LEONARD SUSSKIND AND GEORGE HRABOVSKY

The Theoretical Minimum

What You Need to Know to Start Doing Physics

ALLEN LANE
an imprint
of
PENGUIN BOOKS

Contents

Preface

LECTURE 1 The Nature of Classical Physics

Interlude 1 Spaces, Trigonometry, and Vectors

LECTURE 2 Motion

Interlude 2 Integral Calculus

LECTURE 3 Dynamics

Interlude 3 Partial Differentiation

LECTURE 4 Systems of More Than One
Particle

LECTURE 5 Energy

LECTURE 6 The Principle of Least Action

LECTURE 7 Symmetries and Conservation Laws

LECTURE 8 Hamiltonian Mechanics and
Time-Translation Invariance

LECTURE 9 The Phase Space Fluid and the
Gibbs-Liouville Theorem

LECTURE 10 Poisson Brackets, Angular Momentum,
and Symmetries

LECTURE 11 Electric and Magnetic Forces

Appendix 1 Central Forces and Planetary Orbits

To our spouses—
those who have chosen to put up with us,
and to the
students of Professor Susskind’s
Continuing Education Courses

Preface

I’ve always enjoyed explaining
physics. For me it’s much more than teaching: It’s a way of thinking. Even
when I’m at my desk doing research, there’s a dialog going on in my head.
Figuring out the best way to explain something is almost always the best way to
understand it yourself.

About ten years ago someone asked me if I
would teach a course for the public. As it happens, the Stanford area has a lot of
people who once wanted to study physics, but life got in the way. They had had all kinds
of careers but never forgot their one-time infatuation with the laws of the universe.
Now, after a career or two, they wanted to get back into it, at least at a casual
level.

Unfortunately there was not much opportunity
for such folks to take courses. As a rule, Stanford and other universities don’t
allow outsiders into classes, and, for most of these grown-ups, going back to school as
a full-time student is not a realistic option. That bothered me. There ought to be a way
for people to develop their interest by interacting with active scientists, but there
didn’t seem to be one.

That’s when I first found out about
Stanford’s Continuing Studies program. This program offers courses for people in
the local nonacademic community. So I thought that it might just serve my purposes in
finding someone to explain physics to, as well as their purposes, and it might also be
fun to teach a course on modern physics. For one academic quarter anyhow.

It was fun. And it was very satisfying in a
way that teaching undergraduate and graduate students was sometimes not.
These students were there for only one reason: Not to get credit, not to get a degree,
and not to be tested, but just to learn and indulge their curiosity. Also, having been
“around the block” a few times, they were not at all afraid to ask
questions, so the class had a lively vibrancy that academic classes often lack. I
decided to do it again. And again.

What became clear after a couple of quarters
is that the students were not completely satisfied with the layperson’s courses I
was teaching. They wanted more than the Scientific American experience. A lot
of them had a bit of background, a bit of physics, a rusty but not dead knowledge of
calculus, and some experience at solving technical problems. They were ready to try
their hand at learning the real thing—with equations. The result was a sequence of
courses intended to bring these students to the forefront of modern physics and
cosmology.

Fortunately, someone (not I) had the bright
idea to video-record the classes. They are out on the Internet, and it seems that they
are tremendously popular: Stanford is not the only place with people hungry to learn
physics. From all over the world I get thousands of e-mail messages. One of the main
inquiries is whether I will ever convert the lectures into books? The Theoretical
Minimum is the answer.

The term theoretical minimum was
not my own invention. It originated with the great Russian physicist Lev Landau. The TM
in Russia meant everything a student needed to know to work under Landau himself. Landau
was a very demanding man: His theoretical minimum meant just about everything he knew,
which of course no one else could possibly know.

I use the term differently. For me, the
theoretical minimum means just what you need to know in order to proceed to the next
level. It means not fat encyclopedic textbooks that explain everything,
but thin books that explain everything important. The books closely follow the Internet
courses that you will find on the Web.

Welcome, then, to The Theoretical
Miniumum—Classical Mechanics, and good luck!

Leonard Susskind

Stanford, California, July 2012

I started to teach myself math and physics
when I was eleven. That was forty years ago. A lot of things have happened since then—I
am one of those individuals who got sidetracked by life. Still, I have learned a lot of
math and physics. Despite the fact that people pay me to do research for them, I never
pursued a degree.

For me, this book began with an e-mail.
After watching the lectures that form the basis for the book, I wrote an e-mail to
Leonard Susskind asking if he wanted to turn the lectures into a book. One thing led to
another, and here we are.

We could not fit everything we wanted into
this book, or it wouldn’t be The Theoretical Minimum—Classical Mechanics,
it would be A-Big-Fat-Mechanics-Book. That is what the Internet is for: Taking up large
quantities of bandwidth to display stuff that doesn’t fit elsewhere! You can find
extra material at the website www.madscitech.org/tm. This material will include answers to the problems,
demonstrations, and additional material that we couldn’t put in the book.

I hope you enjoy reading this book as much
as we enjoyed writing it.

George Hrabovsky

Madison, Wisconsin, July 2012

Lecture 1: The Nature of Classical Physics

Somewhere in Steinbeck country two tired
men sit down at the side of the road. Lenny combs his beard with his fingers and says,
“Tell me about the laws of physics, George.” George looks down for a moment,
then peers at Lenny over the tops of his glasses. “Okay, Lenny, but just the
minimum.”

WHAT IS CLASSICAL PHYSICS?

The term classical physics refers to
physics before the advent of quantum mechanics. Classical physics includes
Newton’s equations for the motion of particles, the Maxwell-Faraday theory of
electromagnetic fields, and Einstein’s general theory of relativity. But it is
more than just specific theories of specific phenomena; it is a set of principles and
rules—an underlying logic—that governs all phenomena for which quantum uncertainty is
not important. Those general rules are called classical mechanics.

The job of classical mechanics is to predict
the future. The great eighteenth-century physicist Pierre-Simon Laplace laid it out in a
famous quote:

We may regard the present state of the universe as the effect of its past and the
cause of its future. An intellect which at a certain moment would know all forces
that set nature in motion, and all positions of all items of which nature is
composed, if this intellect were also vast enough to submit these data to analysis,
it would embrace in a single formula the movements of the greatest bodies of the
universe and those of the tiniest atom; for such an intellect
nothing would be uncertain and the future just like the past would be present before
its eyes.

In classical physics, if you know everything
about a system at some instant of time, and you also know the equations that govern how
the system changes, then you can predict the future. That’s what we mean when we
say that the classical laws of physics are deterministic. If we can say the
same thing, but with the past and future reversed, then the same equations tell you
everything about the past. Such a system is called reversible.

SIMPLE DYNAMICAL SYSTEMS AND THE SPACE
OF STATES

A collection of objects—particles, fields,
waves, or whatever—is called a system. A system that is either the entire
universe or is so isolated from everything else that it behaves as if nothing else
exists is a closed system.

Exercise 1: Since the notion is so important to theoretical physics, think
about what a closed system is and speculate on whether closed systems can
actually exist. What assumptions are implicit in establishing a closed system?
What is an open system?

To get an idea of what deterministic and
reversible mean, we are going to begin with some extremely simple closed systems. They
are much simpler than the things we usually study in physics, but they satisfy rules
that are rudimentary versions of the laws of classical mechanics. We begin with an
example that is so simple it is trivial. Imagine an abstract object that has only one
state. We could think of it as a coin glued to the table—forever showing heads. In
physics jargon, the collection of all states occupied by a system is its
space of states, or, more simply, its state-space. The state-space is not
ordinary space; it’s a mathematical set whose elements label the possible states
of the system. Here the state-space consists of a single point—namely Heads (or just
H)—because the system has only one state. Predicting the future of this system is
extremely simple: Nothing ever happens and the outcome of any observation is always
H.

The next simplest system has a state-space
consisting of two points; in this case we have one abstract object and two possible
states. Imagine a coin that can be either Heads or Tails (H or T). See Figure 1.

Figure 1: The space of two states.

In classical mechanics we assume that
systems evolve smoothly, without any jumps or interruptions. Such behavior is said to be
continuous. Obviously you cannot move between Heads and Tails smoothly.
Moving, in this case, necessarily occurs in discrete jumps. So let’s assume that
time comes in discrete steps labeled by integers. A world whose evolution is discrete
could be called stroboscopic.

A system that changes with time is called a
dynamical system. A dynamical system consists of more than a space of
states. It also entails a law of motion, or dynamical law. The
dynamical law is a rule that tells us the next state given the current state.

One very simple dynamical law is that whatever
the state at some instant, the next state is the same. In the case of our example, it
has two possible histories: H H H H H H … and T T T T T T .…

Another dynamical law dictates that whatever
the current state, the next state is the opposite. We can make diagrams to illustrate
these two laws. Figure 2 illustrates the first law, where the arrow from H goes to H and
the arrow from T goes to T. Once again it is easy to predict the future: If you start
with H, the system stays H; if you start with T, the system stays T.

Figure 2: A dynamical law for a two-state
system.

A diagram for the second possible law is
shown in Figure 3, where the arrows lead from H to T and from T to H. You can still
predict the future. For example, if you start with H the history will be H T H T H T H T
H T .… If you start with T the history is T H T H T H T H .…

Figure 3: Another dynamical law for a two-state
system.

We can even write these dynamical laws in
equation form. The variables describing a system are called its degrees of
freedom. Our coin has one degree of freedom, which we can
denote by the greek letter sigma, . Sigma has only two possible values;
= 1 and
=
−1, respectively, for H and T. We also use a symbol to keep track of the time.
When we are considering a continuous evolution in time, we can symbolize it with
t. Here we have a discrete evolution and will use n. The state at
time n is described by the symbol (n), which stands for at
n.

Let’s write equations of evolution for
the two laws. The first law says that no change takes place. In equation form,

In other words, whatever the value of at the nth step, it will have the same value at the next step.

The second equation of evolution has the
form

implying that the state flips during each
step.

Because in each case the future behavior is
completely determined by the initial state, such laws are deterministic. All the basic
laws of classical mechanics are deterministic.

To make things more interesting, let’s
generalize the system by increasing the number of states. Instead of a coin, we could
use a six-sided die, where we have six possible states (see Figure 4).

Now there are a great many possible laws,
and they are not so easy to describe in words—or even in equations. The simplest way is
to stick to diagrams such as Figure 5. Figure 5 says that given the numerical state of
the die at time n, we increase the state one unit at the next instant
n + 1. That works fine until we get to 6, at which point the diagram tells
you to go back to 1 and repeat the pattern. Such a pattern that is repeated endlessly is called a cycle. For example, if we start with 3
then the history is 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, .… We’ll call
this pattern Dynamical Law 1.

Figure 4: A six-state system.

Figure 5: Dynamical Law 1.

Figure 6 shows another law, Dynamical Law 2.
It looks a little messier than the last case, but it’s logically identical—in each
case the system endlessly cycles through the six possibilities. If we relabel the
states, Dynamical Law 2 becomes identical to Dynamical Law 1.

Not all laws are logically the same.
Consider, for example, the law shown in Figure 7. Dynamical Law 3 has two cycles. If you
start on one of them, you can’t get to the other. Nevertheless, this law is
completely deterministic. Wherever you start, the future is determined. For example, if
you start at 2, the history will be 2, 6, 1, 2, 6, 1, … and
you will never get to 5. If you start at 5 the history is 5, 3, 4, 5, 3,
4, … and you will never get to 6.

Figure 6: Dynamical Law 2.

Figure 7: Dynamical Law 3.

Figure 8 shows Dynamical Law 4 with three
Cycles.

Figure 8: Dynamical Law 4.

It would take a long time to write out all of
the possible dynamical laws for a six-state system.

Exercise 2: Can you think of a general way to classify the laws that are
possible for a six-state system?

RULES THAT ARE NOT ALLOWED: THE
MINUS-FIRST LAW

According to the rules of classical physics,
not all laws are legal. It’s not enough for a dynamical law to be deterministic;
it must also be reversible.

The meaning of reversible—in the
context of physics—can be described a few different ways. The most concise description
is to say that if you reverse all the arrows, the resulting law is still deterministic.
Another way, is to say the laws are deterministic into the past as well as the
future. Recall Laplace’s remark, “for such an intellect nothing
would be uncertain and the future just like the past would be present before its
eyes.” Can one conceive of laws that are deterministic into the future, but not
into the past? In other words, can we formulate irreversible laws? Indeed we can.
Consider Figure 9.

Figure 9: A system that is irreversible.

The law of Figure 9 does tell you, wherever
you are, where to go next. If you are at 1, go to 2. If at 2, go to 3. If at 3, go to 2.
There is no ambiguity about the future. But the past is a different
matter. Suppose you are at 2. Where were you just before that? You could have come from
3 or from 1. The diagram just does not tell you. Even worse, in terms of reversibility,
there is no state that leads to 1; state 1 has no past. The law of Figure 9 is
irreversible. It illustrates just the kind of situation that is prohibited
by the principles of classical physics.

Notice that if you reverse the arrows in
Figure 9 to give Figure 10, the corresponding law fails to tell you where to go in the
future.

Figure 10: A system that is not deterministic into
the future.

There is a very simple rule to tell when a
diagram represents a deterministic reversible law. If every state has a single unique
arrow leading into it, and a single arrow leading out of it, then it is a legal
deterministic reversible law. Here is a slogan: There must be one arrow to tell you
where you’re going and one to tell you where you came from.

The rule that dynamical laws must be
deterministic and reversible is so central to classical physics that we sometimes forget
to mention it when teaching the subject. In fact, it doesn’t even have a name. We
could call it the first law, but unfortunately there are already two first
laws—Newton’s and the first law of thermodynamics. There is even a zeroth law of
thermodynamics. So we have to go back to a minus-first law to gain priority for
what is undoubtedly the most fundamental of all physical laws—the conservation of
information. The conservation of information is simply the
rule that every state has one arrow in and one arrow out. It ensures that you never lose
track of where you started.

The conservation of information is not a
conventional conservation law. We will return to conservation laws after a digression
into systems with infinitely many states.

DYNAMICAL SYSTEMS WITH AN INFINITE
NUMBER OF STATES

So far, all our examples have had
state-spaces with only a finite number of states. There is no reason why you can’t
have a dynamical system with an infinite number of states. For example, imagine a line
with an infinite number of discrete points along it—like a train track with an infinite
sequence of stations in both directions. Suppose that a marker of some sort can jump
from one point to another according to some rule. To describe such a system, we can
label the points along the line by integers the same way we labeled the discrete
instants of time above. Because we have already used the notation n for the
discrete time steps, let’s use an uppercase N for points on the track. A
history of the marker would consist of a function N(n), telling you
the place along the track N at every time n. A short portion of this
state-space is shown in Figure 11.

Figure 11: State-space for an infinite system.

A very simple dynamical law for such a
system, shown in Figure 12, is to shift the marker one unit in the positive direction at
each time step.

Figure 12: A dynamical rule for an infinite
system.

This is allowable because each state has one
arrow in and one arrow out. We can easily express this rule in the form of an
equation.

Here are some other possible rules, but not
all are allowable.

Exercise 3: Determine which of
the dynamical laws shown in Eq.s (2) through (5) are allowable.

In Eq. (1), wherever you start, you will
eventually get to every other point by either going to the future or going to the past.
We say that there is a single infinite cycle. With Eq. (3), on the other hand, if you
start at an odd value of N, you will never get to an even value, and vice
versa. Thus we say there are two infinite cycles.

We can also add qualitatively different
states to the system to create more cycles, as shown in Figure 13.

Figure 13: Breaking an infinite configuration space
into finite and infinite cycles.

If we start with a number, then we just keep
proceeding through the upper line, as in Figure 12. On the other hand, if we start at
A or B, then we cycle between them. Thus we can have mixtures
where we cycle around in some states, while in others we move off to infinity.

CYCLES AND CONSERVATION LAWS

When the state-space is separated into
several cycles, the system remains in whatever cycle it started in. Each cycle has its
own dynamical rule, but they are all part of the same state-space because they describe
the same dynamical system. Let’s consider a system with three cycles. Each of
states 1 and 2 belongs to its own cycle, while 3 and 4 belong to the third (see Figure 14).

Figure 14: Separating the state-space into
cycles.

Whenever a dynamical law divides the
state-space into such separate cycles, there is a memory of which cycle they started in.
Such a memory is called a conservation law; it tells us that something is kept intact for all time. To make the conservation law quantitative, we
give each cycle a numerical value called Q. In the example in Figure 15 the
three cycles are labeled Q = +1, Q = −1, and Q = 0.
Whatever the value of Q, it remains the same for all time because the dynamical
law does not allow jumping from one cycle to another. Simply stated, Q is
conserved.

Figure 15: Labeling the cycles with specific values
of a conserved quantity.

In later chapters we will take up the
problem of continuous motion in which both time and the state-space are continuous. All
of the things that we discussed for simple discrete systems have their analogs for the
more realistic systems but it will take several chapters before we see how they all play
out.

THE LIMITS OF PRECISION

Laplace may have been overly optimistic about
how predictable the world is, even in classical physics. He certainly would have agreed
that predicting the future would require a perfect knowledge of the dynamical laws
governing the world, as well as tremendous computing power—what he called an
“intellect vast enough to submit these data to analysis.” But there is
another element that he may have underestimated: the ability to know the initial
conditions with almost perfect precision. Imagine a die with a million faces, each of
which is labeled with a symbol similar in appearance to the usual
single-digit integers, but with enough slight differences so that there are a million
distinguishable labels. If one knew the dynamical law, and if one were able to recognize
the initial label, one could predict the future history of the die. However, if
Laplace’s vast intellect suffered from a slight vision impairment, so that he was
unable to distinguish among similar labels, his predicting ability would be limited.

In the real world, it’s even worse;
the space of states is not only huge in its number of points—it is continuously
infinite. In other words, it is labeled by a collection of real numbers such as the
coordinates of the particles. Real numbers are so dense that every one of them is
arbitrarily close in value to an infinite number of neighbors. The ability to
distinguish the neighboring values of these numbers is the “resolving power”
of any experiment, and for any real observer it is limited. In principle we cannot know
the initial conditions with infinite precision. In most cases the tiniest differences in
the initial conditions—the starting state—leads to large eventual differences in
outcomes. This phenomenon is called chaos. If a system is chaotic (most are),
then it implies that however good the resolving power may be, the time over which the
system is predictable is limited. Perfect predictability is not achievable, simply
because we are limited in our resolving power.

Interlude 1: Spaces, Trigonometry, and Vectors

“Where are we, George?”

George pulled out his map and spread it out
in front of Lenny. “We’re right here Lenny, coordinates 36.60709N,
−121.618652W.”

“Huh? What’s a coordinate
George?”

COORDINATES

To describe points quantitatively, we need to
have a coordinate system. Constructing a coordinate system begins with choosing a point
of space to be the origin. Sometimes the origin is chosen to make the equations
especially simple. For example, the theory of the solar system would look more
complicated if we put the origin anywhere but at the Sun. Strictly speaking, the
location of the origin is arbitrary—put it anywhere—but once it is chosen, stick with
the choice.

The next step is to choose three
perpendicular axes. Again, their location is somewhat arbitrary as long as they are
perpendicular. The axes are usually called x, y, and z but we
can also call them x_1, x_2, and x_3. Such a system of axes is called
a Cartesian coordinate system, as in Figure 1.

Figure 1. A three-dimensional Cartesian coordinate
system.

We want to describe a certain point in space;
call it P. It can be located by giving the x, y, z
coordinates of the point. In other words, we identify the point P with the
ordered triple of numbers (x, y, z) (see Figure 2).

Figure 2. A point in Cartesian space.

The x coordinate represents the
perpendicular distance of P from the plane defined by setting x = 0
(see Figure 3). The same is true for the
y and z coordinates. Because the coordinates represent distances
they are measured in units of length, such as meters.

Figure 3: A plane defined by setting x =
0, and the distance to P along the x axis.

When we study motion, we also need to keep
track of time. Again we start with an origin—that is, the zero of time. We could pick
the Big Bang to be the origin, or the birth of Jesus, or just the start of an
experiment. But once we pick it, we don’t change it.

Next we need to fix a direction of time. The
usual convention is that positive times are to the future of the origin and negative
times are to the past. We could do it the other way, but we won’t.

Finally, we need units for time. Seconds are
the physicist’s customary units, but hours, nanoseconds, or years are also
possible. Once having picked the units and the origin, we can label any time by a number
t.

There are two implicit assumptions about
time in classical mechanics. The first is that time runs uniformly—an interval of 1
second has exactly the same meaning at one time as at another. For example, it took the
same number of seconds for a weight to fall from the Tower of Pisa in Galileo’s
time as it takes in our time. One second meant the same thing then as it does now.

The other assumption is that times can be
compared at different locations. This means that clocks located in
different places can be synchronized. Given these assumptions, the four
coordinates—x, y, z, t—define a reference
frame. Any event in the reference frame must be assigned a value for each of
the coordinates.

Given the function f(t) =
t^2, we can plot the points on a coordinate system.
We will use one axis for time, t, and another for the function,
f(t) (see Figure 4).

Figure 4: Plotting the points of
f(t) = t^2.

We can also connect the dots with curves to
fill in the spaces between the points (see Figure
5).

Figure 5: Joining the plotted points with
curves.

In this way we can visualize functions.

Exercise 1: Using a graphing
calculator or a program like Mathematica, plot each of the following
functions. See the next section if you are unfamiliar with the trigonometric
functions.

TRIGONOMETRY

If you have not studied trigonometry, or if
you studied it a long time ago, then this section is for you.

We use trigonometry in physics all the time;
it is everywhere. So you need to be familiar with some of the ideas, symbols, and
methods used in trigonometry. To begin with, in physics we do not generally use the
degree as a measure of angle. Instead we use the radian; we say that there are
2 radians in 360°, or 1 radian = /180°, thus 90° = /2 radians, and 30° = /6 radians. Thus a radian is
about 57° (see Figure 6).

The trigonometric functions are defined in
terms of properties of right triangles. Figure 7 illustrates the right triangle and its
hypotenuse c, base b, and altitude a. The greek letter theta,
, is defined to be the angle opposite the altitude, and the
greek letter phi, , is defined to be the angle opposite
the base.

Figure 6: The radian as the angle subtended by an
arc equal to the radius of the circle.

Figure 7: A right triangle with segments and angles
indicated.

We define the functions sine (sin), cosine
(cos), and tangent (tan), as ratios of the various sides according to the following
relationships:

We can graph these functions to see how they
vary (see Figures 8 through 10).

Figure 8: Graph of the sine function.

Figure 9: Graph of the cosine function.

Figure 10: Graph of the tangent function.

There are a couple of useful things to know
about the trigonometric functions. The first is that we can draw a triangle within a
circle, with the center of the circle located at the origin of a Cartesian coordinate
system, as in Figure 11.

Figure 11: A right triangle drawn in a circle.

Here the line connecting the center of the
circle to any point along its circumference forms the hypotenuse of a right triangle,
and the horizontal and vertical components of the point are the base and altitude of
that triangle. The position of a point can be specified by two coordinates, x
and y, where

and

This is a very useful relationship between
right triangles and circles.

Suppose a certain angle is the sum or
difference of two other angles using the greek letters alpha, , and beta, , we can write this angle, , as .
The trigonometric functions of can be expressed in
terms of the trigonometric functions of and .

A final—very useful—identity is

(Notice the notation used here: sin^2
= sin sin .) This equation is the Pythagorean theorem in disguise. If we
choose the radius of the circle in Figure 11 to be 1, then the sides a and
b are the sine and cosine of , and the hypotenuse is 1. Equation
(1) is the familiar relation among the three sides of a right triangle: a^2 + b^2 = c^2.

VECTORS

Vector notation is another mathematical
subject that we assume you have seen before, but—just to level the playing
field—let’s review vector methods in ordinary three-dimensional space.

