Feynman Lectures Simplified 2D: Magnetic Matter, Elasticity, Fluids, & Curved Spacetime

By Robert Piccioni

Feynman Simplified gives mere mortals access to the fabled Feynman Lectures on Physics.

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

If you are looking for information about a specific topic, peruse our free downloadable index to the entire Feynman Simplified series found on my website "Guide to the Cosmos . com"

• Language: English
• Publisher: Robert Piccioni
• Release date: Apr 29, 2017
• ISBN: 9781370060764

Robert Piccioni

Dr Robert Piccioni is a physicist, public speaker, educator and expert on cosmology and Einstein's theories. His "Everyone's Guide Series" e-books makes the frontiers of science accessible to all. With short books focused on specific topics, readers can easily mix and match, satisfying their individual interests. Each self-contained book tells its own story. The Series may be read in any order or combination. Robert has a B.S. in Physics from Caltech, a Ph.D. in High Energy Physics from Stanford University, was a faculty member at Harvard University and did research at the Stanford Linear Accelerator in Palo Alto, Calif. He has studied with and done research with numerous Nobel Laureates. At Caltech, one of his professors was Richard Feynman, one of the most famous physicists of the 20th century, and a good family friend. Dr. Piccioni has introduced cutting-edge science to numerous non-scientific audiences, including school children and civic groups. He was guest lecturer on a National Geographic/Lindblad cruise, and has given invited talks at Harvard, Caltech, UCLA, and Stanford University.

Book preview

Feynman Simplified

2D: Magnetic Matter,

Elasticity, Fluids, &

Curved Spacetime

Everyone’s Guide

to the

Feynman Lectures on Physics

by

Robert L. Piccioni, Ph.D.

Second Edition

Copyright © 2017

by

Robert L. Piccioni

Published by

Real Science Publishing

3949 Freshwind Circle

Westlake Village, CA 91361, USA

Edited by Joan Piccioni

All rights reserved, including the right of

reproduction in whole or in part, in any form.

Visit our web site

www.guidetothecosmos.com

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

Curved Spacetime

Magnetic Materials

Diamagnetism & Paramagnetism

Ferromagnetism

Elasticity & Elastic Matter

Viscosity & Liquid Flow

Variational Calculus

To learn more about the Feynman Simplified series, to receive updates, and send us your comments, click here. 

To further Simplify your adventure, learn about my Math for Physicists that explains the math to master Feynman physics.

Looking for information about a specific topic? Peruse our free downloadable index to the entire Feynman Simplified series.

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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: will not be on the exam.

But, that does not mean this is unimportant.

§36.1 Principle of Least Action

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 [that] starts somewhere and moves [freely] to some other point ... in a certain amount of time. Now, you try a different motion. Suppose that [it goes along a very different path, but starts and ends at the same two places] 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. If the potential energy represents all active forces, the principle of least action says:

Objects follow the path of least action.

In a gravitational field, this is:

S = m v² / 2 – m g x

Here, m is the object’s mass is m, v is its velocity, g is the acceleration of gravity, and x is its height above any convenient base elevation, such as sea level.

Let’s consider a ball thrown upward in a uniform gravitational field. The total action from time t=a to time t=b is:

Sab = ∫ab S dt = ∫ab { m v² / 2 – m g x } dt

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. One imaginable alternative is represented by the dashed curve. Both curves start at (xa,ta), and both end at (xb,tb). The alternative path seems 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 [Sab] is least."

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:

Sab = ∫ab { m v² / 2 } dt

Now, we know what the average velocity must be: total distance traveled Δx divided by total travel time Δt. We then write:

Δx = ∫ab v dt = ∫ab (dx/dt) dt = ∫ab dx = xb – xa

Δt = ∫ab dt = tb – ta

= Δx / Δt = (xb – xa) / (tb – ta)

Feynman uses the following argument to show that the action integral is minimized if v always equals :

"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."

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. No one successfully argued against Aristotle for 2000 years.

