dt = – <v>2 Δt
A
The middle term is almost as easy:
2<v> ∫ B v dt = 2<v> ∫ B (dx/dt) dt
A A
2<v> ∫ B v dt = 2<v> Δx = 2<v> (<v> Δt)
A
Therefore the action becomes:
2S /m = <v>2 Δt (+2–1) + ∫ B {(v–<v>)2 dt
AB A
Everything has been reduced to constants except the action S and the last integral. To minimize the
AB
action, we must minimize this integral. Since the integrand is a perfect square, it is always greater
than or equal to zero. The minimum clearly occurs when v=<v> always, just as Feynman argued.
If I were as smart as Feynman, and if I already knew the right answer, I would also be satisfied with
arguments and analogies.
Hence, for a ball subject to no forces whatsoever, the motion of least action is traveling at a constant
velocity from “here to there”, as shown in Figure 36-2.
Figure 36-2 Path Without Forces
Now, let’s make the problem more realistic by adding a conservative force with potential U(x).
(Recall that all fundamental forces are conservative and that only conservative forces have
meaningful potentials; see Feynman Simplified 1A, Chapter 10.) The action equation is:
S = ∫ B { mv2/2 – U(x) } dt
AB A
To minimize S , we would like to reduce the integrand’s positive term (kinetic energy) and maximize
AB
its negative term (potential energy U). Figure 36-3 shows the object’s true path represented by the
solid curve, and an alternative path represented by the dashed curve.
Figure 36-3 Dashed Path With High Potential
Let’s assume U(x) increases with increasing x, as does the gravitational potential. The alternative
path offers the lure of a higher average U than the true path. The problem, however, is that rapidly
increasing x to rapidly increase U(x) requires a large initial velocity v, and therefore a large kinetic
energy mv2/2 that increases the action S .
AB
Finding the minimum action is a puzzle whose solution optimally balances competing effects.
Increasing U too much or too rapidly increases the kinetic energy thus increasing the action. But,
increasing U too little or too slowly fails to reduce the action.
In V2p19-3, Feynman says:
“That is all my teacher told me, because he was a very good teacher and knew when to stop
talking. But I don’t know when to stop talking. So instead of leaving it as an interesting remark, I
am going to horrify and disgust you with the complexities of life by proving that it is so. The kind
of mathematical problem we will have is very difficult and [is of] a new kind.
“You [might say:] ‘Oh, that’s just the ordinary calculus of maxima and minima. You calculate the
action and just differentiate to find the minimum.’
“But watch out. Ordinarily we just have a function of some variable, and we have to find the
value of that variable where the function is least or most. For instance, we have a rod which has
been heated in the middle and the heat is spread around. For each point on the rod we have a
temperature, and we must find the point at which that temperature is largest. But now for each
path in space we have a number—quite a different thing—and we have to find the path in space
for which the number is the minimum. That is a completely different branch of mathematics. It is
not the ordinary calculus. In fact, it is called the calculus of variations.”
Feynman says there are many similar variational problems in physics and mathematics. For example,
we normally define a circle as the locus of points whose distance from a chosen center is r. An
alternative definition is: a circle is the curve of length L that encloses the largest area. From that
definition, one can construct a circle whose radius is L/2π. Feynman suggests you might amuse
yourself by trying to find a circle fulfilling the second definition using integral or differential calculus.
Functions Near Extrema
Before delving into the calculus of variations, let’s first examine carefully the behavior of an arbitrary
function f near an extremum, either a minimum or a maximum. Recall that we can express any function
as a Taylor series. Let’s assume our function f has a minimum, and define the x-axis so that this
minimum occurs at x=0. For some set of constants a, the Taylor series is:
j
f(x) = a + a x + a x2 + a x3 + …
0 1 2 3
We now show that a must be zero. At any extremum, the first derivative of f is zero. This means:
1
at x=0: df/dx = 0 = a + 2a x + 3a x2 + …
1 2 3
Hence a = 0, and the Taylor series reduces to:
1
f(x) = a + a x2 + a x3 + …
0 2 3
For small values of |x|, x2 is much smaller yet. Hence, any function changes very slowly near its
extrema. The more elegant description is:
Near any function’s extrema,