A vector can be thought of as an
object that has both a length (or magnitude) and a direction in space. An
example is displacement. If an object is moved from some particular starting location,
it is not enough to know how far it is moved in order to know where it winds up. One
also has to know the direction of the displacement. Displacement is the simplest example
of a vector quantity. Graphically, a vector is depicted as an arrow with a length and
direction, as shown in Figure 12.

Figure 12: A vector in Cartesian
coordinates.

Symbolically vectors are represented by placing
arrows over them. Thus the symbol for displacement is . The magnitude, or
length, of a vector is expressed in absolute-value notation. Thus the length of is denoted
.

Here are some operations that can be done
with vectors. First of all, you can multiply them by ordinary real numbers. When dealing
with vectors you will often see such real numbers given the special name
scalar. Multiplying by a positive number just multiplies the length of the
vector by that number. But you can also multiply by a negative number, which reverses
the direction of the vector. For example −2 is the vector that is
twice as long as but points in the opposite direction.

Vectors may be added. To add and , place them as shown in Figure 13 to form a quadrilateral
(this way the directions of the vectors are preserved). The sum of the vectors is the
length and angle of the diagonal.

Figure 13: Adding vectors.

If vectors can be added and if they can be
multiplied by negative numbers then they can be subtracted.

Exercise 2: Work out the rule for vector subtraction.

Vectors can also be described in component
form. We begin with three perpendicular axes x, y, z. Next,
we define three unit vectors that lie along these axes and have unit length.
The unit vectors along the coordinate axes are called basis vectors. The three
basis vectors for Cartesian coordinates are traditionally called , , and (see Figure 14). More generally, we write
1, 2, and 3 when we
refer to (x_1, x_2,
x_3), where the symbol known
as a carat) tells us we are dealing with unit (or basis) vectors. The basis vectors are
useful because any vector can be written in terms of them in the following way:

Figure 14: Basis vectors for a Cartesian coordinate
system.

The quantities Vx, Vy, and
Vz are numerical coefficients that are
needed to add up the basis vectors to give . They are also called the
components of . We can say that Eq. (2) is a linear combination of
basis vectors. This is a fancy way of saying that we add the basis vectors along with
any relevant factors. Vector components can be positive or negative. We can also write a
vector as a list of its components—in this case (Vx, Vy, Vz). The magnitude of a vector can be given in terms of its
components by applying the three-dimensional Pythagorean theorem.

We can multiply a vector by a scalar,
, in terms
of components by multiplying each component by .

We can write the sum of two vectors as the
sum of the corresponding components.

Can we multiply vectors? Yes, and there is more
than one way. One type of product—the cross product—gives another vector. For now, we
will not worry about the cross product and only consider the other method, the dot
product. The dot product of two vectors is an ordinary number, a scalar. For
vectors and it is defined as follows:

Here is the angle between the
vectors. In ordinary language, the dot product is the product of the magnitudes of the
two vectors and the cosine of the angle between them.

The dot product can also be defined in terms
of components in the form

This makes it easy to compute dot products
given the components of the vectors.

Exercise 3: Show that the magnitude of a vector satisfies

Exercise 4: Let
(Ax = 2, Ay = −3, Az
= 1) and (Bx = −4,
By = −3, Bz = 2). Compute the magnitude of a and , their dot product, and the
angle between them.

An important property of the dot product is
that it is zero if the vectors are orthogonal (perpendicular). Keep this in
mind because we will have occasion to use it to show that vectors are orthogonal.

Exercise 5: Determine which pair of vectors are orthogonal. (1, 1, 1) (2,
−1, 3) (3, 1, 0) (−3, 0, 2)

Exercise 6: Can you explain why the dot product of two vectors that are
orthogonal is 0?

Lecture 2: Motion

Lenny complained, “George, this jumpy
stroboscopic stuff makes me nervous. Is time really so bumpy? I wish things would go a
little more smoothly.”

George thought for a moment, wiping the
blackboard, “Okay, Lenny, today let’s study systems that do change
smoothly.”

MATHEMATICAL INTERLUDE: DIFFERENTIAL
CALCULUS

In this book we will mostly be dealing with
how various quantities change with time. Most of classical mechanics deals with things
that change smoothly—continuously is the mathematical term—as time changes
continuously. Dynamical laws that update a state will have to involve such continuous
changes of time, unlike the stroboscopic changes of the first lecture. Thus we will be
interested in functions of the independent variable t.

To cope, mathematically, with continuous
changes, we use the mathematics of calculus. Calculus is about limits, so let’s
get that idea in place. Suppose we have a sequence of numbers, l_1, l_2, l_3,…, that get closer and closer to some value L.
Here is an example: 0.9, 0.99, 0.999, 0.9999, .… The limit of this sequence is
1. None of the entries is equal to 1, but they get closer and closer to that value. To
indicate this we write

In words, L is the limit of
li as i goes to infinity.

We can apply the same idea to functions.
Suppose we have a function, f(t), and we want to describe how it
varies as t gets closer and closer to some value, say a. If
f(t) gets arbitrarily close to L as t tends to
a, then we say that the limit of f(t) as t
approaches a is the number L. Symbolically,

Let f(t) be a function of
the variable t. As t varies, so will f(t).
Differential calculus deals with the rate of change of such functions. The idea is to
start with f(t) at some instant, and then to change the time by a
little bit and see how much f(t) changes. The rate of change is
defined as the ratio of the change in f to the change in t. We denote
the change in a quantity with the uppercase greek letter delta, . Let the
change in t be called t.
(This is not × t, this is a change in t.) Over the
interval t,
f changes from f(t) to f(t + t).
The change in f, denoted
f, is then given by

To define the rate of change precisely at
time t, we must let t
shrink to zero. Of course, when we do that f
also shrinks to zero, but if we divide f
by t,
the ratio will tend to a limit. That limit is the derivative of
f(t) with respect to t,

A rigorous mathematician might frown on the
idea that is the ratio of two differentials,
but you will rarely make a mistake this way.

Let’s calculate a few derivatives. Begin
with functions defined by powers of t. In particular, let’s illustrate
the method by calculating the derivative of f(t)= t^2. We apply Eq. (1) and begin by defining f(t
+ t):

We can calculate (t + t)2
by direct multiplication or we can use the binomial theorem. Either way,

We now subtract f(t):

The next step is to divide by t:

Now it’s easy to take the limit t
0. The first
term does not depend on t and survives, but the second term tends to zero and just disappears. This is
something to keep in mind: Terms of higher order in t can be ignored when you calculate derivatives. Thus

So the derivative of t^2 is

Next let us consider a general power,
f(t)= tn. To calculate its
derivative, we have to calculate f(t + t)=(t + t)n.
Here, high school algebra comes in handy: The result is given by the binomial theorem. Given two numbers, a and
b, we would like to calculate (a + b)n. The binomial theorem gives

How long does the expression go on? If
n is an integer, it eventually terminates after n + 1 terms. But
the binomial theorem is more general than that; in fact, n can be any real or
complex number. If n is not an integer, however, the expression never
terminates; it is an infinite series. Happily, for our purposes, only the first two
terms are important.

To calculate (t + t)n, all we have to do is plug in a
= t and b = t to get

All the terms represented by the dots shrink
to zero in the limit, so we ignore them.

Now subtract f(t) (or
tn),

Now divide by t,

and let t
0. The derivative is then

One important point is that this relation
holds even if n is not an integer; n can be any real or complex
number.

Here are some special cases of derivatives:
If n = 0, then f(t) is just the number 1. The derivative is
zero—this is the case for any function that does’t change. If n = 1, then
f(t) = t and the derivative is 1—this is always true when
you take the derivative of something with respect to itself. Here are some derivatives
of powers

For future reference, here are some other
derivatives:

One comment about the third formula in Eq.
(2), = et. The meaning of et is pretty clear
if t is an integer. For example, e^3 =
e × e × e. Its meaning for non-integers is not obvious.
Basically, et is defined by the property that its
derivative is equal to itself. So the third formula is really a definition.

There are a few useful rules to remember
about derivatives. You can prove them all if you want a challenging exercise. The first
is the fact that the derivative of a constant is always 0. This makes sense; the
derivative is the rate of change, and a constant never changes, so

The derivative of a constant times a
function is the constant times the derivative of the function:

Suppose we have two functions,
f(t) and g(t). Their sum is also a function and
its derivative is given by

This is called the sum rule.

Their product of two functions is another
function, and its derivative is

Not surprisingly, this is called the
product rule.

Next, suppose that g(t) is
a function of t, and f(g) is a function of g. That
makes f an implicit function of t. If you want to know what
f is for some t, you first compute g(t). Then,
knowing g, you compute f(g). It’s easy to calculate the
t-derivative of f:

This is called the chain rule. This
would obviously be true if the derivatives were really ratios; in that case, the d
g’s would cancel in the numerator and denominator. In fact, this is one
of those cases where the naive answer is correct. The important thing to remember about
using the chain rule is that you invent an intermediate function,
g(t), to simplify f(t) making it
f(g). For example, if

and we need to find , then the t^3 inside the
logarithm might be a problem. Therefore, we invent the intermediate function g
= t^3, so we have f(g)= ln
g. We can then apply the chain rule.

We can use our differentiation formulas to
note that = and = 3t^2, so

We can substitute g = t^3, to get

That is how to use the chain rule.

Using these rules, you can calculate a lot
of derivatives. That’s basically all there is to differential calculus.

Exercise 1: Calculate the derivatives of each of these functions.

Exercise 2: The derivative of a
derivative is called the second derivative and is written . Take the second derivative
of each of the functions listed above.

Exercise 3: Use the chain rule to find the derivatives of each of the
following functions.

Exercise 4: Prove the sum rule
(fairly easy), the product rule (easy if you know the trick), and the chain rule
(fairly easy).

Exercise 5: Prove each of the formulas in Eq.s (2). Hint: Look up
trigonometric identities and limit properties in a reference
book.

PARTICLE MOTION

The concept of a point particle is an
idealization. No object is so small that it is a point—not even an electron. But in many
situations we can ignore the extended structure of objects and treat them as points. For
example, the planet Earth is obviously not a point, but in calculating its orbit around
the Sun, we can ignore the size of Earth to a high degree of accuracy.

The position of a particle is specified by
giving a value for each of the three spatial coordinates, and the motion of the particle
is defined by its position at every time. Mathematically, we can specify a position by
giving the three spatial coordinates as functions of t: x(t),
y(t), z(t).

The position can also be thought of as a
vector (t) whose components are x, y,
z at time t. The path of the particle—its trajectory—is
specified by (t). The job of classical mechanics is to figure out (t) from some
initial condition and some dynamical law.

Next to its position, the most important
thing about a particle is its velocity. Velocity is also a vector. To define it we need
some calculus. Here is how we do it:

Consider the displacement of the particle
between time t and a little bit later at time t +
t.
During that time interval the particle moves from x(t), y(t),
z(t) to x(t +
t), y(t +
t), z(t +
t), or, in vector notation, from
(t) to
(t +
t).
The displacement is defined as

or

The displacement is the small distance that the
particle moves in the small time t. To get the velocity, we divide the displacement by t
and take the limit as t shrinks to zero. For example,

This—of course—is the definition of the
derivative of x with respect to t.

Placing a dot over a quantity is standard
shorthand for taking the time derivative. This convention can be used to denote the time
derivative of anything, not just the position of a particle. For example, if T
stands for the temperature of a tub of hot water, then Ṫ will represent
the rate of change of the temperature with time. It will be used over and over, so get
familiar with it.

It gets tiresome to keep writing x,
y, z, so we will often condense the notation. The three
coordinates x, y, z are collectively denoted by xi and the velocity components by vi:

where i takes the values x,
y, z, or, in vector notation

The velocity vector has a magnitude ,

this represents how fast the particle is
moving, without regard to the direction. The magnitude is called speed.

Acceleration is the quantity that tells you
how the velocity is changing. If an object is moving with a constant velocity vector, it
experiences no acceleration. A constant velocity vector implies not only a constant
speed but also a constant direction. You feel acceleration only when your velocity
vector changes, either in magnitude or direction. In fact, acceleration is the time
derivative of velocity:

or, in vector notation,

Because vi
is the time derivative of xi and ai is the time derivative of vi, it follows that acceleration is the second time-derivative of
xi,

where the double-dot notation means the second
time-derivative.

EXAMPLES OF MOTION

Suppose a particle starts to move at time
t = 0 according to the equations

The particle evidently has no motion in the
x and y directions but moves along the z axis. The
constants z(0) and v(0) represent the initial values of the position
and velocity along the z direction at t = 0. We also consider
g to be a constant.

Let’s calculate the velocity by
differentiating with respect to time.

The x and y components of
velocity are zero at all times. The z component of velocity starts out at
t = 0 being equal to v(0). In other words, v(0) is the
initial condition for velocity.

As time progresses, the −gt
term becomes nonzero. Eventually, it will overtake the initial value of the velocity,
and the particle will be found moving along in the negative z direction.

Now let’s calculate the acceleration
by differentiating with respect to time again.

The acceleration along the z axis is
constant and negative. If the z axis were to represent altitude, the particle
would accelerate downward in just the way a falling object would.

Next let’s consider an oscillating
particle that moves back and forth along the x axis. Because there is no motion
in the other two directions, we will ignore them. A simple oscillatory motion uses
trigonometric functions:

where the lowercase greek letter omega, , is a constant. The larger , the more rapid the oscillation. This kind of motion is called
simple harmonic motion (see Figure
1).

Figure 1: Simple harmonic motion.

Let’s compute the velocity and
acceleration. To do so, we need to differentiate x(t) with respect to
time. Here is the result of the first time-derivative:

We have the sine of a product. We can relabel
this product as b = t:

Using the chain rule,

or

or

We get the acceleration by similar means:

Notice some interesting things. Whenever the
position x is at its maximum or minimum, the velocity is zero. The opposite is
also true: When the position is at x = 0, then velocity is either a maximum or
a minimum. We say that position and velocity are 90° out of phase. You can see this in
Figure 2, representing x(t), and Figure 3, representing
v(t).

Figure 2: Representing position.

Figure 3: Representing velocity.

The position and acceleration are also related,
both being proportional to sin t.
But notice the minus sign in the acceleration. That minus sign says that
whenever x is positive (negative), the acceleration is negative (positive). In
other words, wherever the particle is, it is being accelerated back toward the origin.
In technical terms, the position and acceleration are 180° out of phase.

Exercise 6: How long does it take for the oscillating particle to go through
one full cycle of motion?

Next, let’s consider a particle moving
with uniform circular motion about the origin. This means that it is moving in a circle at a constant speed. For this purpose, we can ignore the
z axis and think of the motion in the x, y plane. To
describe it we must have two functions, x(t) and
y(t). To be specific we will choose the particle to move in the
counterclockwise direction. Let the radius of the orbit be R.

It is helpful to visualize the motion by
projecting it onto the two axes. As the particle revolves around the origin, x
oscillates between x = −R and x = R. The same
is true of the y coordinate. But the two coordinates are 90° out of phase; when
x is maximum y is zero, and vice versa.

The most general (counterclockwise) uniform
circular motion about the origin has the mathematical form

Here the parameter is called the
angular frequency. It is defined as the number of radians that the angle
advances in unit time. It also has to do with how long it takes to go one full
revolution, the period of motion—the same as we found in Exercise 6:

Now it is easy to calculate the components of
velocity and acceleration by differentiation:

This shows an interesting property of
circular motion that Newton used in analyzing the motion of the moon: The acceleration of a circular orbit is parallel to the position vector,
but it is oppositely directed. In other words, the acceleration vector points radially
inward toward the origin.

Exercise 7: Show that the
position and velocity vectors are orthogonal.

Exercise 8: Calculate the
velocity, speed, and acceleration for each of the following position vectors. If
you have graphing software, plot each position vector, each velocity vector, and
each acceleration vector.

Interlude 2: Integral Calculus

“George, I really like doing things
backward. Can we do differentiation backward?”

“Sure we can, Lenny. It’s called
integration.”

INTEGRAL CALCULUS

Differential calculus has to do with rates of
change. Integral calculus has to do with sums of many tiny incremental quantities.
It’s not immediately obvious that these have anything to do with each other, but
they do.

We begin with the graph of a function
f(t), as in Figure 1.

Figure 1: The behavior of
f(t).

The central problem of integral calculus is
to calculate the area under the curve defined by f(t). To make the
problem well defined, we consider the function between two values that we call
limits of integration, t = a and t = b.
The area we want to calculate is the area of the shaded region in Figure 2.

Figure 2: The limits of integration.

In order to do this, we break the region into
very thin rectangles and add their areas (see
Figure 3).

Figure 3: An illustration of integration.

Of course this involves an approximation, but
it becomes accurate if we let the width of the rectangles tend to zero. In order to
carry out this procedure, we first divide the interval between t = a
and t = b into a number, N, of subintervals—each of width
t.
Consider the rectangle located at a specific value of t. The width
is t
and the height is the local value of f(t). It follows that
the area of a single rectangle, A, is

Now we add up all the areas of the individual
rectangles to get an approximation to the area that we are seeking. The approximate
answer is

where the uppercase greek letter sigma, Σ,
indicates a sum of successive values defined by i. So, if N = 3,
then

Here ti is
the position of the ith rectangle along the t axis.

To get the exact answer, we take the limit
in which t
shrinks to zero and the number of rectangles increases to infinity. That
defines the definite integral of f(t) between the limits
t = a and t = b. We write this as

The integral sign, called summa, ∫,
replaces the summation sign, and—as in differential calculus—t
is replaced by d t. The function f(t) is called
the integrand.

Let’s make a notational change and
call one of the limits of integration T. In particular, replace b by
T and consider the integral

where we are going to think of T as
a variable instead of as a definite value of t. In this case, this integral
defines a function of T, which can take on any value of t. The
integral is a function of T because it has a definite value
for each value of T.

Thus a given function f(t)
defines a second function F(T). We could also let a vary, but
we won’t. The function F(T) is called the indefinite
integral of f(t). It is indefinite because instead of
integrating from a fixed value to a fixed value, we integrate to a variable. We usually
write such an integral without limits of integration,

The fundamental theorem of calculus
is one of the simplest and most beautiful results in mathematics. It asserts a deep
connection between integrals and derivatives. What it says is that if
F(T) = f(t) d t, then

To see this, consider a small incremental
change in T from T to T + t.
Then we have a new integral,

In other words, we have added one more
rectangle of width t
at t = T to the area shaded in Figure 3. In fact, the
difference F(T + t)
− F(T) is just the area of that extra rectangle,
which happens to be f(T) t, so

Dividing by t,

we obtain the fundamental theorem connecting
F and f, when we take the limit where t
0:

We can simplify the notation by ignoring the
difference between t and T,

In other words, the processes of integration
and differentiation are reciprocal: The derivative of the integral is the original
integrand.

Can we completely determine
F(t) knowing that its derivative is f(t)?
Almost, but not quite. The problem is that adding a constant to F(t)
does not change its derivative. Given f(t), its indefinite integral is
ambiguous, but only up to adding a constant.

To see how the fundamental theorem is used,
let’s work out some indefinite integrals. Let’s find the integral of a power
f(t) = tn. Consider,

It follows that

or

All we need to do is find a function
F whose derivative is tn, and that is
easy.

In the last chapter we found that for any
m,

If we substitute m = n + 1,
this becomes

or, dividing by n + 1,

Thus we find that tn is the derivative of . Substituting the relevant values, we get

The only thing missing is the ambiguous
constant that we can add to F. We should write

where c is a constant that has to be
determined by other means.

The ambiguous constant is closely related to
the ambiguity in choosing the other endpoint of integration that we earlier called
a. To see how a determines the ambiguous constant c,
let’s consider the integral

in the limit where the two limits of
integration come together—that is, T = a. In this case, the integral
has to be zero. You can use that fact to determine c.

In general, the fundamental theorem of
calculus is written

Another way to express the fundamental
theorem is by a single equation:

In other words, integrating a derivative
gives back the original function (up to the usual ambiguous constant). Integration and
differentiation undo each other.

Here are some integration formulas:

Exercise 1: Determine the indefinite integral of each of the following
expressions by reversing the process of differentiation and adding a
constant.

Exercise 2: Use the fundamental theorem of calculus to evaluate each
integral from Exercise 1 with limits of integration being t = 0 to
t = T.

Exercise 3: Treat the expressions from Exercise 1 as expressions for the
acceleration of a particle. Integrate them once, with respect to time, and
determine the velocities, and a second time to determine the trajectories.
Because we will use t as one of the limits of integration we will adopt
the dummy integration variable t'. Integrate them from t' = 0
to t' = t.

INTEGRATION BY PARTS

There are some tricks to doing integrals. One
trick is to look them up in a table of integrals. Another is to learn to use
Mathematica. But if you’re on your own and you don’t recognize
the integral, the oldest trick in the book is integration by parts. It’s
just the reverse of using the product rule for differentiation. Recall from Lecture 2
that to differentiate a function, which itself is a product of two functions, you use
the following rule:

Now let’s integrate both sides of this
equation between limits a
and b.

The left side of the equation is easy. The
integral of a derivative (the derivative of f g) is just the function itself.
The left side is

which we often write in the form

Now let’s subtract one of the two
integrals on the right side and shift it to the left side.

Suppose we have some integral that we
don’t recognize, but we notice that the integrand happens to be a product of a
function f(x) and the derivative of another function
g(x). In other words, after some inspection, we see that the
integral has the form of the right side of Eq. (4), but we don’t know how to do
it. Sometimes we are lucky and recognize the integral on the left side of the
equation.

Let’s do an example. Suppose the
integral that we want to do is

That’s not in our list of integrals. But
notice that

so the integral is

Equation (4) tells us that this integral is
equal to

or just

Now it’s easy. The integral sin x d x is on our list:
it’s just cos x. I’ll leave the rest to you.

Exercise 4: Finish evaluating

x cos x d x.

You might wonder how often this trick works.
The answer is quite often, but certainly not always. Good luck.

Lecture 3: Dynamics

Lenny: “What makes things move,
George?

George: “Forces do, Lenny.”

Lenny: “What makes things stop moving,
George?”

George: “Forces do, Lenny.”

ARISTOTLE’S LAW OF MOTION

Aristotle lived in a world dominated by
friction. To make anything move—a heavy cart with wooden wheels, for example—you had to
push it, you had to apply a force to it. The harder you pushed it, the faster
it moved; but if you stopped pushing, the cart very quickly came to rest. Aristotle came
to some wrong conclusions because he didn’t understand that friction is a force.
But still, it’s worth exploring his ideas in modern language. If he had known
calculus, Aristotle might have proposed the following law of motion:

The velocity of any object is proportional to the total applied force.

Had he known how to write vector equations,
his law would have looked like this:

is of course the applied force, and
the response (according to Aristotle) would be the velocity vector, . The factor
m relating the two is some characteristic quantity describing the
resistance of the body to being moved; for a given force, the bigger the m of
the object, the smaller its velocity. With a little reflection, the
old philosopher might have identified m with the mass of the object. It would
have been obvious that heavier things are harder to move than lighter things, so somehow
the mass of the object has to be in the equation.

One suspects that Aristotle never went ice
skating, or he would have known that it is just as hard to stop a body as to get it
moving. Aristotle’s law is just plain wrong, but it is nevertheless worth studying
as an example of how equations of motion can determine the future of a system. From now
on, let’s call the body a particle.