So, let’s do the math, starting with this little trick:

v² = (v – )² + 2v – ²

Sab = ∫ab v² dt m/2

Sab = ab {(v – )² + 2v – ²} dt m/2

The last term is easy:

– ² ∫ab dt = – ² Δt

The middle term is almost as easy:

2 ∫ab v dt = 2 ∫ab (dx/dt) dt

2 ∫ab v dt = 2 Δx = 2 ( Δt)

Therefore the action becomes:

2Sab / m = 2²Δt – ²Δt + ∫AB (v – )² dt

Everything in this equation is constant except the action Sab and the integral. To minimize the 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= 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 between points a and b, as shown in Figure 36-2.

Figure 36-2 Path Without Forces

Now, let’s make the problem more interesting 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, Section §10.4.) To be specific, let U be the gravitational potential. The action equation is:

Sab = ∫ab { m v² / 2 – U(x) } dt

To minimize Sab, we need to reduce the integrand’s positive term (kinetic energy) and maximize 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 Alternate With High 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 that increases the action Sab.

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 that are a distance r from its center. An alternative definition is: a circle is the curve of length L that encloses the largest area. Try using integral or differential calculus to find a circle fulfilling the second definition.

§36.2 Functions Near Extrema

Before delving into the calculus of variations, let’s first carefully examine 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. The Taylor series is:

f(x) = a0 + a1 x + a2 x² + a3 x³ + …

for some set of constants aj.

We now show that a1 must be zero. At any extremum, the first derivative of f is zero. This means:

0 = df/dx = a1 + 2a2 x + 3a3 x² + …

at x=0: 0 = df/dx = a1

Hence the Taylor series reduces to:

f(x) = a0 + a2 x² + a3 x³ + …

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,

the function changes only in second order.

Second order means the change in f(x) is proportional to the change in x to the second or higher power. We write this:

f(x) = a0 + O(x²)

O(x²) denotes any combination of unspecified terms that are proportional to second and higher powers of a small quantity x.

This fact will help us find the path of least action.

§36.3 Variational Calculus

Here is how variational calculus works.

Imagine two paths that both go from point a to point b, as shown in Figure 36-4.

Figure 36-4 Path Differences

The true path (what we seek) is represented by the solid curve, and an alternative path is represented by the dashed curve.

Let’s define x(t) and w(t) as:

true path: x(t)

alternate: x(t) + w(t)

Thus w(t) is the difference between the true path and the alternate path. The length of each vertical line in Figure 36-4 represents the value of w(t) at selected times. In our analysis, let's consider alternate paths that deviate only slightly from the true path, much less than shown in the above image. This means w(t) is small compared with x(t) at all t.

The action along the true path is:

Strue = ∫ab { (dx/dt)² m / 2 – U(x) } dt

The action along the alternative path is:

Salt = ∫ab { (d[x + w]/dt)² m / 2 – U(x + w) } dt

Since the true path has the least action, Salt must be greater than or equal to Strue. We define the variation in S, δS, to be:

δS = Salt – Strue >= 0

δS = ∫ab { [(d[x + w]/dt)² – (dx/dt)²] (m / 2)

                  – U(x + w) + U(x) } dt

Let’s simplify this piece by piece, beginning with the U terms. For small w, we use the Taylor series:

U(x+w) = U(x) + (dU/dx) w + (d²U/dx²) w²/2+ …

U(x+w) = U(x) + (dU/dx) w + O(w²)

– U(x+w) + U(x) = – (dU/dx) w + O(w²)

Now, let’s turn to the more interesting term in [ ]’s.

[(d[x + w]/dt)² – (dx/dt)²]

   = (dx/dt + dw/dt)² – (dx/dt)²

   = (dx/dt)² +2(dw/dt) (dx/dt) +(dw/dt)² –(dx/dt)²

   = 2 (dx/dt) (dw/dt) + (dw/dt)²

Since w(t) is small, the term on the right is much smaller than the term on the left. We can lump that into the other O(w²) terms. We then have:

δS = ∫ab { [m (dx/dt) (dw/dt)]

               – (dU/dx) w } dt + O(w²)

Let’s ignore the clutter for a moment and consider the Big Picture. The integrand has the form:

P dw/dt – Q w

If it were Pw – Qw, we would immediately have a relationship between P and Q. So, the way forward is to turn P(dw/dt) into something of the form Rw. But how?