Consider one-dimensional motion of a
particle along the x axis under the influence of a given force. What I mean by
a given force is simply that we know what the force is at any time. We can call it
F(t) (note that vector notation would be a bit redundant in one
dimension). Using the fact that the velocity is the time derivative of position,
x, we find that Aristotle’s equation takes the form

Before solving the equation, let’s see
how it compares to the deterministic laws of Chapter 1. One obvious difference is that
Aristotle’s equation is not stroboscopic—that is neither t nor x
is discrete. They do not change in sudden stroboscopic steps; they change continuously.
Nevertheless, we can see the similarity if we assume that time is broken up into
intervals of size t and replace the derivative by
Doing so gives

In other words, wherever the particle happens to
be at time t, at the next instant its position will have shifted by a definite
amount. For example, if the force is constant and positive, then in each incremental
step the particle moves forward by an amount .
This law is obviously deterministic. Knowing that the particle was at a point
x(0) at time t = 0 (or x_0), one can
easily predict where it will be in the future. So by the criteria of Chapter 1,
Aristotle did not commit any crime.

Let’s go back to the exact equation of
motion:

Equations for unknown functions that involve
derivatives are called differential equations. This one is a
first-order differential equation because it contains only first
derivatives. Equations like this are easy to solve. The trick is to integrate both sides
of the equation:

The left side of the equation is the integral
of a derivative. That’s where the fundamental theorem of calculus comes in handy.
The left side is just x(t) + c.

The right side, on the other hand, is the
integral of some specified function and, apart from a constant, is also determined. For
example, if F is constant, then the right side is

Note that we included an additive constant.
Putting an arbitrary constant on both sides of the equation is redundant. In this case,
the equation of motion is satisfied by

How do you fix the constant c? The
answer is by the initial condition. For example, if we knew that the particle started at
x = 1, at time t = 3 we would plug these values in, obtaining

and solve for c:

Exercise 1: Given a force that
varies with time according to F = 2t^2, and with the initial condition at time zero, x(0) = , use Aristotle’s law to find
x(t) at all times.

Aristotle’s equations of motion are
deterministic, but are they reversible? In Lecture 1, I explained that
reversible means that if all the arrows were reversed, the resulting new
law of motion would also be deterministic. The analogous procedure to reversing the
arrows when time is continuous is very simple. Everywhere you see time in the equations,
replace it with minus time. That will have the effect of interchanging the future and
the past. Changing t to −t also includes
changing the sign of small differences in time. In other words, every t must be replaced with −t. In fact, you can do it right at the level of the differentials dt.
Reversing the arrows means changing the differential dt to
−dt. Let’s go back to Aristotle’s equation

and change the sign of time. The result
is

The left-hand side of the equation is the
force, but the force evaluated at time −t, not at time t.
However, if F(t) is a known function, then so is
F(−t). In the reversed problem, the force is also a known
function of reversed time.

On the right-hand side of the equation
we’ve replaced dt with −dt, thereby changing the sign of
the whole expression. In fact, one can shift the minus sign to the left-hand side of the
equation:

The implication is simple: The reversed
equation of motion is exactly like the original, but with a different rule for the force
as a function of time. The conclusion is clear: If Aristotle’s equations of motion
are deterministic into the future, they are also deterministic into the past. The
problem with Aristotle’s equations is not that they are inconsistent; they are
just the wrong equations.

It is interesting that Aristotle’s
equations do have an application—not as fundamental laws, but as approximations.
Frictional forces do exist, and in many cases they are so important that
Aristotle’s intuition—things stop if you stop pushing—is almost correct.
Frictional forces are not fundamental. They are a consequence of a body interacting with
a huge number of other tiny bodies—atoms and molecules—that are too small and too
numerous to keep track of. So we average over all the hidden degrees of freedom. The
result is frictional forces. When frictional forces are very strong such as in a stone
moving through mud—then Aristotle’s equation is a very good approximation, but
with a qualification. It’s not the mass that determines the proportionality
between force and velocity. It’s the so-called viscous drag coefficient. But that
may be more than you want to know.

MASS, ACCELERATION, AND FORCE

Aristotle’s mistake was to think that a
net “applied” force is needed to keep an object moving. The right idea is
that one force—the applied force—is needed to overcome another force—the force of
friction. An isolated object moving in free space, with no forces acting on it, requires
nothing to keep it moving. In fact, it needs a force to stop it. This is the law of
inertia. What forces do is change the state of motion of a body. If the body is
initially at rest, it takes a force to start it moving. If it’s moving, it takes a
force to stop it. If it is moving in a particular direction, it takes a force to change
the direction of motion. All of these examples involve a change in the velocity of an
object, and therefore an acceleration.

From experience we know that some objects have
more inertia than others; it requires a larger force to change their velocities. Obvious
examples of objects possessing large and small inertia are locomotives and Ping-Pong
balls, respectively. The quantitative measure of an object’s inertia is its
mass.

Newton’s law of motion involves three
quantities: acceleration, mass, and force. Acceleration we studied in Lecture 2. By
monitoring the position of an object as it moves, a clever observer—with a bit of
mathematics—can determine its acceleration. Mass is a new concept that is actually
defined in terms of force and acceleration. But so far we haven’t defined force.
It sounds like we are in a logical circle in which force is defined by the ability to
change the motion of a given mass, and mass is defined by the resistance to that change.
To break that circle, let’s take a closer look at how force is defined and
measured in practice.

There are very sophisticated devices that
can measure force to great accuracy, but it will suit our purposes best to imagine a
very old-fashioned device, namely, a spring balance. It consists of a spring and a ruler
to measure how much the spring is stretched from its natural equilibrium length (see Figure 1).

Figure 1: A spring balance.

The spring has two hooks, one to attach to
the massive body whose mass is being measured, and one to pull on. In fact, while you
are at it, make several such identical devices.

Let’s define a unit of force by pulling
on one hook, while holding the other hook fixed to some object A, until the
pointer registers one “tick” on the ruler. Thus we are applying a unit of
force to A.

To define two units of force, we could pull
just hard enough to stretch the spring to two ticks. But this assumes that the spring
behaves the same way between one tick and two ticks of stretching. This will lead us
back to a vicious circle of reasoning that we don’t want to get into. Instead, we
define two units of force by attaching two spring balances to A and pulling
both of them with a single unit of force (see
Figure 2).

In other words, we pull both hooks so that
each pointer records a single tick. Three units of force would be defined by using three
springs, and so on.

When we do this experiment in free space, we
discover the interesting fact that object A accelerates along the direction in
which we pull the hook. More exactly, the acceleration is proportional to the
force—twice as big for two units of force, three times as big for three units, and so
on.

Figure 2: Twice the force.

Let us do something to change the inertia of
A. In particular, we will double the inertia by hooking together two
identical versions of object A (see Figure
3).

Figure 3: Twice the mass.

What we find is that when we apply a single unit
of force (by pulling the whole thing with a single spring stretched to one tick) the
acceleration is only half what it was originally. The inertia (mass) is now twice as big
as before.

The experiment can obviously be generalized;
hook up three masses, and the acceleration is only a third as big, and so on.

We can do many more experiments in which we
hook any number of springs to any number of A’s. The observations are
summarized by a single formula, Newton’s second law of motion, which tells us that
force equals mass times acceleration,

This equation can also be written in the
form

In other words, force equals mass
times the rate of change of velocity: no force—no change in velocity.

Note that these equations are vector
equations. Both force and acceleration are vectors because they not only have magnitude
but also direction.

AN INTERLUDE ON UNITS

A mathematician might be content to say that
the length of a line segment is 3. But a physicist or engineer—or even an ordinary
person—would want to know, “Three what?” Three inches, three centimeters, or
three light years?

Similarly, it conveys no information to say
that the mass of an object is 7 or 12. To give the numbers meaning, we must indicate
what units we are using. Let’s begin with length.

Somewhere in Paris rests the defining
platinum meter stick. It is kept in a sealed container at a fixed temperature and away
from other conditions that might affect its length.[1] From here on, we will adopt that
meter stick as our unit of length.

Thus we write

[x] = [length] = meters.

Despite its appearance, this is not an
equation in the usual sense. The way to read it is x has units of length and is
measured in meters.

Similarly, t has units of time and
is measured in seconds. The definition of a second could be given by the amount of time
it takes a certain pendulum to make a single swing:

[t] = [time] = seconds.

The units meters and seconds are abbreviated
as m and s, respectively.

Once we have units for length and time, we can
construct units for velocity and acceleration. To compute the velocity of an object, we
divide a distance by a time. The result has units of length per time, or—in our
units—meters per second.

Similarly, acceleration is the rate of change
of velocity, and its units are velocity per unit time, or length per unit time per unit
time:

The unit of mass that we will use is the
kilogram; it is defined as the mass of a certain lump of platinum, that is also kept
somewhere in France. Thus

Now let’s consider the unit of force.
One might define it in terms of some particular spring made of a specific metal,
stretched a distance of 0.01 meter, or something like that. But in fact, we have no need
for a new unit of force. We already have one—namely the force that it takes to
accelerate one kilogram by one meter per second per second. Even better is to use
Newton’s law F = ma. Evidently, force has units of mass times
acceleration,

There is a name for this unit of force. One
kilogram meter per second squared is called a Newton, abbreviated N. Newton, himself,
being English, probably favored the British unit, namely the pound. There are about 4.4
N to a pound.

SOME SIMPLE EXAMPLES OF SOLVING
NEWTON’S EQUATIONS

The simplest of all examples is a particle
with no forces acting on it. The equation of motion is Eq. (2), but with the force set
to zero:

or, using the dot notation for time
derivative,

We can drop the factor of mass and write the
equation in component form as

The solution is simple: The components of
velocity are constant and can just be set equal to their initial values,

The same goes for the other two components of
velocity. This, incidently, is often referred to as Newton’s first law of
motion:

Every object in a state of uniform motion tends to remain in that state of motion
unless an external force is applied to it.

Equations (1) and (2) are called
Newton’s second law of motion,

The relationship between an object’s mass m, its acceleration a, and the
applied force F is

But, as we have seen, the first law is simply
a special case of the second law when the force is zero.

Recalling that velocity is the derivative of
position, we can express Eq. (3) in the form

This is the simplest possible differential
equation, whose solution (for all components) is

or, in vector notation,

A more complicated motion results from the
application of a constant force. Let’s first carry it out for just the z
direction. Dividing by m, the equation of motion is

Exercise 2: Integrate this
equation. Hint: Use definite integrals.

From this result we deduce

or

This is probably the second simplest
differential equation. It is easy to solve:

Exercise 3: Show by
differentiation that this satisfies the equation of motion.

This simple case may be familiar. If
z represents the height above the surface of the Earth, and is replaced with the acceleration
due to gravity, = −g, then Eq. (4)
is the equation describing the motion of an object falling from height
z_0 with an initial velocity vz(0):

Let’s consider the case of the simple
harmonic oscillator. This system is best thought of as a particle that moves along the
x axis, subject to a force that pulls it toward the origin. The force law
is

The negative sign indicates that at whatever
the value of x, the force pulls it back toward x = 0. Thus, when
x is positive, the force is negative, and vice versa. The equation of
motion can be written in the form

or, by defining ,

Exercise 4: Show by differentiation that the general solution to Eq. (6) is
given in terms of two constants A and B by

Determine the initial position
and velocity at time t = 0 in terms of A and
B.

The harmonic oscillator is an enormously
important system that occurs in contexts ranging from the motion of a pendulum to the oscillations of the electric and magnetic fields in a light wave. It
is profitable to study it thoroughly.

Interlude 3: Partial Differentiation

“Look out there, Lenny. Ain’t
those hills and valleys pretty?”

“Yeah George. Can we get a place over
there when we get some money? Can we?”

George squinted. “Exactly where are
you looking Lenny?”

Lenny pointed. “Right over there
George. That local minimum.”

PARTIAL DERIVATIVES

The calculus of multivariable functions is a
straightforward generalization of single-variable calculus. Instead of a function of a
single variable t, consider a function of several variables. To illustrate,
let’s call the variables x, y, z, although these
don’t have to stand for the coordinates of ordinary space. Moreover, there can be
more or fewer then three. Let us also consider a function of these variables,
V(x, y, z). For every value of x,
y, z, there is a unique value of V(x,
y, z) that we assume varies smoothly as we vary the coordinates.

Multivariable differential calculus revolves
around the concept of partial derivatives. Suppose we are examining the
neighborhood of a point x, y, z, and we want to know the rate
at which V varies as we change x while keeping y and
z fixed. We can just imagine that y and z are fixed
parameters, so the only variable is x. The derivative of V is then
defined by

where
V is defined by

Note that in the definition of
V, only x has been shifted; y and z are kept
fixed.

The derivative defined by Eq. (1) and Eq.
(2) is called the partial derivative of V with respect to x
and is denoted

or, when we want to emphasize that y
and z are kept fixed,

By the same method we can construct the
partial derivative with respect to either of the other variables:

A shorthand notation for the partial
derivatives of V with respect to y is

Multiple derivatives are also possible. If
we think of as itself being a function of
x, y, z, then it can be differentiated. Thus we can
define the second-order partial derivative with respect to x:

Mixed partial derivatives also make sense. For
example, one can differentiate yV with respect to x:

It’s an interesting and important fact
that the mixed derivatives do not depend on the order in which the derivatives are
carried out. In other words,

Exercise 1: Compute all first and second partial derivatives—including mixed
derivatives—of the following functions.

STATIONARY POINTS AND MINIMIZING
FUNCTIONS

Let’s look at a function of y
that we will call F (see Figure
1).

Figure 1: Plot of the function
F(y).

Notice that there are places on the curve where
a shift in y in either direction produces only an upward shift in F.
These points are called local minima. In Figure 2 we have added dots to
indicate the local minima.

Figure 2: Local minima.

For each local minimum, when you go in either
direction along y, you begin to rise above the dot in F(y).
Each dot is at the bottom of a little depression. The global minimum is the
lowest possible place on the curve.

One condition for a local minimum is that
the derivative of the function with respect to the independent variable at that point is
zero. This is a necessary condition, but not a sufficient condition. This condition
defines any stationary point,

The second condition tests to see what the
character of the stationary point is by examining its second
derivative. If the second derivative is larger than 0, then all points nearby will be
above the stationary point, and we have a local minimum:

If the second derivative is less than 0, then
all points nearby will be below the stationary point, and we have a local
maximum:

See
Figure 3 for examples of local maxima.

Figure 3: Local maxima.

If the second derivative is equal to 0, then
the derivative changes from positive to negative at the stationary point, which we call
a point of inflection:

See
Figure 4 for an example of a point of inflection.

Figure 4: Point of inflection.

These are collectively the results of a
second-derivative test.

STATIONARY POINTS IN HIGHER
DIMENSIONS

Local maxima, local minima, and other
stationary points can happen for functions of more than one variable. Imagine a hilly
terrain. The altitude is a function that depends on the two coordinates—let’s say
latitude and longitude. Call it A(x, y). The tops of hills
and the bottoms of valleys are local maxima and minima of A(x,
y). But they are not the only places where the terrain is locally
horizontal. Saddle points occur between two hills. You can see some examples in Figure
5.

Figure 5: A function of several variables.

The very tops of hills are places where no
matter which way you move, you soon go down. Valley bottoms are the opposite; all directions lead up. But both are places where the ground is level.

There are other places where the ground is
level. Between two hills you can find places called saddles. Saddle points are
level, but along one axis the altitude quickly increases in either direction. Along
another perpendicular direction the altitude decreases. All of these are called
stationary points.

Let’s take a slice along the
x axis through our space so that the slice passes through a local minimum
of A, see Figure 6.

Figure 6: A slice along the x axis.

It’s apparent that at the minimum, the
derivative of A with respect to x vanishes, we write this:

On the other hand, the slice could have been
oriented along the y axis, and we would then conclude that

To have a minimum, or for that matter to have
any stationary point, both derivatives must vanish. If there were more directions of
space in which A could vary, then the condition for a stationary point is given
by:

for all xi.

There is a shorthand for summarizing these
equations. Recall that the change in a function when the point x is varied a
little bit is given by

The set of Equations (3) are equivalent to
the condition that

for any small variation of x.

Suppose we found such a point. How do we
tell whether it is a maximum, a minimum, or a saddle. The answer is a generalization of
the criterion for a single variable. We look at the second derivatives. But there are
several second derivatives. For the case of two dimensions, we have

and

the last two being the same.

These partial derivatives are often arranged
into a special matrix called the Hessian matrix.

Important quantities, called the determinant
and the trace, can be made out of such a matrix. The determinant is given by

and the trace is given by

Matrices, determinants, and traces may not
mean much to you beyond these definitions, but they will if you follow these lectures to
the next subject—quantum mechanics. For now, all you need is the definitions and the
following rules.

If the determinant and the trace of the Hessian is positive then the point is a
local minimum.

If the determinant is positive and the trace negative the point is a local
maximum.

If the determinant is negative, then irrespective of the trace, the point is a
saddle point.

However: One caveat, these rules
specifically apply to functions of two variables. Beyond that, the rules are more
complicated. None of this is obvious for now, but it still enables you
to test various functions and find their different stationary points. Let’s take
an example. Consider

Differentiating, we get

Take the point x = , y = Since cos = 0, both derivatives are
zero and the point is a stationary point.

Now, to find the type of stationary point,
compute the second derivatives. The second derivatives are

Since sin = 1 we see that both the determinant and the trace of the Hessian are positive. The point is therefore a minimum.

Exercise 2: Consider the points
,

,

. Are these points
stationary points of
the following functions? Is so, of what
type?.

Lecture 4: Systems of More Than One Particle

It’s a lazy, warm evening. Lenny and
George are lying in the grass looking up at the sky.

“Tell me about the stars George. Are
they particles?”

“Kind of, Lenny.”

“How come they don’t
move?”

“They do, Lenny. It’s just that
they’re very far away.”

“There’s an awful lot of them,
George. Do you think that guy Laplace could really figure them all out?”

SYSTEMS OF PARTICLES

If—as Laplace believed—natural systems are
composed of particles, then the laws of nature must be the dynamical laws of motion that
determine the motion of those systems of particles. Again, Laplace: “An intellect
which at a certain moment would know all forces … and all
positions .…” What is it that determines the force on a given particle? It is
the positions of all the other particles.

There are many types of forces—such as
friction, the drag force exerted by the wind, and the force exerted by the floor that
keeps you from falling to the basement—that are not fundamental. They originate from the
microscopic interactions between atoms and molecules.

The fundamental forces are those that act
between particles, like gravity and electric forces. These depend on a number of things:
Gravitational forces between particles are proportional to the product
of their masses, and electric forces are proportional to the product of their electric
charges. Charges and masses are considered to be intrinsic properties of a particle, and
specifying them is part of specifying the system itself.

Apart from the intrinsic properties, the
forces depend on the location of the particles. For example, the distance between
objects determines the electric and gravitational force that one particle exerts on
another. Suppose that the locations of all the particles are described by their
coordinates: x_1, y_1,
z_1 for the first particle, x_2, y_2, z_2 for the second particle, x_3,
y_3, z_3 for the
third particle, and so on up to the last, or the Nth, particle. Then the force
on any one particle is a function of its location as well as the location of all the
others. We can write this in the form

What this equation means is that the force on
the ith particle is a function of the positions of all the particles. The
symbol stands for the collective location
of every particle in the system. Another way of saying this is that the symbol
represents the set of all position vectors.

Once we know the force on any particle—for
example, particle number 1—we can write Newton’s equation of motion for that
particle:

where m_1
and 1 are the mass and acceleration of
particle 1. When we express the acceleration as the second derivative of the position,
the equation becomes

In fact, we can write such an equation for each
particle:

or, in condensed form,

We can also write these equations in
component form:

In this set of equations, (Fx)i, (Fy)i, and (Fz)i mean the
x, y, and z components of the force on the ith
particle, and the symbols {x}, {y}, and {z}
represent the sets of all the x coordinates, all the y coordinates,
and all the z coordinates of all the particles.

This last set of equations makes it clear
that there is an equation for each coordinate of every particle, which would tell
Laplace’s vast intellect how every particle moves if the initial conditions were
known. How many equations are there in all? The answer is three for each particle, so if
there are N particles the grand total is 3N equations.

THE SPACE OF STATES OF A SYSTEM OF
PARTICLES

The formal meaning of the state of a system
is, “Everything you need to know (with perfect accuracy) to predict its future,
given the dynamical law.” Recall from Lecture 1, that the space of states, or
state-space, is the collection of all possible states of the system. In the examples of
Lecture 1, the state-space was typically a discrete collection of possibilities: H or T
for the coin, 1 through 6 for the die, and so forth. In Aristotelian mechanics, assuming
that the forces on an object are known, the state is specified by simply knowing the
location of the object. In fact, from Aristotle’s law, the force determines the
velocity, and the velocity tells you where the particle will be at the next instant.

But Newton’s law is different from
Aristotle’s: It tells you the acceleration, not the velocity. This means that to
get started, you need to know not only where the particles are but also their
velocities. Knowing the velocity tells you where the particle will be at the next
instant, and knowing the acceleration tells you what the velocity will be.

All of this means that the state of a system
of particles consists of more than just their current locations; it also includes their
current velocities. For example, if the system is a single particle,
its state consists of six pieces of data: the three components of its position and the
three components of its velocity. We may express this by saying that the state is a
point in a six-dimensional space of states labeled by axes x, y,
z, vx, vy, vz.

Now let’s consider the motion of the
particle. At each instant of time, the state is specified by the values of the six
variables x(t), y(t), z(t),
vx(t), vy(t), vz(t). The
history of the particle can be pictured as a trajectory through the six-dimensional
state-space.

Next, consider the space of states of a
system of N particles. To specify the state of the system, we need to specify
the state of every particle. This obviously means that the space of states is
6N-dimensional: three position components and three velocity components for
each of the N particles. One may even say that the motion of the system is a
trajectory through a 6N-dimensional space.

But wait. If the state-space is
6N-dimensional, why is it that 3N components in Equations (1) are
enough to determine how the system evolves? Are we missing half the equations?
Let’s go back to a system of a single particle with specified forces and write
Newton’s equations, using the fact that acceleration is the rate of change of
velocity:

Since there is no expression for the velocity
here, let’s add to this another equation expressing the fact that velocity is the
rate of change of position:

When we include this second equation, we have a
total of six components that tell us how the six coordinates of the state-space change
with time. The same idea, applied to each individual particle, gives us 6N
equations governing the motion through the space of states:

Thus, in answer to the question posed above,
we were missing half the equations.

Wherever you happen to be in the
6N-dimensional space of states, Equations (2) tell you where you will be next.
They also tell you where you were an instant ago. Thus, Equations (2) are suitable
dynamical laws. We now have our 6N equations for the N particles.

MOMENTUM AND PHASE SPACE

If you are struck by a moving object, the
result depends not only on the velocity of the object but also on its mass. Obviously, a
Ping-Pong ball at 30 miles per hour (about 13 meters per second) will have much less of
a mechanical effect than a locomotive moving at the same speed. In fact, the effect is
proportional to the momentum of the object, which for now we shall define as
the product of the velocity and the mass. Since the velocity is a vector, so is the momentum, denoted by the letter p. Thus

or

Since velocity and momentum are so closely
linked, we can use momentum and position instead of velocity and position to label the
points of the state-space. When the state-space is described this way, it has a special
name—phase space. The phase space of a particle is a six-dimensional space
with coordinates xi and pi (see Figure 1).

Figure 1: A point in phase space.

Why didn’t we call this space
configuration space? Why the new term phase space? The reason is that the term
configuration space is used for something else, namely, the three-dimensional space of
positions: Just the ri’s. It might have been
called position space; then we could have said, “Position space plus momentum
space equals phase space.” In fact, we do say that, but we also use the term
configuration space interchangeably with position space. Therefore the slogan is

Configuration space plus momentum space equals phase space.

You may wonder why we go to the trouble of
replacing the intuitive concept of velocity with the more abstract concept of momentum in describing the state of a particle. The answer should
become clear as we develop the basic framework of classical mechanics in later chapters.
For now, let’s just reexpress Equations (2) in terms of momentum instead of
velocity. To do so, we first note that

is nothing but the time rate of change of
momentum—that is, , or
in the condensed dot notation,

The full set of equations becomes

This simple, elegant set of equations is
exactly what Laplace imagined the laws of nature to be: For each coordinate of phase
space we have a single equation to tell you how it changes over an infinitesimal
interval of time.

ACTION, REACTION, AND THE CONSERVATION
OF MOMENTUM

The principle of the conservation of momentum
is a profound consequence of abstract general principles of classical mechanics that we
have yet to formulate. But it can also be understood at an elementary level from
Newton’s third law of motion:

For every action there is an equal and opposite reaction.

The simplest way to think of the third law is
to suppose first that particles interact in pairs. Each particle j exerts a
force on each other particle i, and the total force on any particle is the sum
of the forces on it exerted by all the other particles. If we denote the force on
particle i due to particle j by the symbol ij, then the total force
acting on particle i is

The left side represents the total force on
particle i, and the right side is the sum of the forces acting on i
due to all the other particles.

Newton’s law of action and reaction is
about the force between pairs of particles, ij. What it says is simple:
The force due to one particle j on another particle i is equal and
opposite to the force due to particle i acting on particle j.
As an equation, the third law says that for every i and j,

Let’s rewrite the first of Equations
(3), plugging in Eq. (4):

In other words, the rate of change of the
momentum of any particle is the sum of the forces due to all the other particles. Now
let’s add up all these equations to see how the total momentum changes.

The left-hand side of this equation is the
sum of the rates of change of all the momenta (the plural of momentum). In
other words, it is the rate of change of the total momentum. The right-hand side of the
equation is zero. That’s because when you write it out, each pair of particles
contributes two terms: the force on i due to j, and the force on
j due to i. The law of action and reaction, Eq. (5), ensures that
these cancel. Thus we are left with an equation that we can write in the form

This equation is precisely the mathematical
expression of the “conservation” of momentum: The total momentum of an
isolated system never changes.

Let’s consider the
6N-dimensional space of p’s and x’s. At every
point the entire collection of momenta are specified, so it follows that every point in
the phase space is (partially) characterized by a value of the total momentum. We could
go through the phase space, labeling each point with its total momentum. Now imagine
starting the system of particles at some point. As time evolves, the phase point sweeps
out a path in phase space. Every point on that path is labeled with the same value of
total momentum; the point never jumps from one value to another. This is entirely
similar to the idea of a conservation law that we explained in Lecture 1.

Lecture 5: Energy

“Old timer, what are you looking for
under the locomotive?”

Lenny loved the big steam locomotives, so
now and then, on their days off, George took him down to the train yard. Today, they
found a confused old man who looked as if he had lost something.

“Where’s the horse that pulls
this thing?” the old timer asked George.

“We’ll, it don’t need no
horse. Here, I’ll show you how it works. You see this place over here,” he
said, pointing. “That’s the fire box where they burn the coal to get out the
chemical energy. Then this, right next to it, is the boiler where the heat boils the
water to make steam. The steam pressure does work against the piston in this here box.
Then, the piston pushes against these rods, and they make the wheels turn.” The
old timer grinned, shook George’s hand, and took his leave.

Lenny had been standing aside while George
explained the locomotive. Now, with a look of sheer admiration, he came over to George
and said, “George, I loved the way you explained things to that guy. And I
understood all of it. The fire box, the boiler, the piston. Just one thing I
didn’t get.”

“What’s that, Lenny?”

“Well, I was just wondering.
Where’s the horse?”

FORCE AND POTENTIAL ENERGY

One often learns that there are many forms of
energy (kinetic, potential, heat, chemical, nuclear, … ) and that the sum total of
all of them is conserved. But when reduced to the motion of
particles, classical physics really has only two forms of energy: kinetic and potential.
The best way to derive the conservation of energy is to jump right into the formal
mathematical principles and then step back and see what we have.

The basic principle—call it the
potential energy principle—asserts that all forces derive from a potential
energy function denoted V({x}). Recall that {x} represents
the entire set of 3N coordinates—the configuration space—of all particles in
the system. To illustrate the principle, let’s begin with the simplest case of a
single particle moving along the x axis under the influence of a force
F(x). According to the potential energy principle, the force on
the particle is related to the derivative of the potential energy,
V(x):

In the one-dimensional case, the potential
energy principle is really just a definition of V(x). In fact, the
potential energy can be reconstructed from the force by integrating Eq. (1):

We can think of Eq. (1) in the following way:
The force is always directed in a way that pushes the particle toward lower potential
energy (note the minus sign). Moreover, the steeper V(x), the stronger
the force. The slogan that captures the idea is Force pushes you down the
hill.

Potential energy by itself is not conserved.
As the particle moves, V(x) varies. What is conserved is the sum of
potential energy and kinetic energy. Roughly speaking, as the particle rolls down the hill (in other words, as it moves toward lower potential
energy), it picks up speed. As it rolls up the hill, it loses speed. Something is
conserved.

Kinetic energy is defined in terms of the
velocity v and mass m of the particle. It is denoted by
T:

The total energy E of the particle
is the sum of the kinetic and potential energies:

As the particle rolls along the x
axis, the two types of energy individually vary, but always in such a way that the sum
is conserved. Let’s prove it by showing that the time derivative of E is
zero.

First let’s calculate the rate of
change of the kinetic energy. The mass is assumed constant, but v^2 can vary. The time derivative of v^2 is

Exercise 1: Prove Eq. (3).
Hint: Use the product rule for differentiation.

It follows that the time derivative of the
kinetic energy is

where the time derivative of the velocity has
been replaced by the acceleration.

Next, let’s calculate the rate of change
of potential energy. The key is to realize that V(x) changes with time
because x changes. Here is the formula that expresses this:

(It’s okay to think of derivatives as
ratios and to cancel the factors of d x in the numerator and denominator.)
Another way to write this equation is to replace with the velocity v:

(Be careful to not confuse V and
v.)

Now we can calculate the rate of change of
the total energy:

Note that since both terms contain a factor
of v, we can factor it out:

Now look at the expression in parentheses.
Use the fact that the derivative of V is related to the force. Recalling the
minus sign in Eq. (1), we see that the rate of change of E is given by

We now have what we need to prove energy
conservation: Newton’s law, F = m a, is exactly the condition
that the factor in parentheses vanishes, which in turn tells us that the total energy is
constant.

One point before we go on to
many-dimensional motion. We have shown that energy is conserved, but why is it that
momentum is not conserved in this case? After all, in the previous chapter we showed
that for an isolated system of particles, Newton’s third law implies that total
momentum does not change. The answer is that we have left something out of the
system—namely, the object that exerts the force on the one-dimensional particle. For
example, if the problem has to do with a particle falling in a gravitational field, the
gravitational force is exerted by the Earth. When the particle falls, its momentum
changes, but that change is exactly compensated for by a tiny change in the motion of
the Earth.

MORE THAN ONE DIMENSION

It is a fact that the components of force are
derivatives of potential energy, but it is not a definition. This is so when there is
more than one x to worry about—because space has more than one dimension, or
because there is more than one particle, or both. It is quite possible to imagine force
laws that do not come from differentiating a potential energy function, but nature does
not make use of such nonconservative forces.

Let’s be a little more abstract than
we have so far been. Call the coordinates of configuration space xi (remember, configuration space is the same as position space). For
now, the subscript i will not refer to which particle we are
talking about or which direction of space. It runs over all these possibilities. In
other words, for a system of N particles there are 3N values of
i. Let’s forget where they come from; we are simply considering a
system of abstract coordinates labeled i.

Now let’s write the equations of
motion:

For each coordinate, there is a mass
mi and a component of force Fi. Each component of force can depend on all positions
{x}.

We have seen in the one-dimensional case
that the force is minus the derivative of the potential energy, as in Eq. (1). This was
a definition of V, not a special condition on the force. But when there is more
than one dimension, things get more complicated. It is generally not true that
if you have a set of functions Fi({x}), that
they can all be derived by differentiating a single function V({x}).
It would be a brand-new principle if we asserted that the components of force can be
described as (partial) derivatives of a single potential energy function.

Indeed this principle is not hypothetical.
It is a basic mathematical expression of one of the most important principles of
physics:

For any system there exists a
potential
V({x}) such that

What law of nature does Eq. (5) represent?
You may have already guessed that it is the conservation of energy. We’ll see that
shortly, but first let’s try to visualize what it means.

Picture a terrain with the function
V({x}) representing the height or altitude at
each point. First of all, the minus sign in Eq. (5) means that the force points in the
downhill direction. It also says that the force is greater along directions where the
slope is steeper. For example, on a contour map, there is no force pushing along the
contour lines. The force vector points perpendicular to the contours.

Now let’s come back and derive energy
conservation. To do that, we plug Eq. (5) into the equations of motion (4):

The next step is to multiply each of the
separate equations in Eq. (6) by the corresponding velocity i and sum them all,

Now we have to manipulate both sides of the
equation in the same way that we did in the one-dimensional example. We define the
kinetic energy to be the sum of all the kinetic energies for each coordinate:

Here is what the two sides of Eq (7) give.
First the left-hand side:

Now the right-hand side:

Thus we can rewrite Eq. (7) as

Precisely as in the one-dimensional case, Eq.
(8) says that the time derivative of the total energy is zero—energy is conserved.

To picture what is going on, imagine that
the terrain has a frictionless ball rolling on it. Whenever the ball rolls toward a
lower altitude it picks up speed, and whenever it rolls uphill it loses speed. The
calculation tells us this happens in a special way that conserves the sum of the kinetic
and potential energies.

You might wonder why the forces of nature
are always gradients (derivatives) of a single function. In the next chapter we will
reformulate classical mechanics using the principle of least action. In this
formulation, it is “built in” from the very beginning that there is a
potential energy function. But then why the principle of least action? Ultimately, the
answer can be traced to the laws of quantum mechanics and to the origin of forces in
field theory—subjects that, for the moment, are still out of range for us. So, why
quantum field theory? At some point we have to give up and say that’s just the way
it is. Or, not give up and push on.

Exercise 2: Consider a particle in two dimensions, x and
y. The particle has mass m, equal in both directions. The
potential energy is . Work out the equations of motion. Show that there are
circular orbits and that all orbits have the same period. Prove explicitly that
the total energy is conserved.

Exercise 3: Rework Exercise 2 for the potential . Are there circular orbits?
If so, do they all have the same period? Is the total energy conserved?

Before moving on to the principle of least
action, I want to list a few of the different kinds of energy that we talk about in
physics, and review how they fit into the picture. Let’s consider

- mechanical energy

- heat

- chemical energy

- atomic/nuclear energy

- electrostatic energy

- magnetic energy

- radiation energy

Some, but not all, of these distinctions are
a bit old-fashioned. Mechanical energy usually refers to the kinetic and
potential energy of large visible objects such as planets or weights being hoisted by a
crane. It often refers to gravitational potential energy.

The heat contained in a gas or other
collection of molecules is also kinetic and potential energy. The only difference is
that it involves the large and chaotic motion of so many particles
that we don’t even try to follow it in detail. Chemical energy is also a special
case: The energy stored in chemical bonds is a combination of the potential energy and
kinetic energy of the constituent particles that make up the molecules. It’s
harder to understand because quantum mechanics has to replace classical mechanics, but
nonetheless, the energy is the potential and kinetic energy of particles. The same goes
for atomic and nuclear energy.

Electrostatic energy is just
another word for the potential energy associated with the forces of attraction and
repulsion between electrically charged particles. In fact, apart from gravitational
energy, it is the primary form of potential energy in the ordinary, classical world. It
is the potential energy between charged particles in atoms and molecules.

Magnetic energy is tricky, but the force
between the poles of magnets is a form of potential energy. The tricky part comes when
we think about the forces between magnets and charged particles. Magnetic forces on
charged particles are a new kind of beast called velocity-dependent forces. We will come
back to this later in the book.

Finally, there is the energy stored in
electromagnetic radiation. It can take the form of heat from the sun, or the energy
stored in radio waves, laser light, or other forms of radiation. In some very general
sense, it is a combination of kinetic and potential energy, but it is not the energy of
particles (not until we get to quantum field theory, anyway) but, of fields. So we will
set electromagnetic energy aside until a later book.

Lecture 6: The Principle of Least Action

Lenny was frustrated—not a good sign
considering his size and strength—and his head hurt. “George, I can’t
remember all this stuff! Forces, masses, Newton’s equations, momentum, energy. You
told me that I didn’t need to memorize stuff to do physics. Can’t you make
it just one thing to remember?”

“Okay, Lenny. Calm down. I’ll
make it simple. All you have to remember is that the action is always
stationary.”

THE TRANSITION TO ADVANCED
MECHANICS

The principle of least action—really the
principle of stationary action—is the most compact form of the classical laws of
physics. This simple rule (it can be written in a single line) summarizes everything!
Not only the principles of classical mechanics, but electromagnetism, general
relativity, quantum mechanics, everything known about chemistry—right down to the
ultimate known constituents of matter, elementary particles.

Let’s begin with a general observation
about the basic problem of classical mechanics, namely this problem is to determine the
trajectories (or orbits) of systems from their equations of motion. We usually express
the problem by postulating three things: the masses of the particles, a set of forces
F({x}) (or, even better, a formula for the potential energy), and
an initial condition. The system begins with some values of the coordinates and
velocities and then moves, according to Newton’s second law, under the influence
of the given forces. If there are a total of N coordinates, (x_1, x_2, …, xN), then the initial conditions
consist of specifying the 2N positions and velocities. For example, at an
initial time t_0, we can specify the positions
{x} and velocities {} and then solve the equations to find
out what the positions and velocities will be at the later time t_1. In the process, we will usually determine the whole
trajectory between t_0 and t_1 (see Figure 1).

Figure 1: A trajectory from time t_0 to time t_1.

But we can formulate the problem of
classical mechanics in another way that also involves specifying 2N items of
information. Instead of providing the initial positions and velocities, we provide the
initial and final positions. Here is the way to think about it: Suppose that an
outfielder wants to throw a baseball (from x_0 at time
t_0) and he wants it to arrive at second base
(x_1) after exactly 1.5 seconds (t_1). How does the ball have to move in between? Part of the
problem in this case will be to determine what the initial velocity of the ball has to
be. The initial velocity is not part of the input data in this way of posing the
question; it is part of the solution.

Let’s draw a space-time picture to
illustrate the point (see Figure 2). The
horizontal axis shows the position of a particle (or the baseball), and the vertical
axis denotes the time. The beginning and end of the trajectory are a pair of points on
the space-time diagram, and the trajectory itself is a curve
connecting the points.

Figure 2: A trajectory of the baseball.

The two ways of posing the problem of motion
are analogous to two ways of formulating the problem of fixing a straight line in space.
One thing we could ask is to construct a straight line from the origin that begins in
some particular direction. That’s like asking for the trajectory given the initial
position and velocity. On the other hand, we could ask to construct a straight line that
connects two particular points. That is like finding the trajectory that begins at one
position and arrives at another position after a specified time. In this form, the
problem is similar to asking how we have to aim a line from some initial point so that
it passes through another point. The answer: Find the shortest path between the points.
In the problems of classical mechanics, the answer is to find the path of stationary
action.

ACTION AND THE LAGRANGIAN

Formulating the action principle involves
exactly the same parameters as formulating Newton’s equations. You have to know
the masses of the particles, and you have to know the potential
energy. The action for a trajectory is an integral from the start of the trajectory at
t_0 to the end of the trajectory at t_1. I’ll just tell you what the integral is—no
motivation—and then we’ll explore the consequences of minimizing it.[1] We’ll
end up with Newton’s equations. Once we see how that works, any further motivation
will be unnecessary. If it’s equivalent to Newton’s equations, what more
motivation do we need?

Before being general, let’s illustrate
the idea for a single particle moving on a line. The position of the particle at time
t is x(t), and its velocity is (t).
The kinetic and potential energies are

respectively. The action of a trajectory is
written

You might think that there is a typo in Eq.
(1). The energy is the sum of T and V, but the integral involves the
difference. Why the difference and not the sum? You can try the derivation with
T + V, but you’ll get the wrong answer. The quantity
T − V is called the Lagrangian of
the system, and it’s denoted by the symbol L. The things you need to know
to specify L are the mass of the particle (for the kinetic energy) and the
potential V(x). It is, of course, no accident that these are the same
things you need to know to write Newton’s equation of motion.

Think of the Lagrangian as a function of the
position x and the velocity . It’s a function of position
because the potential energy depends on x, and it’s a function of
velocity because the kinetic energy depends on . So we write

We can rewrite the action as the integral of
the Lagrangian:

The principle of stationary action is really
very remarkable. It almost seems that the particle must have supernatural powers to feel
out all the possible trajectories and pick the one that makes the action stationary.
Let’s pause to consider what we are doing and where we are going.

The process of minimizing the action is a
generalization of minimizing a function. The action is not an ordinary function of a few
variables. It depends on an infinity of variables: All the coordinates at every
instant of time. Imagine replacing the continuous trajectory by a
“stroboscopic” trajectory consisting of a million points. Each point is
specified by a coordinate x, but the whole trajectory is specified only when a
million x’s are specified. The action is a function of the whole
trajectory, so it is a function of a million variables. Minimizing the
action involves a million equations.

Time is not really stroboscopic, and a real
trajectory is a function of a continuously infinite number of variables. To put it
another way, the trajectory is specified by a function x(t), and the
action is a function of a function. A function of a function—a quantity that depends on
an entire function—is called a functional. Minimizing a functional is the
subject of a branch of mathematics called the calculus of variations.

Nevertheless, despite the differences from
ordinary functions, the condition for a stationary action strongly resembles the
condition for a stationary point of a function. In fact, it has exactly the same form as
Eq. (4) in Interlude 3, namely

Now, however, the variations are not just
small shifts of a few coordinates, but all the possible small variations of a whole
trajectory.

Later in this lecture we will work out the
equations for minimizing the action. They are called the Euler-Lagrange equations. For
the case of a single degree of freedom, there is one equation at each point along the
trajectory. In fact, the equations become differential equations
that tell the system how to move from one instant to the next. Thus the particle does
not have to have supernatural powers to test out all future trajectories—at least no
more so than it needs to follow Newton’s equations of motion.

We will derive the Euler-Lagrange equations
later in this lecture. To do you a flavor, I will write down their
form. If you are the independent type, you can
try to plug in the Lagrangian and see if you can derive Newton’s equation of
motion. Here, then, is the Euler-Lagrange equation for a single degree of freedom:

DERIVATION OF THE EULER-LAGRANGE
EQUATION

Let’s see if we can derive the
Euler-Lagrange equation for a single degree of freedom. Start by replacing continuous
time with stroboscopic time. The instants can be labeled by integers n. The
time between neighboring instants is very small. Call it t. The action is an integral, but, as always, an integral is the limit of a
sum. In this case, we are going to think of the sum as being over the intervals between
successive instants.

Here are the replacements that we do when we
approximate the integral by the sum:

The first replacement is just the usual
approximation of replacing the integral by a discrete sum of terms, each weighted with
the small time interval t. The second is also familiar. It replaces the velocity with the difference of
neighboring positions divided by the small time interval.

The last replacement is a bit more subtle.
Since we are going to think of the sum as being over the small
intervals between neighboring instants, we need an expression for the position halfway
between the instants. That’s easy. Just replace x(t) with the average
position between neighboring instants:

Notice that everywhere occurred in
the Lagrangian I replaced it with , and
everywhere that x occurred I substituted

The total action is found by adding up all
the incremental contributions:

I have very explicitly taken the action apart
into its components, almost like writing a computer program to evaluate it.

Now suppose we want to minimize the action
by varying any one of the xn and setting the result
equal to zero. Let’s pick one of them, say x_8.
(Any other one would have been just as good.) This sounds very complicated, but notice
that x_8 appears only in two of the terms in Eq. (3).
The two terms that contain x_8 are

Now all we have to do is differentiate with
respect to x_8. Notice that x_8 appears in two ways in each term. It appears through the velocity dependence and through the x dependence. The
derivative of A with respect to x_8 is

The symbol |n=8 is an instruction to evaluate the function at the discrete time
n = 8.

To minimize the action with respect to
variations of x_8, we set d A/d x equal to
zero. But before we do, let’s see what happens to d A/d x in the limit
when t tends to zero. Start with the first term,

This has the form of the difference between a
quantity evaluated at two neighboring times, n = 8 and n = 9, divided
by the small separation between them. This obviously tends to a derivative, namely

The second term,

also has a simple limit. It is half the sum
of evaluated at neighboring times. As
the separation between the points shrinks to zero, we just get .

The condition that = 0 becomes the Euler-Lagrange equation,

Exercise 1: Show that Eq. (4) is just another form of Newton’s
equation of motion F = m a.

The derivation is essentially the same for
many degrees of freedom. There is an Euler-Lagrange equation for each coordinate
xi:

What this derivation shows is that there is
no magic involved in the ability of the particle to feel out the entire path before
deciding which way to go. At each stage along the trajectory, the particle has only to
minimize the action between a point in time and a neighboring point in time. The
principle of least action just becomes a differential equation at each instant that
determines the immediate future.

MORE PARTICLES AND MORE DIMENSIONS

All together, let there be N
coordinates that we call xi. The motion of the system
is described by a trajectory, or orbit, through an N-dimensional
space. For an even better description we can add time, thinking of the orbit as a path
through N + 1 dimensions. The starting point of the
trajectory is the set of points xi(t_0), and the endpoint is another set of points xi(t_1). The orbit
through the (N + 1)-dimensional space is described by giving all the
coordinates as functions of time xi(t).

The principle of least action for more
degrees of freedom is essentially no different than the case with only a single degree
of freedom. The Lagrangian is the kinetic energy minus the potential energy:

The action is also exactly as before, the
integral of the Lagrangian,

and the principle of least (stationary)
action is that the trajectory minimizes this action.

When there are many variables, we can vary
the trajectory in many ways, for example we can vary x_1(t), or x_2(t), and so
on. It’s like minimizing a function of many variables: There is an equation for
each variable. The same is true for the Euler-Lagrange equations: There is one for each
variable xi. Each one has the same general form as Eq.
(4)

Exercise 2: Show that Eq. (6) is just another form of Newton’s
equation of motion Fi = mi
i.

WHAT’S GOOD ABOUT LEAST ACTION?

There are two primary reasons for using the
principle of least action. First, it packages everything about a system in a very
concise way. All the parameters (such as the masses and forces), and all the equations
of motion are packaged in a single function—the Lagrangian. Once you know the
Lagrangian, the only thing left to specify is the initial conditions. That’s
really an advance: a single function summarizing the behavior of any number of degrees
of freedom. In future volumes, we will find that whole theories—Maxwell’s theory
of electrodynamics, Einstein’s theory of gravity, the Standard Model of elementary
particles—are each described by a Lagrangian.

The second reason for using the principle of
least action is the practical advantage of the Lagrangian formulation of mechanics.
We’ll illustrate it by an example. Suppose we want to write Newton’s
equations in some other coordinates, or in some frame of reference that is moving or
accelerating.

Take the case of a particle in one dimension
that, from the point of view of someone standing at rest, satisfies Newton’s laws.
The physicist at rest—call him Lenny—uses the coordinate x to locate the
object.

A second physicist—George—is moving relative
to Lenny, and he wants to know how to describe the object relative to his own
coordinates. First of all, what does it mean to talk about George’s coordinates?
Because George moves relative to Lenny, the origin of his coordinate frame moves
relative to Lenny’s origin. This is easily described by changing coordinates from
Lenny’s x to George’s coordinate system X.