The first key step in this problem is employing integration by parts. Recall the procedure: for any two functions u and v:

d (uv) /dt = u dv/dt + v du/dt

∫ab [d(uv)/dt] dt = ∫ab u [dv/dt] dt + ∫ab v [du/dt] dt

∫ab u [dv/dt] dt = ∫ab d(uv) – ∫ab v [du/dt] dt

∫ab u [dv/dt] dt = u v |ab – ∫ab v [du/dt] dt

For our problem, set v = w and u = dx/dt. Integration by parts yields:

∫ab (dx/dt) (dw/dt) dt

   = (dx/dt) w |ab – ∫ab w (d²x/dt²) dt

Now comes the second key step. The alternative path and the true path both start at (xa,ta) and both end at (xb,tb). We vary the path between the endpoints, but not at the endpoints. This means:

w(a) = w(b) = 0

∫ab (dx/dt) (dw/dt) dt = – ∫ab w (d²x/dt²) dt

Dropping O(w²), the equation for the variation of the action is becoming more manageable.

δS = ∫ab { [–m (d²x/dt²) w – (dU/dx) w } dt

δS = – ∫ab { m (d²x/dt²) + (dU/dx) } w dt

As we described earlier, any function changes very slowly near its extrema. In our case, the function is S. When the alternative path is the same as the true path, when w=0 everywhere, the variation δS equals zero. When the alternative path is close to the true path, δS is very close to zero, deviating from zero only in second order.

Our problem boils down to finding the x(t) for which δS=0 for any small path deviation w(t).

The third key step is realizing that, along the true path, the above term in { }’s is zero everywhere. Why?

Consider a function w(t) that is nonzero only over a very small range of t, say between t* and t*+Δt. If Δt is small enough, we can assume w(t) is constant over the time interval Δt. This reduces the integrand to a constant, making the integral trivial.

δS = – { m [d²x(t*)/dt²] + [dU(t*)/dx] } w(t*) Δt

On the true path δS = 0, which means:

On the true path: 0 = m [d²x(t*)/dt²] + [dU(t*)/dx]

This relationship must hold for every value of t*.

In fact, this is a general result: the principle of least action is local, not merely global. It applies not just to the complete time interval a to b, but rather it applies separately to every infinitesimal time interval. Just as momentum is conserved at every instant, motion is governed by least action at every instant.

We finally have our solution:

for all t: m d²x(t)/dt² = – dU(t)/dx

since in one dimension, F = – dU/dx. We obtain the familiar Newtonian equation of motion:

m a = F

Variational calculus proves that, for a conservative force, every body moves according to Newton’s second law: F=ma.

Should we quibble about the need to be conservative? Newton’s law applies to any force, even to a non-conservative force like friction. The principal of least action, however, applies only to conservative forces, because only for those can we define a corresponding potential energy.

But non-conservative forces are only non-conservative when we fail to account for all the action. Friction heats atoms increasing their kinetic and potential energy. Energy is not lost, it is still conserved, but the accounting becomes more complex. At a fundamental level, all forces are conservative.

In V2p19-5, Feynman provide sage advice on this variational method:

"It turns out that the whole trick of the calculus of variations consists of writing down the variation of S and then integrating by parts so that the derivatives of [w] disappear. It is always the same in every problem in which derivatives appear. … [Next] comes something which always happens—the integrated part disappears [(dx/dt)w|ab in this case]. (In fact, if the integrated part does not disappear, you restate the principle, adding conditions to make sure it does!)"

§36.4 Least Action or Most Action

Feynman Simplified 1C, Chapters 30 and 31, explore optics, the classical theory of light and refracting media.

Optics demonstrates that the propagation of light is governed by wave interference: light takes the paths in which waves interfere constructively. The dominant paths, those with the most constructive interference, are those with nearby paths that have nearly the same travel times.

The dominant paths