Here is how we do it. At any time
t, Lenny locates George’s origin at x + f(t),
where f is some function that describes how George moves relative to Lenny. An
event (at time t) that Lenny assigns a coordinate x,
George assigns coordinate X where

When Lenny sees a particle moving on the
trajectory x(t), George sees the same particle moving on the
trajectory X = x(t) − f(t). If George
does not want to keep asking Lenny what the trajectory is, then he wants his own laws of
motion to describe the object from his coordinates. The easiest way to do this is to
transform the equations of motion from one coordinate system to another is
to use the principle of least action, or the Euler-Lagrange equations.

According to Lenny, the action of a
trajectory is

But we can also write the action in terms of
George’s coordinates. All we have to do is to express in terms of
:

So we plug this into Eq. (7) to get

The potential energy V(X)
simply means the potential energy that Lenny would use, evaluated at the object’s
location, but expressed in George’s coordinates—same point, different label. And
now we know the Lagrangian in the X frame of reference,

where we can expand the square:

What does George do with Eq. (8)? He writes
the Euler-Lagrange equation. Here is what he gets:

or, with a small rearrangement,

The result is nothing surprising. George sees
an extra, “fictitious” force on the object equal to −m. What is interesting is the procedure: Instead of transforming
the equation of motion, we worked directly with the Lagrangian.

Let’s do another example. This time
George is on a rotating carousel. Lenny’s coordinates are x and
y. George’s coordinate frame is X and Y, and it
rotates with the carousel. Here is the connection between the two frames:

Both observers see a particle moving in the
plane. Let’s assume that Lenny observes that the particle moves with no forces acting on it. He describes the motion using the action principle with
Lagrangian

What we want to do is express the action in
George’s rotating frame and then use the Euler-Lagrange equations to figure out
the equations of motion. Since we already know the action in Lenny’s frame, all we
need to do is express the velocity in his frame in terms of George’s variables.
Just differentiate Equations (9) with respect to time:

After a little bit of algebra using sin^2 + cos^2 = 1, here is what we get for
2 + 2

Now all we have to do is plug Eq. (11) into
Lenny’s Lagrangian, Eq. (10), to get George’s Lagrangian. It’s the
same Lagrangian except expressed in George’s coordinates:

Let’s examine the various terms. The
first term, , is familiar—it’s just what
George would call the kinetic energy. Notice that if angular velocity were zero,
that’s all there would be. The next term, m
2(X^2 + Y^2), is something new due to the rotation. What it looks like to
George is a potential energy,

which can easily be seen to create an outward
force proportional to the distance from the center of rotation:

This is nothing but the centrifugal
force.

The last term in Eq. (12) is a little less
familiar. It is called the Coriolis force. To see how it all works, we can work
out the Euler-Lagrange equations. Here is what we get:

This looks exactly like Newton’s
equations with centrifugal and Coriolis forces. Notice that there is something new in
the form of the force law. The components of the Coriolis force,

depend not only on the position of the particle
but also on its velocity. The Coriolis force is a velocity-dependent force.

Exercise 3: Use the Euler-Lagrange equations to derive the equations of
motion from this Lagrangian.

The main point of this exercise was not so
much to derive the centrifugal and Coriolis forces as to show you how to transform a
mechanics problem from one coordinate system to another by simply rewriting the
Lagrangian in the new coordinates. This is, by far, the easiest way to do the
transformation—a lot easier than trying to transform Newton’s equations
directly.

Another example, which we will leave to you,
is to transform George’s equations to polar coordinates:

Exercise 4: Work out
George’s Lagrangian and Euler-Lagrange equations in polar
coordinates.

GENERALIZED COORDINATES AND MOMENTA

There is really nothing very general about
Cartesian coordinates. There are many coordinate systems that we can choose to represent
any mechanical system. For example, suppose we want to study the motion of an object
moving on a spherical surface—say, the Earth’s surface. In this
case, Cartesian coordinates are not of much use: The natural coordinates are two angles,
longitude and latitude. Even more general would be an object rolling on a general curved
surface like a hilly terrain. In such a case, there may not be any special set of
coordinates. That’s why it is important to set up the equations of classical
mechanics in a general way that applies to any coordinate system.

Consider an abstract problem in which a
system is specified by a general set of coordinates. We usually reserve the notations
xi for Cartesian coordinates. The notation for
a general system of coordinates is called qi. The
qi could be Cartesian coordinates, or polar
coordinates, or anything else we can think of.

We also need to specify the velocities,
which in the abstract situation means the time derivatives of the qi generalized coordinates. An initial condition
consists of the set of generalized coordinates and velocities (qi, i)
.

In a general coordinate system, the
equations of motion may be complicated, but the action principle always applies. All
systems of classical physics—even waves and fields—are described by a Lagrangian.
Sometimes the Lagrangian is calculated from some previous knowledge. An example is
calculating George’s Lagrangian, knowing Lenny’s. Sometimes the Lagrangian
is guessed on the basis of some theoretical prejudices or principles, and sometimes we
deduce it from experiments. But however we get it, the Lagrangian neatly summarizes all
the equations of motion in a simple package.

Why are all systems described by action
principles and Lagrangians? It’s not easy to say, but the reason is very closely
related to the quantum origins of classical physics. It is also closely related to the
conservation of energy. For now, we are going to it take as given that
all known systems of classical physics can be described in terms of the action
principle.

The Lagrangian is always a function of the
coordinates and the velocities, L = L(qi, i),
and the action principle is always

This means that the equations are of the
Euler-Lagrange form. Here, then, is the most general form of classical equations of
motion. There is an equation for each qi

That’s it, all of classical physics in
a nutshell! If you know what the qi’s are, and
if you know the Lagrangian, then you have it all.

Let’s look a little closer at the two
sides of Eq. (13). Begin with the expression .
Suppose for a moment that the qi’s are the
ordinary Cartesian coordinates of a particle and L is the usual kinetic energy
minus potential energy. In this case, the Lagrangian would contain and then would just be m—in
other words, the component of momentum in the x direction. We then call the generalized momentum conjugate to qi or just the conjugate
momentum to qi.

The concept of conjugate momentum transcends
the simple example in which momentum comes out to be mass times velocity. Depending on
the Lagrangian, the conjugate momentum may not be anything you recognize, but it is
always defined by

The notation for generalized momentum is
pi.

With that definition, the Euler-Lagrange
equations are

Let’s do a couple of examples starting
with a particle in polar coordinates. In this case the qi’s are the radius, r, and the angle, . We can use the result from Exercise 4 to get the Lagrangian:

The generalized momentum conjugate to
r (the r momentum) is

and the corresponding equation of motion
is

Using = m and canceling
m from both sides, we can write this equation in the form

The equation of motion for the angle is especially interesting. First consider the conjugate momentum to :

This quantity should be familiar. It is the
angular momentum of the particle. Angular momentum and p are exactly the same thing.

Now consider the equation of motion for . Since
itself does not appear in the Lagrangian, there is no right-hand side, and we have

In other words, angular momentum is
conserved. Another way to write Eq. (14) is

We can see that r^2
is a constant. That’s why angular velocity increases as
a particle gets closer to the origin.

Exercise 5: Use these results
to predict the motion of a pendulum of length l.

CYCLIC COORDINATES

As we’ve just seen, it sometimes
happens that some coordinate does not appear in the Lagrangian though its velocity does.
Such coordinates are called cyclic (I don’t know why.)

What we do know is that the
Lagrangian does not change when you shift the value of a cyclic coordinate. Whenever a
coordinate is cyclic, its conjugate momentum is conserved. Angular
momentum is one example. Another is ordinary (linear) momentum. Take the case of a
single particle with Lagrangian

None of the coordinates appear in the
Lagrangian, so they are all cyclic. Again, there is nothing particularly cyclic about
them—it’s just a word. Therefore, all of the components of momentum are conserved.
This would not be true if there were a potential energy that depended on the
coordinates.

Let’s take another case: two particles
moving on a line with a potential energy that depends on the distance between them. For
simplicity I’ll take the masses to be equal, but there is nothing special about
that case. Let’s call the positions of the particles x_1 and x_2. The Lagrangian is

Now the Lagrangian depends on both
x_1 and x_2, and
neither is cyclic. Neither momentum is conserved.

But that’s missing an important point.
Let’s make a change of coordinates. Define x+
and x− as

We can easily rewrite the Lagrangian. The
kinetic energy is

Exercise 6: Explain how we
derived this.

The important point is that the potential
energy depends only on x−. The Lagrangian is
then

In other words, there was a hidden cyclic
coordinate, namely x+. This means that the conjugate
momentum to x+ (call it p+) is conserved. It is easy to see that p+
is nothing but the total momentum,

The real point that we will come to in the
next lecture is not so much about cyclic coordinates but about symmetries.

Lecture 7: Symmetries and Conservation Laws

Lenny had trouble reading maps. It always
seemed like whichever way he was facing must be north. He wondered why he had more
trouble with NSEW than he did with up and down. He could almost always get up and down
right.

PRELIMINARIES

The relationship between symmetries and
conservation laws is one of the big main themes of modern physics. We’re going to
begin by giving some examples of conservation laws for some simple systems. At first,
the fact that certain quantities are conserved will seem somewhat accidental—hardly
things of deep principle. Our real goal, however, is not to identify accidental
conserved quantities, but to identify a set of principles connecting them to something
deeper.

We’ll begin with the system that we
studied at the end of Lecture 6 in Eq. (16), but let’s free it from the
interpretation of particles moving on a line. It could be any system with two
coordinates: particles, fields, rotating rigid bodies, or whatever. To emphasize the
broader context, let’s call the coordinates q instead of x and
write a Lagrangian of similar—but not quite identical—form:

The potential is a function of one combination
of variables, namely (q_1 − q_2). Let’s denote the derivative of the potential
V by V'. Here are the equations of motion:

Exercise 1: Derive Equations
(2) and explain the sign difference.

Now add the two equations together to see
that the sum p_1 + p_2
is conserved.

Next, let’s do something slightly more
complicated. Instead of the potential being a function of (q_1 − q_2), let’s have it be a
function of a general linear combination of q_1 and
q_2. Call the combination (a q_1 − b q_2). The potential
then has the form

For this case the equations of motion are

It seems that we’ve lost the
conservation law; adding the two equations does not give the conservation of
p_1 + p_2.

But the conservation law has not been lost;
it just changed a little bit. By multiplying the first equation by b and the
second by a and then adding them, we can see that b p_1 + a p_2 is conserved.

Exercise 2: Explain this
conservation.

On the other hand, suppose the potential is
a function of some other, more general combination of the q’s, such as
q_1 + q_2^2. Then there is no conserved combination of the
p’s. So, then, what is the principle? What determines whether there are
conservation laws and what they are? The answer has been known for almost 100 years from
the work of the German mathematician Emmy Noether.

EXAMPLES OF SYMMETRIES

Let’s consider a change of coordinates
from qi to a new set qi'. Each qi' is a function of all of
the original q coordinates:

There are two ways to think about a change of
coordinates. The first way is called passive. You don’t do anything to
the system—just relabel the points of the configuration space.

For example, suppose that the x
axis is labeled with tick marks, x = …, −1, 0, 1, 2, … and
there is a particle at x = 1. Now suppose you are told to perform the
coordinate transformation

According to the passive way of thinking, the
transformation consists of erasing all the labels and replacing them with new ones. The
point formerly known as x = 0 is now called x' = 1. The point formerly
known as x = 1 is now called x' = 2, and so on. But
the particle is left where it was (if it was at x = 1, then the new labeling
puts it at x' = 2); only the label has changed.

In the second way of thinking about
coordinate transformations, which is called active, you don’t relabel the
points at all. The transformation x' = x + 1 is interpreted as an
instruction: Wherever the particle is, move it one unit to the right. In other words, it
is an instruction to actually move the system to a new point in the configuration
space.

In what follows, we will adopt the active
point of view. Whenever I make a change of coordinates, it means that the system is
actually displaced to the new point in the configuration space. In general, when we make
a transformation, the system actually changes. If, for example, we move an object, the
potential energy—and therefore the Lagrangian—may change.

Now I can explain what a symmetry means. A
symmetry is an active coordinate transformation that does not change the
value of the Lagrangian. In fact, no matter where the system is located in the
configuration space, such a transformation does not change the Lagrangian.

Let’s take the simplest example: a
single degree of freedom with Lagrangian

Suppose we make a change in the coordinate
q by shifting it an amount . In other words, any configuration
is replaced by another in which q has been shifted (see Figure 1).

Figure 1: Shifting the coordinate of a point,
q, by .

If the shift does not depend on time
(as we will assume), then the velocity does not change, and—most
important—neither does the Lagrangian. In other words, under the change

the change in the Lagrangian is
L = 0.

In Eq. (5) the quantity can be any
number. Later, when we consider transformations by infinitesimal steps, the symbol will be used
to represent infinitesimal quantities, but for now it doesn’t matter.

We could consider a more complicated
Lagrangian with a potential energy V(q). Unless the potential is a
constant independent of q, then the Lagrangian will change as q is
shifted. In that case there is no symmetry. The symmetry of moving a system in space by
adding a constant to the coordinates is called translation symmetry, and we
will spend a lot of time discussing it.

Now look at Equations (2). Suppose we shift
q_1 but not q_2.
In that case the Lagrangian will change because the potential energy changes. But if we
shift both coordinates by the same amount so that q_1
− q_2 does not change, then the value of the
Lagrangian is unchanged. We say that the Lagrangian is invariant under the
change

We say that the Lagrangian is symmetric with
respect to the transformation in Equations (6). Again this is a case of translation symmetry, but in this case, to have a symmetry we must translate both
particles so that the distance between them is unchanged.

For the more complicated case of Eq. (3),
where the potential depends on a q_1 + b q_2, the symmetry is less obvious. Here is the transformation:

Exercise 3: Show that the
combination a q_1 + b q_2, along with the Lagrangian, is invariant under
Equations (7).

If the potential is a function of a more
complicated combination, it is not always clear that there will be a symmetry. To
illustrate a more complex symmetry, let’s revert to Cartesian coordinates for a
particle moving on the x, y plane. Let’s say this particle is
under the influence of a potential energy that depends only on the distance from the
origin:

It’s pretty obvious that Eq. (8) has a
symmetry. Imagine rotating the configuration about the origin by an angle (see Figure 2).

Figure 2: Rotation by .

Since the potential is a function only of the
distance from the origin, it doesn’t change if the system is rotated through an
angle. Moreover, the kinetic energy is also unchanged by a rotation.
The question is how we express such a change. The answer is obvious: Just rotate
coordinates

where is any angle.

Now we come to an essential point about the
transformations of translation and rotation. You can do them in small
steps—infinitesimal steps. For example, instead of moving a particle from x to
x + 1, you can move it from x to x + . Now I am
using to
denote an infinitesimal. In fact, you can build up the original displacement x

x + 1 by many tiny steps of size . The same is true for rotations:
You can rotate through an infinitesimal angle and, by repeating the process,
eventually build up a finite rotation. Transformations like this are called
continuous: They depend on a continuous parameter (the angle of rotation),
and, moreover, you can make the parameter infinitesimal. This will prove to be a good
thing, because we can explore all the consequences of continuous symmetries by
restricting our attention to the infinitesimal case.

Since finite transformations can be
compounded out of infinitesimal ones, in studying symmetries it’s enough to
consider transformations with very small changes in the coordinates, the so-called
infinitesimal transformations. So let’s consider what happens to
Equations (9) when the angle is replaced by an infinitesimal
angle . To
first order in ,

(Recall that for small angles, sin = and cos = 1 − , so
the first-order shift in the cosine vanishes and the first-order
shift in sine is .)

Then the rotation represented by Equations
(9) simplifies to

You can also see that the velocity components
change. Just differentiate Equations (10) with respect to time:

Another way to express the effect of the
infinitesimal transformation is to concentrate on the changes in the coordinates and
write

Now it’s a simple calculus exercise to
show that the Lagrangian does not change to first order in .

Exercise 4: Show this to be
true.

One thing worth noting is that if the
potential is not a function of distance from the origin, then the Lagrangian is not
invariant with respect to the infinitesimal rotations. This very important point should
be checked by examining some explicit examples. A simple example is a
potential that depends only on x and not on y.

MORE GENERAL SYMMETRIES

Before we get to the connection between
symmetries and conservation laws, let’s generalize our notion of symmetry. Suppose
the coordinates of an abstract dynamical system are qi. The general idea of an infinitesimal transformation is that it is a small
shift of the coordinates, that may itself depend on the value of the coordinates. The
shift is parameterized by an infinitesimal parameter , and it has the
form

In other words, each coordinate shifts by an
amount proportional to , but the proportionality factor
depends on where you are in configuration space. In the example of Equations (6) the
value of f_1 and of f_2 are both 1. In the example of Equations (7) the f-functions are
f_1 = a and f_2 = −b. But in the more complicated example of the rotations
of Equations (12), the f’s are not constant:

If we want to know the change in
velocities—in order, for example, to compute the change in the Lagrangian—we need only
to differentiate Eq. (13). A little calculus exercise gives

For example, from Equations (12),

Now we can re-state the meaning of a symmetry
for the infinitesimal case. A continuous symmetry is an infinitesimal transformation of
the coordinates for which the change in the Lagrangian is zero. It is particularly easy
to check whether the Lagrangian is invariant under a continuous symmetry: All you have
to do is to check whether the first order variation of the Lagrangian is zero. If it is,
then you have a symmetry.

Now let’s see what the consequences of
a symmetry are.

THE CONSEQUENCES OF SYMMETRY

Let’s calculate how much
L(q, ) changes when we do a transformation
that shifts qi by the amount in Eq. (13) and, at the
same time, shifts i by the amount in Eq. (14).
All we have to do is compute the change due to varying the ’s and add it to the change due
to varying the q’s:

Now we do a bit of magic. Watch it carefully.
First, we remember that is the momentum conjugate to
qi, which we denote pi. Thus the first term in Eq. (16) is .
Hold on to that while we study the second term,
qi. To evaluate terms of this type, we assume the
system is evolving along a trajectory that satisfies the Euler-Lagrange equations

Combining the terms, here is what we get for
the variation of the Lagrangian:

The final piece of magic is to use the
product rule for derivatives:

Thus we get the result

What does all of this have to do with
symmetry and conservation? First of all, by definition, symmetry means that the
variation of the Lagrangian is zero. So if Eq. (13) is a symmetry, then
L = 0 and

But now we plug in the form of the symmetry
operation, Eq. (13), and get

That’s it: The conservation law is
proved. What Eq. (17) states is that a certain quantity,

does not change with time. In other words, it
is conserved. The argument is both abstract and powerful. It did not depend on the
details of the system, but only on the general idea of a symmetry. Now let’s turn
back to some particular examples in light of the general theory.

BACK TO EXAMPLES

Let’s apply Eq. (18) to the examples we
studied earlier. In the first example, Eq. (1), the variation of the coordinates in
Equations (12) defines both f_1 and f_2 to be exactly 1. Plugging f_1
= f_2 = 1 into Eq. (18) gives exactly what we found
earlier: (p_1 + p_2)
is conserved. But now we can say a far more general thing: For any system of
particles, if the Lagrangian is invariant under simultaneous translation of the
positions of all particles, then momentum is conserved. In fact, this can be
applied separately to each spatial component of momentum. If L is invariant
under translations along the x axis, then the total x component of
momentum is conserved. Thus we see that Newton’s third law—action equals
reaction—is the consequence of a deep fact about space: Nothing in the laws of
physics changes if everything is simultaneously shifted in space.

Next let’s look at the second example,
in which the variations of Equations (12) imply f_1 =
b, f_2 = −a. Again,
plugging this result into Eq. (18), we find that the conserved quantity is b
p_1 + a p_2.

The last example—rotation—is more
interesting. It involves a new conservation law that we haven’t met yet. From Eq.
(14) we obtain fx = y, fy = −x. This time the
conserved quantity involves both coordinates and momenta. It is called l, or
angular momentum. From Eq. (18) we get

l = y px − x py.

Again, as in the case of translations, there
is a deeper thing involved than just the angular momentum of a single particle:

For any system of particles, if the
Lagrangian is invariant under simultaneous rotation of the position of all
particles, about the origin, then angular momentum is conserved.

Exercise 5: Determine the equation of motion for a simple pendulum of length
l swinging through an arc in the x, y plane from
an initial angle of .

So far, our examples have been very trivial.
The Lagrangian formulation is beautiful, elegant, blah blah, but is it really good for
solving hard problems? Couldn’t you just use F = m a?

Try it. Here is an example: the double
pendulum. A pendulum swings in the x, y plane supported at the origin.
The rod of the pendulum is massless, and the bob (weight at the end) is M. To
make it simple, let the rod be 1 meter in length and let the bob be 1 kilogram in mass.
Next, take another identical pendulum, but suspend it from the bob of
the first pendulum, as shown in Figure 3. We can study two cases: with and without a
gravitational field.

Figure 3: The double pendulum.

Our goal will not be to solve the equations
of motion. That we can always do, even if we have to put them on a computer and do it
numerically. The goal is to find those equations. It’s a tricky problem if you try
to do it by F = M a. Among other things, you have to worry about the
forces transmitted through the rod. The Lagrangian method is much easier. There is a
more or less mechanical procedure for doing it. The steps are the following:

1. Choose some coordinates that uniquely specify
the configuration of the components. You can choose them however you like—just make
sure that you have just enough to determine the configuration—and keep them as
simple as possible.
In the double
pendulum example, you need two coordinates. I will choose the first one to be
the angle of the first pendulum from the vertical. Call it .
Next, I have a choice. Should I choose the second angle (the angle of the second
rod) also to be measured from the vertical, or should I measure it relative to
the angle of the first rod? The answer is that it does not matter. One choice
may make the equations a little simpler, but either will get you to the answer. I will choose the angle to be measured
relative to the first rod rather than to the vertical.

2. Work out the total kinetic energy. In this case it
is the kinetic energy of the two bobs.
The easiest way to do this is to refer temporarily
to Cartesian coordinates x, y. Let x_1, y_1 refer to the
first bob and x_2, y_2 to the second bob. Here are some relations among the angles , α and x, y: For bob 1,

and for bob 2,

Now, by differentiating with respect
to time, you can compute the Cartesian velocity components in terms of the
angles and their time derivatives.

Finally, work out the kinetic energy
for each bob and add them.
It should take a couple of minutes. Remember that we have chosen the masses and
rod lengths to be 1.

Here is the result: The kinetic
energy of the first bob is:

and the kinetic energy of the second
bob is

If there is no gravitational field,
then the kinetic energy is the Lagrangian:

If there is gravity, then we have to
calculate the gravitational potential energy. That’s easy: For each bob we
add its altitude times m g. This gives a potential energy

3. Work out the Euler-Lagrange equations for each
degree of freedom.

4. For later purposes, work out the conjugate
momenta for each coordinate, pi = .

Exercise 6: Work out the
Euler-Lagrange equations for

and .

There is more that you may want to do. In
particular, you may want to identify the conserved quantities. Energy is usually the
first one. The total energy is just T + V. But there may be more.
Finding symmetries is not always a mechanical procedure; you may have to do some pattern
recognition. In the double pendulum case without any gravity, there is another
conservation law. It follows from rotation symmetry. Without a gravitational field, if
you rotate the whole system about the origin, nothing changes. This implies conservation
of angular momentum, but to find the form of the angular momentum, you have to go
through the procedure that we derived. That involves knowing the conjugate momenta.

Exercise 7: Work out the form of the angular momentum for the double
pendulum, and prove that it is conserved when there is no gravitational
field.

Lecture 8: Hamiltonian Mechanics and Time-Translation
Invariance

Doc was sitting at the bar drinking his
usual—a beer milk shake—and reading the paper, when Lenny and George walked in.
“What are you reading about, Doc?”

Doc looked up at Lenny over his glasses.
“I see where this guy Einstein says, ‘Insanity is doing the same thing over
and over and expecting different results.’ What do you think about
that?”

Lenny thought for a minute. “You mean
like every time I eat here I order chili, and then I get a stomachache?”

Doc chuckled, “Yeah, that’s the
idea. I see you’re beginning to understand Einstein.”

TIME-TRANSLATION SYMMETRY

You may wonder what happened to energy
conservation and whether it fits the pattern relating symmetries to conservation laws.
Yes, it does, but in a slightly different way than the examples in Lecture 7. In all of
those examples the symmetry involved shifting the coordinates qi. For instance, a translation is a symmetry that simultaneously
shifts the Cartesian coordinates of all the particles in a system by the same amount.
The symmetry connected with energy conservation involves a shift of time.

Imagine an experiment involving a closed
system far from any perturbing influences. The experiment begins at time t_0
with a certain initial condition, proceeds for a definite period, and
results in some outcome. Next, the experiment is repeated in exactly the same way but at
a later time. The initial conditions are the same as before, and so is the duration of
the experiment; the only difference is the starting time, which is pushed forward to
t_0 + t. You might expect that the outcome will be exactly the same, and that the
shift t would make no difference. Whenever this is true, the system is said to be
invariant under time translation.

Time-translation invariance does not always
apply. For example, we live in an expanding universe. The effect of the expansion on
ordinary laboratory experiments is usually negligible, but it’s the principle that
counts. At some level of accuracy, an experiment that begins later will have a slightly
different outcome than one which begins earlier.

Here is a more down-to-earth example.
Suppose the system of interest is a charged particle moving in a magnetic field. If the
magnetic field is constant then the motion of the particle will be time-translation
invariant. But if the current that generates the field is being slowly increased, then
the same initial condition for the particle—but starting at different times—will result
in a different outcome. The description of the particle will not be time-translation
invariant.

How is time-translation symmetry, or the
lack of it, reflected in the Lagrangian formulation of mechanics? The answer is simple.
In those cases where there is such symmetry, the Lagrangian has no explicit dependence
on time. This is a subtle point: The value of the Lagrangian may vary with time, but
only because the coordinates and velocities vary. Explicit time dependence
means that the form of the Lagrangian depends on time. For example, take the
harmonic oscillator with Lagrangian

If m and k are
time-independent then this Lagrangian is time-translation invariant.

But one can easily imagine that the spring
constant k might, for some reason, change with time. For example, if the
experiment took place in a changing magnetic field, this could have a subtle effect on
the atoms of the spring, which in turn could cause k to vary. In that case, we
would have to write

This is what we mean by an explicit time
dependence. More generally, we can write

where the t dependence is due to the
time variation of all the parameters controlling the behavior of the system.

With this idea in hand, we can now give a
very succinct mathematical criterion for time-translation symmetry: A system is
time-translation invariant if there is no explicit time dependence in its
Lagrangian.

ENERGY CONSERVATION

Let’s consider how the actual value of
the Lagrangian, Eq. (1), changes as a system evolves. There are three sources of time
dependence of L. The first and second are due to the time dependence of the
coordinates q and the velocities . If that were all, we would
write

But if the Lagrangian has explicit time
dependence, then there is another term:

Let’s examine the various terms in Eq.
(2) using the Euler-Lagrange equations of motion. The first type of term, , can be written

The second type of term , takes the form

If we combine everything, we get

The first two terms can be simplified. We use
the identity

to get

Notice that even if there is no explicit time
dependence in L, the Lagrangian will nevertheless depend on time through the
first term . The upshot is that there is no
such thing as conservation of the Lagrangian.

Inspection of Eq. (3) reveals something
interesting. If we define a new quantity H by

then Eq. (3) has a very simple form:

The steps leading to Eq. (5) may seem a bit
complicated, but the result is very simple. The new quantity H varies with time
only if the Lagrangian has an explicit time dependence. An even more
interesting way to say it is if a system is time-translation invariant, then the
quantity H is conserved.

The quantity H is called the
Hamiltonian, and, as you might expect, it is important because (among other
reasons) it is the energy of a system. But it is more than important; it is the central
element in an entirely new formulation of mechanics called the Hamiltonian
formulation. But for now, let’s consider its meaning by returning to a
simple example, the motion of a particle in a potential. The Lagrangian is

and the canonical momentum is just the usual
momentum

Let’s plug Eq. (6) and Eq. (7) into Eq.
(4), the definition of H:

Notice what happens: Two terms proportional
to m^2 combine to give the usual
kinetic energy, and the potential term becomes + V(x). In other words,
H just becomes the usual total energy, kinetic plus potential.

This is the general pattern that you can
check for any number of particles. If the Lagrangian is kinetic energy minus potential
energy, then

There are systems for which the Lagrangian
has a more intricate form than just T − V. For some of those
cases, it is not possible to identify a clear separation into kinetic and potential
energy. Nonetheless, the rule for constructing the Hamiltonian is the same. The general
definition of energy for these systems is

Energy equals Hamiltonian.

Moreover, if there is no explicit time
dependence in the Lagrangian, then the energy H is conserved.

If, however, the Lagrangian is explicitly
time-dependent, then Eq. (5) implies that the Hamiltonian is not conserved. What
happened to the energy in that case? To understand what is going on, let’s
consider an example. Suppose that a charged particle, with unit electric charge, is
moving between the plates of a capacitor. The capacitor has a uniform electric field
due to the charges on the plates. (The reason we are using
for
electric field, instead of the more conventional E, is to avoid confusing it
with energy.) You don’t have to know anything about electricity. All you need to
know is that the capacitor creates a potential energy equal to
x. The Lagrangian is

As long as the field is constant, the energy
is conserved. But suppose the capacitor is being charged up so that is also ramping
up. Then the Lagrangian has an explicit time dependence:

Now the energy of the particle is not
conserved. Depending on the momentary location x of the particle, the energy
varies according to

Where did that energy come from? The answer
is that it came from the battery that was charging the capacitor. I won’t go into details, but the point is that when we defined the system to
consist of just the particle, we narrowed our focus to just a part of a bigger system
that includes the capacitor and the battery. These additional items are also made of
particles and therefore have energy.

Consider the entire experiment, including
the battery, capacitor, and particle. The experiment begins with an uncharged capacitor
and a particle at rest, somewhere between the plates. At some moment we close a circuit,
and current flows into the capacitor. The particle experiences a time-dependent field,
and, at the end of the experiment, the capacitor is charged and the particle is
moving.

What if we did the entire experiment an hour
later? The outcome, of course, would be the same. In other words, the entire closed
system is time-translation invariant, so the entire energy of all items is conserved. If
we treated the entire collection as a single system, it would be time-translation
invariant, and the total energy would be conserved.

Nevertheless, it is often useful to divide a
system into parts and to focus on one part. In that case, the energy of part of the
system will not be conserved if the other parts are varying with time.

PHASE SPACE AND HAMILTON’S
EQUATIONS

The Hamiltonian is important because (among
other reasons) it is the energy. But its significance is far deeper: It is the basis for
a complete revamping of classical mechanics, and it is even more important in quantum
mechanics.

In the Lagrangian—or action—formulation of
mechanics, the focus is on the trajectory of a system through the configuration space. The trajectory is described in terms of the coordinates
q(t). The equations are second-order differential equations, so it
is not enough to know the initial coordinates; we also have to know the initial
velocities.

In the Hamiltonian formulation, the focus is
on phase space. Phase space is the space of both the coordinates qi and the conjugate momenta pi. In
fact, the q’s and p’s are treated on the same footing, the
motion of a system being described by a trajectory through the phase space.
Mathematically, the description is through a set of functions qi(t), pi(t).
Notice that the number of dimensions of phase space is twice that of configuration
space.

What do we gain by doubling the number of
dimensions? The answer is that the equations of motion become first-order differential
equations. In less technical terms, this means that the future is laid out if we know
only the initial point in phase space.

The first step in constructing the
Hamiltonian formulation is to get rid of the ’s and replace them with the
p’s. The goal is to express the Hamiltonian as a function of
q’s and p’s. For particles in ordinary Cartesian
coordinates, the momenta and velocities are almost the same thing, differing only by a
factor of the mass. As usual, the particle on a line is a good illustration.

We start with the two equations

When we replace the velocity with p /
m the Hamiltonian becomes a function of p and x:

One last point before we write the equations of
motion in Hamiltonian form: The partial derivative of H with respect to
x is just , or
minus the force. Thus the equation of motion (F = m a) takes the form

We noted earlier that in the Hamiltonian
formulation, the coordinates and momenta are on the same footing. From that you might
guess that there is another equation similar to Eq. (9), with p and x
interchanged. That is almost true, but not quite. The correct equation is

with a plus sign instead of a minus sign.

To see why Eq. (10) is true, just
differentiate the expression for H with respect to p. From the second
of Equations (8) we get

which from the first equation is just .

So now we see a very simple symmetric
packaging of the equations. We have two equations of motion instead of one, but each is
of first-order:

These are Hamilton’s equations for a
particle on a line. Soon we will derive the general form for any system, but for now I
will tell you what it is. We start with a Hamiltonian that is a function of all the
q’s and p’s:

H = H(qi, pi).

We can use this to generalize Equations
(11),

So we see that for each direction in phase
space, there is a single first-order equation.

Let’s stop to consider how these
equations are related to the very first chapter of this book, in which we described how
deterministic laws of physics predict the future. What Equations (12) say is this:

If at any time you know the exact values of all the coordinates and momenta, and you
know the form of the Hamiltonian, Hamilton’s equations will tell you the
corresponding quantities an infinitesimal time later. By a process of successive
updating, you can determine a trajectory through phase space.

THE HARMONIC OSCILLATOR HAMILTONIAN

The harmonic oscillator is by far the most
important simple system in physics. It describes all sorts of oscillations in which some
degree of freedom is displaced and then oscillates about an equilibrium position. To see
why it is so important, let’s suppose a degree of freedom q has a
potential energy V(q) that has a minimum. The minimum describes a
stable equilibrium, and when the degree of freedom is displaced, it will tend to return
to the equilibrium position. Without any real loss of generality, we can locate the
minimum at q = 0. The generic function that has a minimum at this point can be
approximated by the quadratic function

where V(0) and c are
constants. The reason why there is no linear term proportional to q is that the
derivative must be zero at the minimum. We can
also drop the term V(0) since adding a constant to the potential energy has no
effect.

The form of Eq. (13) is not very general;
V could contain terms of all orders—for example, q^3 or q^4. But as long as the
system deviates from q = 0 by only a small amount, these higher-order terms
will be negligible compared to the quadratic term. This reasoning applies to all sorts
of systems: springs, pendulums, oscillating sound waves, electromagnetic waves, and on
and on.

I will write the Lagrangian in what may seem
like a special form involving a single constant called :

Exercise 1: Start with the Lagrangian and show that if you make the change in variables
q = (k m)1/4x, the
Lagrangian has the form of Eq. (14). What is the connection among k,
m, and ?

Exercise 2: Starting with Eq.
(14), calculate the Hamiltonian in terms of p and
q.

The Hamiltonian corresponding to Eq. (14) is
very simple:

It was in order to get H into such
a simple form that we changed variables from x to q in Exercise 1.

One of the hallmarks of the Hamiltonian
formulation is how symmetric it is between the q’s and
p’s. In the case of the harmonic oscillator, it is almost completely
symmetric. The only asymmetry is a minus sign in the first of Equations (12). For a
single degree of freedom, Hamilton’s equations are Equations (11). If we plug our
Hamiltonian, Eq. (15), into Equations (12), we get,

How do these two equations compare with
Lagrange’s equations that we would derive from Eq. (14)? First of all, there is
only one Lagrangian equation:

Second, this equation is second-order,
meaning that it involves second time-derivatives. By contrast, the Hamiltonian equations
are each first-order. This somehow means that two first-order equations are equivalent
to one second-order equation. We can see this by differentiating the second equation in
Equations (16) with respect to time,

and then using the first equation in
Equations (16). This enables us to replace with −
q, which gives us Eq. (17): The Euler-Lagrange equation of motion.

Is one formulation better than the other?
Did Lagrange have the final word or did Hamilton? You can decide for yourself, but wait
a while before you do. We still have a couple of courses on relativity and quantum
mechanics before the real meanings of the Lagrangian and Hamiltonian become completely
clear.

Let’s return to Equations (16). We
usually “think” in configuration space. The harmonic oscillator is a system
that moves back and forth along a single axis. But it is also an excellent starting
point for getting used to “thinking” in phase space. Phase space (for the
oscillator) is two-dimensional. It is easy to see that the
trajectories of the oscillator in phase space are concentric circles about the origin.
The argument is very simple. Go back to the expression for the Hamiltonian, Eq. (15).
The Hamiltonian, being the energy, is conserved. It follows that q^2 + p^2 is constant with time.
In other words, the distance from the origin of phase space is constant, and the phase
point moves on a circle of fixed radius. In fact Eq. (16) is the equation for a point
moving with constant angular velocity about the origin. Especially
interesting is the fact that the angular velocity in phase space is the same for all
orbits, independent of the energy of the oscillator. As the phase point circles the
origin, you can project the motion onto the horizontal q axis, as shown in
Figure 1. It moves back and forth in an oscillatory motion, exactly as expected.
However, the two-dimensional circular motion through phase space is a more comprehensive
description of the motion. By projecting onto the vertical p axis, we see that
the momentum also oscillates.

Figure 1: The harmonic oscillator in phase
space.

The harmonic oscillator is especially
simple. In general, the motion of a system through phase space is more complicated and
less symmetric. But the fact that the phase point stays on a contour of constant energy
is universal. Later we will discover more general properties of motion in phase
space.

DERIVATION OF HAMILTON’S
EQUATIONS

Let’s complete a piece of business that
we left unfinished: the general derivation of Hamilton’s equations. The Lagrangian
is some general function of the coordinates and velocities,

and the Hamiltonian is

The change in the Hamiltonian is

Now if we use the definition of
pi, namely pi = , we
see that the first and last terms exactly cancel, leaving

Let’s compare this with the general
rule for a small change in a function of several variables:

By matching the terms proportional to
qi and
pi, we arrive at

There is only one last step, and that is to
write Lagrange’s equations in the form

Inserting this in the second of Equations
(18) we get Hamilton’s equations,

Lecture 9: The Phase Space Fluid and the
Gibbs-Liouville Theorem

Lenny loved watching the river, especially
watching little bits of floating debris making their way downstream. He tried to guess
how they would move between the rocks or get caught in eddys. But the river as a
whole—the large-scale current, the volume of water, the shear, and the divergence and
convergence of the flow were beyond him.

THE PHASE SPACE FLUID

Focusing on a particular initial condition
and following it along its specific trajectory through phase space are very natural
things to do in classical mechanics. But there is also a bigger picture that emphasizes
the entire collection of trajectories. The bigger picture involves visualizing all
possible starting points and all possible trajectories. Instead of putting your pencil
down at a point in phase space and then following a single trajectory, try to do
something more ambitious. Imagine you had an infinite number of pencils and used them to
fill phase space uniformly with dots (by uniformly, I mean that the density of dots in
the q, p space is everywhere the same). Think of the dots as particles
that make up a fictitious phase-space-filling fluid.

Then let each dot move according to the
Hamiltonian equations of motion,

so that the fluid endlessly flows through the
phase space.

The harmonic oscillator is a good example to
start with. In Lecture 8 we saw that each dot moves in a circular orbit with uniform
angular velocity. (Remember, we are talking about phase space, not coordinate space. In
coordinate space, the oscillator moves back and forth in one dimension.) The whole fluid
moves in a rigid motion, uniformly circulating around the origin of phase space.

Now let’s return to the general case.
If the number of coordinates is N, then the phase space, and the fluid, are
2N-dimensional. The fluid flows, but in a very particular way. There are
features of the flow that are quite special. One of these special features is that if a
point starts with a given value of energy—a given value of H(q,
p)—then it remains with that value of energy. The surfaces of fixed energy
(for example, energy E) are defined by the equation

For each value of E we have a single
equation for 2N phase-space variables, thus defining a surface of dimension
2N − 1. In other words, there is a surface for each value of
E; as you scan over values of E, those surfaces fill up the phase
space. You can think of the phase space, along with the surfaces defined in Eq. (2) as a
contour map (see Figure 1), but, instead of
representing altitude, the contours denote the value of the energy. If
a point of the fluid is on a particular surface, it stays on that
surface forever. That’s energy conservation.

Figure 1: Contour plot of energy surfaces of a
harmonic oscillator in phase space.

For the harmonic oscillator, the phase space
is two-dimensional and the energy surfaces are circles:

For a general mechanical system, the energy
surfaces are far too complicated to visualize, but the principle is the same: The
energy surfaces fill the phase space like layers and the flow moves so that the
points stay on the surface that they begin on.

A QUICK REMINDER

We want to stop here and remind you of the
very first lecture, where we discussed coins, dice, and the simplest idea of a law of
motion. We described those laws by a set of arrows connecting dots that represented the
states of the system. We also explained that there are allowable laws and unallowable
laws, the allowable laws being reversible. What is it that
characterizes an allowable law? The answer is that every point must have exactly one
incoming arrow and one outgoing arrow. If at any point the number of incoming arrows
exceeds the number of outgoing arrows (such a situation is called a
convergence), then the law is irreversible. The same is true if the number of
outgoing arrows exceeds the number of incoming arrows (such a situation is called a
divergence). Either a convergence or divergence of the arrows violates
reversibility and is forbidden. So far we have not returned to that line of reasoning.
Now is the time.

FLOW AND DIVERGENCE

Let’s consider some simple examples of
fluid flow in ordinary space. Forget about phase space for the moment, and just consider
an ordinary fluid moving through regular three-dimensional space labeled by axes
x, y, z. The flow can be described by a velocity
field. The velocity field (x, y,
z) is defined by going to each point of space and specifying the velocity
vector at that point (see Figure 2).

Figure 2: Velocity field.

Or we may describe the velocity field to be the
components of the velocity: vx (x,
y, z), vy(x, y,
z), vz(x, y,
z). The velocity at a point might also depend on time, but let’s suppose
that it doesn’t. In that case the flow is called stationary.

Now let’s suppose the fluid is
incompressible. This means that a given amount of the fluid always occupies the same
volume. It also means that the density of the fluid—the number of molecules per unit
volume—is uniform and stays that way forever. By the way, the term incompressible also
means indecompressible. In other words, the fluid cannot be stretched out, or
decompressed. Consider a small cubic box defined by

Incompressibility implies that the number of
fluid points in every such box is constant. It also means that the net flow of fluid
into the box (per unit time) must be zero. (As many points flow in as flow out.)
Consider the number of molecules per unit time coming into the box across the face
x = x_0. It will be proportional to the
flow velocity across that face, vx(x_0).

If vx were
the same at x_0 and at x_0 + d x, then the flow into the box at x = x_0 would be the same as the flow out of the box at x =
x_0 + d x. However, if vx varies across the box, then the two flows will not
balance. Then the net flow into the box across the two faces will be proportional to

Exactly the same reasoning applies to the
faces at y_0 and y_0 +
d y, and also at z_0 and z_0 + d z. In fact, if you add it all up, the net flow of molecules into the box (inflow minus outflow) is
given by

The combination of derivatives in the
parentheses has a name: It is the divergence of the vector field (t) and is denoted by

The divergence is aptly named; it represents
a spreading out of the molecules, or an increase in the volume occupied by the
molecules. If the fluid is incompressible, then the volume must not change, and this
implies that the divergence must be zero.

One way to think about incompressibility is
to imagine that each of the molecules, or points, of the fluid occupies a volume that
cannot be compromised. They cannot be squeezed into a smaller volume, nor can they
disappear or appear from nowhere. With a little bit of thought, you can see how similar
incompressibility is to reversibility. In the examples that we examined in Lecture 1,
the arrows also defined a kind of flow. And in a sense the flow was incompressible, at
least if it was reversible. The obvious question that this raises is whether the flow
through phase space is incompressible. The answer is yes, if the system satisfies
Hamilton’s equations. And the theorem that expresses the incompressibility is
called Liouville’s theorem.

LIOUVILLE’S THEOREM

Let’s go back to the fluid flow in
phase space and consider the components of the velocity of the fluid at every point of
the phase space. Needless to say, the phase-space fluid is not
three-dimensional with coordinates x, y, z. Instead it is a
2N-dimensional fluid with coordinates pi,
qi. Therefore, there are 2N
components of the velocity field, one for each q and one for each p.
Let’s call them vqi and
vpi.

The concept of a divergence in Eq. (4) is
easily generalized to any number of dimensions. In three dimensions it is the sum of the
derivatives of the velocity components in the respective directions. It’s exactly
the same in any number of dimensions. In the case of phase space, the divergence of a
flow is the sum of 2N terms:

If the fluid is incompressible, then the
expression in Eq. (5) must be zero. To find out, we need to know the components of the
velocity field—that being nothing but the velocity of a particle of the phase space
fluid.

The flow vector of a fluid at a given point
is identified with the velocity of a fictitious particle at that point. In other
words,

Moreover, i and i are exactly the quantities
that Hamilton’s equations, Equations (1), give:

All we have to do is plug Equations (6) into
Eq. (5) and see what we get:

Recalling that a second derivative like does not depend on the order of
differentiation, we see that the terms in Eq. (7) exactly cancel in pairs:

Thus the phase space fluid is incompressible.
In classical mechanics, the incompressibility of the phase space fluid is called
Liouville’s theorem, even though it had very little to do with the
French mathematician Joseph Liouville. The great American physicist Josiah Willard Gibbs
first published the theorem in 1903, and it is also known as the Gibbs-Liouville
theorem.

We defined the incompressibity of a fluid by
requiring that the total amount of fluid that enters every small box be zero. There is
another definition that is exactly equivalent. Imagine a volume of fluid at a given
time. The volume of fluid may have any shape—a sphere, a cube, a blob, or whatever. Now
follow all the points in that volume as they move. After a time the fluid blob will be
at a different place with a different shape. But if the fluid is incompressible, the
volume of the blob will remain what it was at the beginning. Thus we
can rephrase Liouville’s theorem: The volume occupied by a blob of phase space
fluid is conserved with time.

Let’s take the example of the harmonic
oscillator in which the fluid moves around the origin in circles. It’s obvious
that a blob maintains its volume since all it does is rigidly rotate. In fact, the shape
of the blob stays the same. But this latter fact is special to the harmonic oscillator.
Let’s take another example. Suppose the Hamiltonian is given by

You probably don’t recognize this
Hamiltonian, but it is completely legitimate. Let’s work out its equations of
motion:

What these equations say is that q
increases exponentially with time, and p decreases exponentially at the same
rate. In other words, the flow compresses the fluid along the p axis, while
expanding it by the same amount along the q axis. Every blob gets stretched
along q and squeezed along p. Obviously, the blob undergoes an extreme
distortion of its shape—but its phase space volume does not change.

Liouville’s theorem is the closest
analogy that we can imagine to the kind of irreversibility we discussed in Lecture 1. In
quantum mechanics, Liouville’s theorem is replaced by a quantum version called
unitarity. Unitarity is even more like the discussion in Lecture 1—but
that’s for the next installment of The Theoretical Minimum.

POISSON BRACKETS

What were the nineteenth-century French
mathematicians thinking when they invented these extremely beautiful—and extremely
formal—mathematical ways of thinking about mechanics? (Hamilton himself was an
exception—he was Irish.) How did they get the action principle, Lagrange’s
equations, Hamiltonians, Liouville’s theorem? Were they solving physics problems?
Were they just playing with the equations to see how pretty they could make them? Or
were they devising principles by which to characterize new laws of physics? I think the
answer is a bit of each, and they were incredibly successful in all these things. But
the really astonishing degree of success did not become apparent until the twentieth
century when quantum mechanics was discovered. It almost seems as if the earlier
generation of mathematicians were clairvoyant in the way they invented exact parallels
of the later quantum concepts.

And we are not finished. There is one more
formulation of mechanics that seems to have been very prescient. We owe it to the French
mathematician Poisson, whose name means “fish” in French. To motivate the
concept of a Poisson bracket, let’s consider some function of qi and pi. Examples
include the kinetic energy of a system that depends on the p’s, the
potential energy that depends on the q’s, or the angular momentum that
depends on products of p’s and q’s. There are, of course,
all sorts of other quantities that we might be interested in. Without specifying the
particular function, let’s just call it F(q, p).

We can think of F(q,
p) in two ways. First of all, it is a function of position in the phase
space. But if we follow any point as it moves through the phase space—that is, any
actual trajectory of the system—there will be a value of F that varies along
the trajectory. In other words, the motion of the system along a
particular trajectory turns F into a function of time. Let’s compute how
F varies for a given point as it moves, by computing the time derivative of
F:

By now the routine should be obvious—we use
Hamilton’s equations for the time derivatives of q and p:

I don’t know exactly what Poisson was
doing when he invented his bracket, but I suspect he just got tired of writing the right
hand side of Eq. (8) and decided to abbreviate it with a new symbol. Take any two
functions of phase space, G(q, p) and F(q,
p). Don’t worry about their physical meaning or whether one of them
is the Hamiltonian. The Poisson bracket of F and G is defined as

Poisson could now save himself the trouble of
writing Eq. (8). Instead, he could write

The amazing thing about Eq. (10) is that it
summarizes so much. The time derivative of anything is given by the Poisson bracket of
that thing with the Hamiltonian. It even contains Hamilton’s equations themselves. To see that, let F(q, p) just be one
of the q’s:

Now, if you work out the Poisson bracket of
qi and H, you will discover that it
has only one term—namely, the one where you differentiate qk with respect to itself. Since = 1, we find that the Poisson bracket {qk, H} is just equal to , and
we recover the first of Hamilton’s equations. The second equation is easily seen
to be equivalent to

Notice that in this formulation the two
equations have the same sign, the sign difference being buried in the definition of the
Poisson bracket.

The French obsession with elegance really
paid off. The Poisson bracket turned into one of the most basic quantities of quantum
mechanics: the commutator.

Lecture 10: Poisson Brackets, Angular Momentum, and
Symmetries

Lenny asked, “Hey George, can we hang
fish on a Poisson Bracket?”

George smiled. “Only if they’re
theoretical.”

AN AXIOMATIC FORMULATION OF
MECHANICS

Let’s abstract a set of rules that
enable one to manipulate Poisson Brackets (from now on I’ll use the abbreviation
PB) without all the effort of explicitly calculating them. You can check (consider it
homework) that the rules really do follow from the definition of PB’s. Let
A, B, and C be functions of the p’s and
q’s. In the last lecture, I defined the PB:

- The first property is antisymmetry:
If you interchange the two functions in the PB it changes sign:

In particular, that means that the PB
of a function with itself is zero:

- Next is linearity in either entry.
Linearity entails two properties. First, if you multiply A (but not
C) by a constant k, the PB gets multiplied by the same
constant:

Second if you add A +
B and take the PB with C, the result is additive:

Eq.s (2) and (3) define the linearity
property of PB’s.

- Next we consider what happens when we multiply
A and B and then take the PB with C. To figure it
out, all you need to do is go back to the definition of the PB and apply the rule
for differentiating a product. For example,

The same thing is true for
derivatives with respect to p. Here is the rule:

- Finally, there are some specific PB’s
that you need in order to get started. Begin by noting that any q or any
p is a function of the p’s and q’s. Since
every PB involves derivatives with respect to both p’s and
q’s, the PB of any q with any other q is zero. The
same is true for the PB of two p’s:

But a PB of a q with a
p is not zero. The rule is that {qi, pj} is one if i =
j and zero otherwise. Using the Kronecker symbol,

Now we have everything we need to calculate any
PB. We can forget the definition and think of Eq.s (2, 3, 4, 5, 6, 7, and 8) as a set of
axioms for a formal mathematical system.

Suppose we want to compute

where for simplicity I have assumed a system
with just one q and one p. I will tell you the answer and then prove
it. The answer is

The way to prove this kind of formula is to
use mathematical induction. That takes two steps. The first step is to assume
the answer for n (assume the induction hypothesis, Eq. (10)) and show that it
follows for n + 1. The second step is to explicitly show that the induction
hypothesis holds for n = 1.

Thus, replacing n with n +
1, we can write Eq. (9) using Eq. (6):

Next use Eq. (8), which in this case is just
{q, p} = 1:

We now use the induction hypothesis—Eq.
(10)—and get

Equation (11) is exactly the induction
hypothesis for n + 1. Therefore, all we need to do is show that Eq. (10) holds
for n = 1. But all it says is that {q, p} =
1, which is of course true. Thus Eq. (10) is true.

We can write this example in another way
that has far-reaching consequences. Notice that n q(n-1) is nothing but the derivative of qn.
Thus, for this case,

Now take any polynomial (even an infinite
power series) of q. By applying Eq. (12) to each term in the polynomial and
using linearity to combine the results, we can prove

Since any smooth function can be arbitrarily
well approximated by a polynomial, this enables us to prove Eq. (13) for any function of
q. In fact, it even goes further. For any function of q and
p, it is easy to prove that

Exercise 1: Prove Eq.
(14).

Thus we have discovered a new fact about
Poisson Brackets: Taking the PB of any function with pi has
the effect of differentiating the function with respect to qi. We could have proved that directly from the definition of the
PB, but I wanted to show you that it follows from the formal axioms.

What about taking the Poisson bracket of
F (q, p) with qi? You
may be able to guess the answer from the symmetric way in which the p’s
and q’s enter all the rules. By now you may even guess the sign of the
answer:

Exercise 2: Hamilton’s
equations can be written in the form = {q, H} and
=
{p, H}. Assume that the Hamiltonian has the
form H =

p^2 + V(q). Using only the PB
axioms, prove Newton’s equations of motion.

ANGULAR MOMENTUM

In Lecture 7, I explained the relationship
between rotation symmetry and the conservation of angular momentum. Just to remind you,
I will briefly review it for the case of a single particle moving in the x,
y plane. We wrote the formula for an infinitesimal rotation in the form

Then, assuming that the Lagrangian is
invariant, we derived a conserved quantity

with a change of sign, we call it the angular
momentum L,

Now I want to go to three-dimensional space,
where angular momentum has the status of a vector. Equation (16) is still true, but it
takes on a new meaning: It becomes the rule for rotating a system about the z
axis. In fact, we can fill it out with a third equation that expresses the fact that
z is unchanged by a rotation about the z axis:

Equation (17) is also unchanged, except that
we interpret the left-hand side as the z component of the angular momentum. The
other two components of angular momentum are also easily computed, or you can guess them
just by cycling the equation x

y, y

z, z

x:

As you might expect, each component of the
vector is conserved if the system is rotationally symmetric about
every axis.

Now let’s consider some Poisson
Brackets involving angular momentum. For example, consider the PB’s of x,
y, and z with Lz:

You can work out these PB’s using the
definition Eq. (1), or you can use the axioms.

Exercise 3: Using both the
definition of PB’s and the axioms, work out the PB’s in Equations
(19). Hint: In each expression, look for things in the parentheses that have
nonzero Poisson Brackets with the coordinate x, y, or z. For example, in the
first PB, × has a nonzero PB with px.

Here are the results:

If we compare this with Equations (18) we see
a very interesting pattern. By taking the PB’s of the coordinates with Lz we reproduce (apart from the ) the expressions
for the infinitesimal rotation about the z axis. In other words,

where means “apart from the factor .”

The fact that taking a PB with a conserved
quantity gives the transformation behavior of the coordinates under a symmetry—the symmetry related to the conservation law—is not an accident. It is very
general and gives us another way to think about the relationship between symmetry and
conservation. Before we pursue this relationship further, let’s explore other
PB’s involving angular momentum. First of all, it is easy to generalize to other
components of L. Again, you can do it by cycling x

y, y

z, z

x. You’ll get six more equations, and you might wonder whether there is a
nice way to summarize them. In fact there is.

MATHEMATICAL INTERLUDE—THE LEVI-CIVITA
SYMBOL

A good notation can be worth a lot of
symbols, especially if it appears over and over. An example is the Kronecker delta
symbol ij. In this section I will
give you another one, the Levi-Civita symbol, which is also called the symbol ijk. As in the Kronecker case, the indices i,
j, k represent the three directions of space, either x,
y, z or 1, 2, 3. The Kronecker symbol takes on two values: either
1 or 0, depending on whether i = j or i

j. The
symbol takes on one of three values: 0, 1, or −1. The rules for ijk are a little more complicated than those for ij.

First of all, ijk = 0 if any two indices are the same.—for example, 111 and 223 are both zero. The only time ijk is not zero is when all three indices are
different. There are six possibilities: 123, 231, 312, 213, 132, 321. The first three have value 1, and
the second three have value −1.

What is the difference between the two
cases? Here is one way to describe it: Arrange the three numbers 1, 2, 3 on a circle,
like a clock with only three hours (see Figure
1).

Figure 1: A circular arrangement of the numbers 1,
2, and 3.

Start at any of the three numbers and go
around clockwise. You get (123), (231), or (312), depending on where you start. If you
do the same going counterclockwise, you get (132), (213), or (321). The rule for the
Levi-Civita symbol is that ijk
= 1, for the clockwise sequences, and ijk = −1 for the
counterclockwise sequences.

BACK TO ANGULAR MOMENTUM

Now, with the aid of the symbol, we can
write the PB’s for all the coordinates and all the components of :

For example, suppose that you want to know
{y, Lx}. Identifying 1, 2, 3 with
x, y, z and plugging these into Eq. (20) we get

Since 213 is a counterclockwise sequence,
213 = −1, so

Let’s consider another set of
PB’s—namely, the PB’s of pi with the
components of . They are easy to work out, and with the aid of the symbol, we
get

For example,

The thing to notice is that the PB’s of
the p’s and L’s have exactly the same form as those of the
x’s and L’s. That is interesting because the
p’s and x’s transform exactly the same way under a
rotation of coordinates. Just as x
− y for a rotation about z, the
variation of px is proportional to −py.

The meaning of this is quite deep. It says
that to compute the change in any quantity when the coordinates are rotated, we compute
the Poisson bracket of the quantity with the angular momentum. For a rotation about the
ith axis,

The angular momentum is the
generator of rotations.

We will come back to this theme, and to the
intimate relationship connecting symmetry transformations, Poisson Brackets, and
conserved quantities, but first I want to explain how PB’s can be useful in
formulating and solving problems.

ROTORS AND PRECESSION

One thing we haven’t done yet is to
compute the PB’s between different components of the angular momentum. The PB of
anything with itself is always zero, but the PB of one component of with another
is not zero. Consider

Either by using the definition of PB’s or
by using the axioms, we will get

Try it.

The general relation can be read off by
cycling through x, y, z. Here it is using the Levi-Civita
symbol:

That’s very pretty, but what can we do
with it? To illustrate the power of relations such as Eq. (22), let’s consider a
small, rapidly spinning ball in outer space. Call it a rotor. At any instant
there is an axis of rotation, and the angular momentum is along that axis. If the rotor
is isolated from all influences, then its angular momentum will be conserved, and the
axis of rotation will not change.

Now suppose the rotor has some electric
charge. Because the rotor is rapidly spinning, it behaves like an electromagnet with its
north and south poles along the rotation axis. The strength of the dipole is
proportional to the rate of rotation—or, better yet—to the angular momentum. This
won’t make any difference unless we put the whole thing in a magnetic field . In that case, there will be some energy associated with any
misalignment between and (see Figure 2).

Figure 2: A rotor aligned at an angle to a magnetic
field.

That energy is proportional to the cosine of
the angle between the two vectors and to the product of their magnitudes. In other
words, the alignment energy is proportional to the dot product

I’ve used the notation H for
energy because later we will identify it with the Hamiltonian of the system.

Let’s take the magnetic field to be
along the z axis so that H is proportional to the z component
of . Lumping the magnetic field, the electric charge, the radius
of the sphere, and all the other unspecified constants into a single constant , the energy
of alignment takes the form

Let’s pause for some perspective on
what we are doing and where we are going. It’s obvious that without the magnetic
field, the system is rotationally symmetric in the sense that the energy does not change
if you rotate the axis of the rotor. But with the magnetic field, there is something to
rotate relative to. Therefore, the rotational symmetry is ruined. Eq.s
(23) and (24) represent the rotational asymmetry. But what is the effect? The answer is
obvious: The angular momentum is no longer conserved—no symmetry, no conservation. That
means the direction of the spin will change with time, but exactly how?

One can try to guess the answer. The rotor
is a magnet—like a compass needle—and intuition suggests that the angular momentum will
swing toward the direction of , like a pendulum. That’s
wrong if the spin is very rapid. What does happen is that the angular momentum
precesses, exactly like a gyroscope, around the magnetic field. (A gyroscope would
precess about the gravitational field.) To see that, let’s use the Poisson Bracket
formulation of mechanics to work out the equations of motion for the vector .

First, recall that the time derivative of
any quantity is the PB of that quantity with the Hamiltonian. Applying this rule to the
components of gives

or, using Eq. (24)

Now we can see the point. Even if we know
nothing about the material that the rotor is made of, where the charge resides, or how many particles are involved, we can solve the problem: We know
the PB’s between all components of . First we take the equation for
z. Since it involves the PB of Lz with itself,

The z component of does not change. That immediately precludes the idea that swings like
a pendulum about .

Next we use Eq. (22) to work out x and y:

This is exactly the equation of a vector in
the x, y plane rotating uniformly about the origin with angular
frequency . In other words, precesses about the magnetic field.
The magic of Poisson Brackets allows us to solve the problem knowing very little other
than that the Hamiltonian is proportional to · .

SYMMETRY AND CONSERVATION

Let’s go back to Eq. (21), the meaning
of which is that the variation of any quantity, under the action of a rotation, is
proportional to the PB of that quantity with Li.
Moreover, Li happens to be the quantity that
is conserved by virtue of invariance with respect to rotation. That’s an
interesting connection, and one wonders how general it is. Let me give a couple of other
examples of the same thing. Consider a particle on a line. If there is
translation invariance, then the momentum p is conserved. Now take the PB of
any function of x with p:

What is the change in F(x)
under an infinitesimal translation by distance ? The answer is

or

Here’s another example: If a system has
time-translation invariance, then the Hamiltonian is conserved. What is the small change
in a quantity under a time translation? You guessed it—the time derivative of the PB of
the quantity with H.

Let’s see if we can generalize the
connection. Let G(q, p) be any function of the coordinates
and momenta of a system. I use the letter G because I am going to call it a
generator. What it generates is small displacements of the phase space
points. By definition, we will shift every point in phase space by the amount
qi,
pi, where

Equations (25) generate an infinitesimal transformation of phase space. The transformation generated by
G may or may not be a symmetry of the system. What exactly does it mean to
say that it is a symmetry? It means that no matter where you start, the transformation
does not change the energy. In other words, if H =
0 under the transformation generated by G, then the transformation
is a symmetry. We can therefore write that the condition for a symmetry is

But Eq. (26) can be read another way. Since
interchanging the order of the two functions in a PB changes only the sign, Eq. (26) may
be expressed as

which is exactly the condition that
G is conserved. One can say it this way: The same Poisson Bracket that
tells us how H changes under the transformation generated by G also
tells us how G changes with time.

Lecture 11: Electric and Magnetic Forces

He kept a magnet in his coat pocket. How it
attracted nails and other bits of metal was an endless source of fascination, and the
way it spun the needle of his compass, round and round the world. What magic was inside
that horseshoe-shaped bit of iron? Whatever it was, Lenny never tired of playing with
his favorite toy.

What he didn’t know was that the whole
Earth is a magnet. Or that the earth-magnet was a providential force, that protected him
from deadly solar radiation, bending the paths of charged particles into safe orbits.
For the moment such things were beyond Lenny’s imagination.

“Tell me about magnets,
George.”

VECTOR FIELDS

A field is nothing but a function of space
and time that usually represents some physical quantity that can vary from point to
point and from time to time. Two examples taken from meteorology are the local
temperature and the air pressure. Since the temperature can vary, it makes sense to
think of it as a function of space and time, T(x, y,
z, t), or, more simply, T(x, t). The
temperature and air pressure are obviously not vector fields. They have no sense of
direction, nor do they have components in different directions. Asking for the
y component of temperature is nonsense. A field that consists of only one
number at each point of space is called a scalar field. The temperature field
is a scalar.

There are, however, vector fields such as the
local wind velocity. It has a magnitude, a direction, and components. We can write it as
(x, t), or we can write its components
vi(x, t). Other examples of
vector fields are the electric and magnetic fields created by electric charges and
currents.

Because such fields vary in space, we can
construct new fields by differentiating the original fields. For example, the three
partial derivatives of temperature, can
be considered to be the components of a vector field called the temperature
gradient. If the temperature increases from north to south then the
gradient points toward the south. Let’s spend a little time going over the tricks
used to create new fields from old ones by differentiating.

MATHEMATICAL INTERLUDE: DEL

Let’s invent a fake vector called . The verbal name of is “del,” standing,
I suppose, for delta, although an honest delta is written as . The components of are not numbers.
They are derivative symbols:

At first sight Equations (1) look like
nonsense. The components of vectors are numbers, not derivative symbols. And anyway the
derivative symbols don’t make sense—derivatives of what? The
point is that never stands alone. Just like the derivative symbol , it must act on something—it must have a function of some sort
to differentiate. For example, can act on a scalar such as the temperature. The components of
t are

and they indeed form the components of a
genuine vector field—the gradient of the temperature. In a similar way, we can form the
gradient of any scalar field.

Next, let’s define the
divergence of a vector field. The divergence is defined in analogy with the
dot product of two vectors · = Vx
Ax + Vy Ay + Vz Az, which, by the way, is a scalar. The divergence of a vector is
also a scalar. Let the vector field be (x). The divergence of is the dot
product of and —in other words, · . The meaning of this symbol is easy
to guess by analogy with the usual dot product:

Then consider the cross product of two vectors
and which gives another vector. The components of the cross
product are

Here is another way to write them using the
Levi-Civita symbol:

Exercise 1: Confirm Eq. (3). Also prove that

Now let’s substitute the fake vector
for in Eq. (3):

More explicitly

What we have done is to start with a vector
field (x) and generate another vector field × by differentiating
A in a particular way. The new vector field × is called the curl of .

Here is a theorem that takes a few seconds
to prove. For any starting field (x), the curl of A
has no divergence,

The theorem actually has a stronger form that
is harder to prove. A field has zero divergence if and only if it is the curl of another
field.

Here is another theorem that is easy to
prove. Let a vector field be defined by the gradient of a scalar field:

where V is the scalar. Then it
follows that the curl of is zero:

Exercise 2: Prove Eq.
(4).

MAGNETIC FIELDS

Magnetic fields (called (x))
are vector fields, but not just any vector field can represent a magnetic field. All
magnetic fields have one characteristic feature: Their divergence is
zero. Thus it follows that any magnetic field can be expressed as a curl of some
auxiliary field:

where is called the
vector potential. In component form,

The vector potential is a peculiar field. In
a sense it does not have the same reality as magnetic or electric fields. It’s
only definition is that its curl is the magnetic field. A magnetic or electric field is
something that you can detect locally. In other words, if you want to know whether there
is an electric/magnetic field in a small region of space, you can do an experiment in
that same region to find out. The experiment usually consists of seeing whether there
are any forces exerted on charged particles in that region. But vector potentials cannot
be detected locally. First of all, they are not uniquely defined by the magnetic field
they are representing. Suppose is given by a vector potential, as
in Eq. (5). We can always add a gradient to to define a new vector potential
without changing . The reason is that the curl of a gradient is always zero.
Thus if two vector potentials are related by

for some scalar s, then they produce
identical magnetic fields and cannot be distinguished by any experiment.

This is not the first time we have seen an
ambiguity having to do with one thing being defined by a derivative of another. Remember
that the force on a system is minus the gradient of the potential energy:

The potential energy is not unique: You can
always add a constant without changing the force. This means that you can never directly
measure the potential, but only its derivative. The situation is similar with the vector
potential; indeed, that’s why it is called a potential.

Let’s consider an example of a
magnetic field and its associated vector potential. The simplest case is a uniform
magnetic field pointing, say, along the z axis:

where b is a number representing the
strength of the field. Now define a vector potential by

When the curl of is computed,
there is only one term, namely
= b. Thus the only component of the magnetic field is
the z component, and it has value b.

Now, there is something funny about
Equations (8). The uniform magnetic field seems to be completely symmetric with respect
to rotations in the x, y plane. But the vector potential has only a
y component. However, we could have used a different vector potential ′—one with only an x component—to generate the
very same magnetic field:

Exercise 3: Show that the
vector potentials in Equations (8) and Equations (9) both give the same uniform
magnetic field. This means that the two differ by a gradient. Find the scalar
whose gradient, when added to Equations (8), gives Equations (9).

The operation of changing from one vector
potential to another to describe the same magnetic field has a name. It is called a
gauge transformation. Why “gauge”? It’s a historical
glitch. At one time it was wrongly thought to reflect ambiguities in gauging lengths at
different locations.

If the vector potential is ambiguous but the
magnetic field quite definite, why bother with the vector potential at all? The answer
is that without it, we could not express the principle of stationary action, or the
Lagrangian, Hamiltonian, and Poisson formulations of mechanics for particles in magnetic
fields. It’s a weird situation: The physical facts are gauge
invariant, but the formalism requires us to choose a gauge (a particular choice
of vector potential).

THE FORCE ON A CHARGED PARTICLE

Electrically charged particles are influenced
by electric and magnetic fields and . The force due to the electric
field is simple and of the form that we studied in earlier chapters; specifically, it is
the gradient of a potential energy. In terms of the electric field,

where e is the charge of the
particle. It is a rule of electromagnetic theory that a static (not time-dependent)
electric field has no curl so it must be a gradient. The usual notation is

so we can write the force as

The potential energy is e

V, and everything is completely conventional.

Magnetic forces on charged particles are
different and a little more complicated. They depend not only on the position of the
particle through the value of the magnetic field, but also on the velocity of the
particle. They are referred to as velocity-dependent forces. The magnetic force
on a charged particle was first written down by the great Dutch physicist H. A. Lorentz
and is called the Lorentz force. It involves the velocity
vector of the particle and the speed of light c:

Notice that the Lorentz force is
perpendicular to both the velocity and the magnetic field. Combining Eq. (10) with
Newton’s = m
, we
find that the equations of motion for a particle in a magnetic field are

The Lorentz force is not the first
velocity-dependent force we have encountered. Recall that in a rotating frame, there are
two so-called fictitious forces: The centrifugal force and (more to the point) the
Coriolis force. The Coriolis force is given by

where is the vector
representing the angular velocity of the rotating frame. The Coriolis and Lorentz forces
are very similar, with the magnetic field and the angular velocity playing the same
role. Of course, not all magnetic fields are uniform, so the magnetic situation can be
far more complex than the Coriolis case.

THE LAGRANGIAN

All of this raises the question of how to
express magnetic forces in the action, or Lagrangian, form of mechanics. One source of
confusion is that the symbol for action and the symbol for vector potential are both A. In what follows, we will use A for the action,
and , or Ai, as the vector
potential. Let’s ignore or set equal to zero the electric field and concentrate on
the magnetic, or Lorentz force. Begin with the action for a free particle with no
forces:

with

Here i refers to the direction of
space, and the summation sign for summing over x, y, z has
been left implicit. Get used to it.

What can we add to the action or to the
Lagrangian to give rise to a Lorentz force? The answer is not obvious. However, we know
that whatever the additional ingredient is, it should be proportional to the electric
charge, and it should also involve the magnetic field in some form.

You can experiment around with it and get
frustrated. There is nothing you can do directly involving that will
give the Lorentz force. The key is the vector potential. The simplest thing we can do
with the vector potential is to dot it into the velocity vector. Remember that the
Lagrangian involves only the positions and the velocities. You might also try dot
products of the position vector with , but that doesn’t work very
well. So let’s try adding to the Lagrangian the term

The reason for including the speed of light is
that it occurs together with the charge in Lorentz force. Thus we try out the
action:

Now you might object that the equation of
motion is not supposed to involve the vector potential, but only the magnetic field. We
know that the vector potential is not unique, so won’t we get another answer if we
make a gauge transformation
Let’s see what happens to the action if we do so.

The important part of the action is the term
arising from Eq. (13):

or, more explicitly,

In this equation, AL is the part of the action that we are adding to try to account for
the Lorentz force—hence the subscript L. Suppose we change by adding

s. At first sight, it would seem to change AL
by adding the term

If you look at this carefully, you will see
that it all boils down to a simple expression. The d t’s in the numerator
and denominator cancel:

And then the whole thing is just the
difference between the value of s at the beginning and its value at the end of
the trajectory. In other words, the gauge transformation added a term s_1 − s_0 to the action,
where s_0 and s_1 are
the values of s at the initial and final positions of the trajectory,
respectively. In other words, the change in the action due to the gauge transformation
is

Does such a change make any difference to the
equations of motion? Let’s recall exactly what the action principle actually says.
Given any two points in space and time, x_0,
t_0 and x_1,
t_1, there are many trajectories that connect them,
but only one is the true trajectory taken by a particle. The true trajectory is the one
that minimizes, or makes stationary, the action. So what we do is explore all
trajectories that connect the points until we find the stationary-action solution. From
that principle we derived the Euler-Lagrange equations of motion.

As we see in Eq. (15), a gauge
transformation changes the action, but only if we vary the endpoints.
If the endpoints are kept fixed, the change in the action has no effect. The stationary
point has to do only with changing the trajectory without moving the endpoints. Though
the action changes, the equations of motion do not, and neither do the solutions. We say
that the equations of motion and their solutions are gauge invariant.

One more bit of jargon: Since there are many
possible choices of vector potentials that describe the same physical situation, a
specific choice is simply called a gauge. For example, Equations (8) and
Equations (9) are two different gauges describing the same uniform magnetic field. The
physical principle that the result of any experiment should not depend on the gauge
choice is called gauge invariance.

EQUATIONS OF MOTION

Let’s return to the action, Eq. (14).
And let’s be very explicit about the Lagrangian:

Starting with x, the Lagrange
equation of motion is

First the canonical momenta: You might think
that the momenta are just the usual mass times velocity, but that’s not right. The
correct definition is that the momenta are the derivatives of the Lagrangian with
respect to the components of velocity. This does give p = m v with the
usual particle Lagrangians, but not with a magnetic field. From Eq. (16) we get

This may worry you. It indicates that the
canonical momentum is not gauge invariant. This is true, but we are not finished yet. We have two more things to do. We must compute the time derivative of
px and also compute the right-hand side of Eq.
(17). Maybe, if we are lucky, all the gauge-dependent stuff will cancel.

The left-hand side of Eq. (17) is

where ax is
the x component of acceleration.

The right-hand side side of Eq. (17) is:

Now let’s combine the left and right
sides:

Equation (19) looks complicated, but note
that the combinations of derivatives

and

are things we saw in Equations (7)—namely,
the z and y
components of the magnetic field. We can rewrite Eq. (19) in a much
simpler form:

Take a careful look at Eq. (20). You should
be impressed by a number of things. First of all, the equation is gauge invariant: On
the right-hand side, the vector potential has completely disappeared in favor of the
magnetic field. The left-hand side is the mass times the acceleration—that is the
left-hand side of Newton’s equation. In fact, Eq. (20) is nothing but the
x component of the Newton-Lorentz equation of motion, Eq. (12).

One might wonder why we bothered introducing
the vector potential at all. Why not just write the gauge-invariant Newton-Lorentz
equation? The answer is that we can, but then we lose any possibility of formulating the
equations as an action principle, or as Hamilton’s equations of motion. That might
not be such a tragedy for the classical theory, but it would be a disaster for quantum
mechanics.

THE HAMILTONIAN

Before discussing the Hamiltonian of a
charged particle in a magnetic field, let’s go back to the definition of the
particle’s momentum. You may still find it confusing. The reason is that there are
two separate concepts: mechanical momentum and canonical momentum. Mechanical momentum
is what you learn about in elementary mechanics (Momentum equals mass times
velocity) and in advanced mechanics (Canonical momentum equals derivative
of the Lagrangian with respect to velocity). In the simplest situations where
the Lagrangian is just the difference of kinetic and potential energy,
the two kinds of momentum are the same. That’s because the only dependence on
velocity is
m v^2.

But once the Lagrangian gets more
complicated, the two kinds of momentum may not be the same. In Eq. (18) we see such an
example. The canonical momentum is the mechanical momentum plus a term proportional to
the vector potential. We can write it in vector notation:

The mechanical momentum is not only familiar;
it is gauge invariant. It is directly observable, and in that sense it is
“real.” The canonical momentum is unfamiliar and less “real”; it
changes when you make a gauge transformation. But whether or not it is real, it is
necessary if you want to express the mechanics of charged particles in Lagrangian and
Hamiltonian language.

To pass to the Hamiltonian, we recall the
definition

which in this case is

Let’s work it out. First we will need
to get rid of the velocities; the Hamiltonian is always thought of as a function of
coordinates and momenta. That’s easy. We just solve Eq. (21) for velocity in terms
of p:

Now wherever you see a velocity component in
Eq. (22), substitute Eq. (23) and then do a little rearranging. Here is what you will
get:

Exercise 4: Using the
Hamiltonian, Eq. (24), work out Hamilton’s equations of motion and show
that you just get back the Newton-Lorentz equation of motion.

If you carefully look at Eq. (24), you will
see something a little surprising. The combination is the mechanical momentum m
vi. The Hamiltonian is nothing but

In other words, its numerical value is the
same as the naive kinetic energy. That proves (among other things) that the energy is
gauge invariant. Since it conserved, the naive kinetic energy is also conserved, as long
as the magnetic field does not change with time. But that does not mean the particle
motion does not sense the magnetic field. If you want to use the Hamiltonian to find the
motion, you must express it in terms of the canonical momentum, not the velocity, and
then use Hamilton’s equations. Alternatively, you can work with velocities and use
the Lagrangian form of the equations, but in that case the Lagrangian is not the naive
kinetic energy. In either case, if you work it all out, you will
discover that the charged particle experiences a gauge-invariant Lorentz magnetic
force.

MOTION IN A UNIFORM MAGNETIC FIELD

Motion in a uniform magnetic field is easy
enough to solve, and it illustrates a lot of the principles we have been discussing.
Let’s take the field to lie in the z direction and to have magnitude
b. This is the situation described in Equations (6, 7, 8). The choice
between the vector potentials in Equations (7, 8) is an example of the ambiguity
associated with gauge transformations. Let’s first choose Equations (7) and write
the Hamiltonian, Eq. (24), using (Ax = 0,
Ay = b x, Az = 0). We get

As always, the first thing to do is to look
for conservation laws. We already know one: energy conservation. As we’ve seen,
the energy is the old-fashioned kinetic energy
m v^2. It follows that the magnitude of the velocity is
constant.

Next, notice that the only coordinate that
appears in H is x. This means that when we work out Hamilton’s
equations, we will find that px is not conserved but
that both pz and py
are conserved. Let’s see what the implications are. First the z
component. Since Az = 0, pz = m vz, and the
conservation of pz tells the familiar story that the
z component of velocity is constant.

Next look at the conservation of py. This time py is
not equal to m vy but, rather, to m
vy + b x. The conservation of py
then tells us that

or

Notice that the conservation of py does not imply that the y component
of velocity is conserved.

What about px? It does not seem to be conserved since H has explicit
dependence on x. We could use Hamilton’s equations to determine the
x component of acceleration, but I’m going to do it another way.
Instead of using Equations (8) I’m going to change the gauge in midstream and use
Equations (7). Remember that the physical phenomena should not change. The new
Hamiltonian that goes with Equations (7) is,

Now the Hamiltonian does not depend on
x, which implies that px is conserved.
How can that be? We previously showed the x-component of momentum px is conserved when we used Equations (8). The answer
is that when we make a gauge transformation the components of p change. In the
two cases, px does not have the same meaning.

Let us see the implication of px conservation in the new gauge. Using Equations (7) we
find that px = m vx −
b y. Thus the conservation of px is expressed
as

By now you may have already realized that Eq.
(25) and Eq. (26) are familiar. They are the Newton-Lorentz equations for motion in a
uniform magnetic field.

Exercise 5: Show that in the
x, y plane, the solution to Eq. (25) and the solution to
Eq. (26) are a circular orbit with the center of the orbit being anywhere on the
plane. Find the radius of the orbit in terms of the velocity.

GAUGE INVARIANCE

The reason why I left magnetic forces for the
last lecture is that I want you to remember the lessons when, in future study, we come
to quantum mechanics and field theory. Gauge fields and gauge invariance are not minor
artifacts of writing the Lorentz force in Lagrangian form. They are the central guiding
principles that underlie everything, from quantum electrodynamics to general relativity
and beyond. They play a leading role in condensed matter physics—for example, in
explaining all sorts of laboratory phenomena such as superconductivity. I will close
these lectures on classical mechanics by reviewing the meaning of the gauge idea, but
its real importance will become clear only in later lectures.

The simplest meaning of a gauge field—the
vector potential is the most elementary example—is that it is an auxiliary device that
is introduced to make sure certain constraints are satisfied. In the case of a magnetic
field, not any (x) is allowed. The constraint is
that (x) should have no divergence:

To ensure that, we write the magnetic field
as the curl of something—(x)—because curls
automatically have no divergence. It’s a trick to avoid having to worry explicitly
about the fact that (x) is constrained.

But we soon discover that we cannot get
along without (x). There is no way to derive Lorentz’s force
law from a Lagrangian without the vector potential. That is a pattern: To write the
equations of modern physics in either Lagrangian or Hamiltonian form, auxiliary gauge
fields have to be introduced.

But they are also nonintuitive and abstract.
Despite their being indispensable, you can change them without changing the physics.
Such changes are called gauge transformations, and the fact that physical
phenomenal do not change is called gauge invariance. Gauge fields cannot be
“real,” because we can change them without disturbing the gauge invariant
physics. On the other hand, we cannot express the laws of physics without them.

I am not about to give you a sudden insight
that will resolve this tension. I will just say that’s the way it is: The laws of
physics involve gauge fields, but objective phenomena are gauge invariant.

GOOD BYE FOR NOW

We are now finished with classical mechanics.
If you have followed along, you know the Theoretical Minimum—all you need to know about
classical mechanics to move on to the next thing. See you in Quantum Mechanics!

Appendix 1: Central Forces and Planetary Orbits

Lenny stooped and peered through the
eyepiece of the telescope. It was the first time he had ever done that. He saw the rings
of Saturn and whistled at their beauty. “George, have you seen the
rings?”

George nodded and said, “Yup, I seen
’em.”

Lenny looked up and pressed his friend.
“Where do they come from?”

George said, “It’s like the
Earth goin’ round the Sun.”

Lenny nodded. “How does it go
around?”

THE CENTRAL FORCE OF GRAVITY

A central force field is a force that points
toward a center—in other words, toward a point of space (see Figure 1). In addition, for a force to be a
central force, the magnitude of the force must be the same in every direction.

Figure 1: A central force.

Other than the obvious symmetry—rotational
symmetry—there is nothing very special about central forces from a mathematical viewpoint. But their role in physics and in the history of physics is
very special. The first problems solved by Newton—the problems of planetary orbits—were
central force problems. The motion of an electron orbiting a hydrogen nucleus is a
central force problem. Two atoms orbiting one another to form a simple molecule can be
reduced to a central force problem in which the center is the center of mass. Since
there was not enough time to cover this subject in the lectures, we’ll add it here
as a supplement.

Let’s focus on the motion of the Earth
as it orbits the much heavier Sun. According to Newton’s laws, the force exerted
by the Sun on the Earth is equal and opposite to that exerted by the Earth on the Sun.
Moreover, the direction of those forces is along the line connecting the two bodies.
Because the Sun is so much heavier than the Earth, the motion of the Sun is negligible,
and it can be considered to be at a fixed location. We can choose our coordinates so
that the Sun is at the origin, x = y = z = 0. The Earth, by
contrast, moves in an orbit about the origin. Let’s denote the location of the
Earth by the vector with components x,
y, z. Since the Sun is located at the origin, the force on the
Earth points toward the origin, as shown in Figure 1. Moreover, the magnitude of the
force depends only on the distance r from the origin. A force with these
properties—pointing toward the origin and depending only on the distance—is called a
central force.

Let’s rewrite the unit vector from
Interlude 1:

In equation form, the definition of a central
force is

where determines two things. First, the magnitude of is the magnitude of the force when the Earth is at distance
r. Second, the sign of
determines whether the force is toward or away from the Sun—in other words, whether the
force is attractive or repulsive. In particular, if is
positive the force is away from the Sun (repulsive), and if it is negative the force is
toward the Sun (attractive).

The force between the Sun and the Earth is
of course gravitational. According to Newton’s law of gravitation, the
gravitational force between two objects of mass m_1 and
m_2 has the following properties.

N1: The force is attractive and
proportional to the product of the objects’ masses and a constant called G.
Today we refer to G as Newton’s constant. Its value is G
6.673 m^3 kg−1s−2.

N2: The force is inversely proportional the square of the distance between the
masses.

To summarize, the force is attractive and
has magnitude .

In other words, the function is given by

and

For the case of the Earth-Sun system
let’s denote the Sun’s mass by M and the Earth’s mass by
m. The force on the Earth is

The equation of motion for the Earth’s
orbit is the usual F = m a, or, using the gravitational force,

Notice an interesting fact: The mass of the
Earth cancels from both sides of the equation, so the equation of motion does not depend
on the mass of the Earth:

An object of very different mass, such as a
satellite, could orbit the Sun in the same orbit as the Earth. One caveat about this
fact: It is true only if the Sun is so massive, compared with the Earth or satellite,
that its motion can be ignored.

GRAVITATIONAL POTENTIAL ENERGY

The gravitational force can be derived from a
potential energy function. Recall that the force associated with a potential energy is
minus the gradient of the potential:

It’s not too hard to guess the form of
V for the gravitational case. First of all, since the force is proportional
to the constant G M m, one expects the potential energy also to have this
factor.

Next, because the magnitude of the force
only depends on the distance r, one may expect the potential energy
V(r) also to depend only on r. Finally, since we have to
differentiate V(r) to get the force, and since the force is
proportional to 1 / r^2, the potential energy must be
proportional to −1 / r. Thus it is natural to try

In fact, this is exactly right.

THE EARTH MOVES IN A PLANE

Earlier, we mentioned that the central force
problem has a symmetry. You probably recognize it as rotational symmetry about the
origin. The implication of the symmetry, explained in Lecture 7, is the conservation of
angular momentum. Suppose that at some instant the earth has location and velocity . We can place these two
vectors and the position of the Sun in a plane—the momentary plane of the Earth’s
orbit.

The angular momentum vector is proportional to the cross product × , so it is
perpendicular to both and (see Figure 2). In other words, the angular
momentum is perpendicular to the plane of the orbit. This is a powerful fact when
combined with the conservation of angular momentum. The conservation
tells us that the vector never changes. From that we conclude that the orbital plane
never changes. To put it simply, the Earth’s orbit and the Sun permanently lie in
a fixed plane that does not vary. Knowing this, we may rotate our coordinates so that
the orbit is in the x, y plane. The entire problem is then
two-dimensional, the third coordinate z playing no role.

Figure 2: The relationship among the angular
momentum , the position vector , and the velocity .

POLAR COORDINATES

We could work with the Cartesian coordinates
x, y, but central force problems are much easier to solve in polar
coordinates r, :

In polar coordinates the kinetic energy of the
earth is simple enough:

The potential energy is even simpler—it does
not involve at all:

EQUATIONS OF MOTION

As is usually the case, the easiest route to
the equations of motion is through the Lagrangian method. Recall that the Lagrangian is
the difference of the kinetic and potential energies, L = T −
V. Using Eq. (2) and Eq. (3), the Lagrangian in polar coordinates is

The equations of motion,

take the explicit form

and

This last equation has the form of a
conservation law. Not surprisingly, it is conservation of angular momentum. (To be
precise, it is the conservation of the z component of angular momentum.) It is
traditional to denote the angular momentum by the symbol L, but we are using
that as the Lagrangian, so we will use p
instead. If we know p
at any
particular instant, then we know it for all time. We may write

and just treat p
as a known constant.

This enables us to express the angular
velocity in terms of the distance of the Earth from the Sun. We just solve the equation
for :

We will come back to this relation between
angular velocity and radial distance, but first let’s return to the equation for
r, namely

In Eq. (9) the angular velocity appears, but
we can use Eq. (8) to replace it:

The equation for r has an interesting
interpretation. It looks like the equation for a single coordinate r under the
influence of a combined “effective” force:

The term − is just the gravitational force, but at first sight the second
term may be a surprise. It is, in fact, nothing but the fictitious centrifugal force
experienced by any particle that has an angular motion about the origin.

It’s useful to pretend that Eq. (11)
really does describe a particle moving under a total force that includes both the real
gravitational force and the centrifugal force. Of course, for each value of the angular
momentum, we must readjust p
, but since p
is
conserved, we may regard it as a fixed number.

Given the effective force, one can also
construct an effective potential energy function that includes the effect of gravity and
the effect of centrifugal force:

You can easily check that

For all practical purposes, we can pretend
the r motion is just that of a particle whose kinetic energy has the usual
form, , whose potential energy is
Veffective, and whose Lagrangian is

EFFECTIVE POTENTIAL ENERGY DIAGRAMS

In getting a feel for a problem, it is often
a good idea to make a graph of the potential energy. For example, the equilibrium points
(where the system may be at rest) can be identified as the stationary points (minima,
maxima) of the potential. In understanding central force motion, we do exactly the same,
except that we apply it to the effective potential. Let’s first plot the two terms
in Veffective separately, as shown in Figure 3. Note
that the two terms are of opposite sign; the centrifugal term is positive and the
gravitational term negative. The reason is that the gravitational force is attractive,
whereas the centrifugal force pushes the particle away from the origin.

Figure 3: The potential energy diagram for the
centrifugal and gravitational terms.

Near the origin the centrifugal term is the
most important, but at large values of r the gravitational
term has the larger magnitude. When we combine them, we get a graph of Veffective that looks like Figure 4.

Figure 4: The potential energy diagram for the
combination of centrifugal and gravitational terms.

Note that when the two terms are combined,
the graph has a minimum. That may seem odd; we don’t expect an equilibrium point
where the Earth can stand still. But we have to remember that we are discussing only the
behavior of r and ignoring the angular coordinate . The point is
that for each angular momentum, there are orbits that maintain a constant radial
distance while moving around the Sun. Such orbits are circular. On the graph of
Veffective a circular orbit is represented by a
fictitious particle sitting at rest at the minimum.

Let’s compute the value of r
at the minimum. All we have to do is differentiate Veffective and set the derivative equal to zero. It’s an easy
calculation that I will leave to you. The result is that the minimum occurs at

Equation (14) yields the radius of the
Earth’s orbit (assuming it is circular, which is not quite right) given its
angular momentum.

KEPLER’S LAWS

Tycho Brahe was a sixteenth-century Danish
astronomer before the age of telescopes. With the help of a long rod and some
instruments to measure angles, he made the best tables and records of the motion of the
Solar System before telescopes were invented. As a theoretician, he was somewhat
confused. His legacy was his tables.

It was Tycho’s assistant Johannes
Kepler who put the tables to good use. Johannes Kepler took those records and fit the
observed data to simple geometric and mathematical facts. He had no idea why the planets
moved according to his laws—by modern standards, his theories of why were at
best, odd—but he got the facts right.

Newton’s great achievement—in a sense
the start of modern physics—was to explain Kepler’s laws of planetary motion
through his own laws of motion, including the inverse-square law of gravity. Let’s
recall Kepler’s three laws.

K1: The orbit of every planet is an ellipse with the Sun at one of the two
foci.

K2: A line joining a planet and the Sun sweeps out equal areas during equal
intervals of time.

K3: The square of the orbital period of a planet is directly proportional to the
cube of the radius of its orbit.

Begin with K1, the ellipse law. Earlier we
explained that circular orbits correspond to being in equilibrium at the minimum of the
effective potential. But there are motions of the effective one-dimensional system in
which it oscillates back and forth near, but not at, the minimum. A motion of this type
would have the Earth periodically getting closer to and farther from the Sun. Meanwhile,
because it has angular momentum L, the Earth must also be moving around the
Sun. In other words the angle is increasing with time. The
resulting trajectory, in which the distance oscillates and the angular position changes,
is elliptical. Figure 5 shows just such an elliptical orbit. If you follow the orbit and
keep track only of the radial distance, the position of the Earth periodically moves in
and out as if it were oscillating in the effective potential.

Figure 5: The elliptical orbit of the Earth around
the Sun.

To prove the orbit is exactly an ellipse is a
bit difficult, and we will not prove it now.

Let’s take another look at the motion
of a particle in the effective potential. Imagine a particle with so
much energy that it would completely escape from the dip in the potential energy. In
such an orbit the particle comes in from infinity, bounces off the potential near
r = 0, and goes back out, never to return. Such orbits certainly exist;
they are called unbounded hyperbolic orbits.

Now let’s move on to K2. According to
Kepler’s second law, as the radial vector sweeps out the ellipse, the area that it
sweeps per unit time is always the same. This sounds like a conservation law, and indeed
it is—the conservation of angular momentum. Go back to Eq. (7) and divide it by the mass
m:

Imagine the radial line as it sweeps out an
area. In a small time t,
the area changes by
.

Figure 6: The area swept out by the line connecting
the Earth to the Sun in a short time t.

The small triangle swept out in Figure 6 has
area

You can check this using the fact that the area
of a triangle is one-half the base (r) times the height (r

). If we
divide by the small time interval t, we get

But now we use angular momentum conservation
in the form of Eq. (15), and we get the final equation

Since p
(and also m) do not vary, we see that the rate of
swept out area is constant, and, moreover, it is just proportional to the angular
momentum of the orbit.

Finally, we come to K3: The square of
the orbital period of a planet is directly proportional to the cube of the radius of
its orbit.

Kepler’s formulation was very general,
but we will work it out only for circular orbits. There are a number of ways we can do
this, but the simplest is just to use Newton’s law, F = m a. The
force on the orbiting Earth is just the gravitational force, whose magnitude is

On the other hand, in Lecture 2 we calculated
the acceleration of an object moving in a circular orbit,

where is the angular
velocity.

Exercise 1: Show that Eq. (17), above, is a consequence of Equations (3)
from Lecture 2.

Newton’s law becomes

We can easily solve this for 2:

The last step is to note that the period of
the orbit—the time to make one circuit—is simply related to the angular velocity.
Denoting the period by the Greek letter tau, , we have

Traditionally we would use T for the
period, but we are already using T for the kinetic energy. Putting it all
together we get

Indeed, the square of the period is
proportional to the cube of the radius.

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ISBN: 978-0-141-97623-5

LECTURE 3: DYNAMICS

[1] There is a more modern definition of
the meter in terms of the wavelength of light emitted by atoms jumping from one quantum
level to another. For our purposes the Paris meter stick will do just fine.

LECTURE 6: THE PRINCIPLE OF LEAST ACTION

[1] I use the term minimizing because, to
my knowledge, there is no verb to express making a quantity stationary. I tried
stationaryizing, stationizing, and a few others, but I eventually gave up and took the
path of least action. But remember, least action really means stationary action.