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00:00:08,000 --> 00:00:14,000
Well, today is the last day on
Laplace transform and the first

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day before we start the rest of
the term, which will be spent on

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00:00:16,000 --> 00:00:22,000
the study of systems.
I would like to spend it on one

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00:00:20,000 --> 00:00:26,000
more type of input function
which, in general,

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00:00:23,000 --> 00:00:29,000
your teachers in other courses
will expect you to have had some

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00:00:28,000 --> 00:00:34,000
acquaintance with.
It is the kind associated with

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an impulse, so an input
consisted of what is sometimes

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called a unit impulse.
Now, what's an impulse?

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It covers actually a lot of
things.

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It covers a situation where you
withdraw from a bank account.

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For example,
take half your money out of a

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bank account one day.
It also would be modeled the

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same way.
But the simplest way to

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understand it the first time
through is as an impulse,

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if you know what an impulse is.
If you have a variable force

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acting over time,
and we will assume it is acting

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along a straight line so I don't
have to worry about it being a

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vector, then the impulse,
according to physicists,

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the physical definition,
the impulse of f of t

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over some time interval.
Let's say the time interval

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running from a to b is,
by definition,

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the integral from a to b of f
of t dt.

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Actually, I am going to do the

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most horrible thing this period.
I will assume the force is

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actually a constant force.
So, in that case,

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I wouldn't even have to bother
with the integral at all.

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If f of t is a constant,
let's say capital F,

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then the impulse is --
Well, that integral is simply

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the product of the two,
the impulse over that time

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interval is simply F times b
minus a.

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Just the product of those two.
The force times the length of

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time for which it acts.
Now, that is what I want to

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calculate, want to consider in
connection with our little mass

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system.
So, once again,

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I think this is probably the
last time you'll see the little

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spring.
Let's bid a tearful farewell to

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it.
There is our little mass on

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wheels.
And let's make it an undamped

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mass.
It has an equilibrium point and

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all the other little things that
go with the picture.

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And when I apply an impulse,
what I mean is applying a

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constant force to this over a
definite time interval.

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And that is what I mean by
applying an impulse over that

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time interval.
Now, what is the picture of

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00:03:06,000 --> 00:03:12,000
such a thing?
Well, the force is only going

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to be applied,
in other words,

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I am going to push on the mass
or pull on the mass with a

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constant force.
With a little electromagnet

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here, we will assume,
there is a pile of iron filings

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or something inside there.
I turn on the electromagnet.

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It pulls with a constant force
just between time zero and time

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00:03:30,000 --> 00:03:36,000
two seconds.
And then I stop.

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That is going to change the
motion of the thing.

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First it is going to start
pulling it toward the thing.

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And then, when it lets go,
it will zoom back and there

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will be a certain motion after
that.

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What the question is,
if I want to solve that problem

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of the motion of that in terms
of the Laplace transform,

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00:03:53,000 --> 00:03:59,000
how am I going to model this
force?

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00:03:55,000 --> 00:04:01,000
Well, let's draw a picture of
it first.

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It starts here.
It is zero for t,

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let's say the force is applied
between time zero to time h.

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And then its force is turned
on, it stays constant and then

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it is turned off.
And those vertical lines

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shouldn't be there.
But, since in practice,

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it takes a tiny bit of time to
turn a force on and off.

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It is, in practice,
not unrealistic to suppose that

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there are approximately vertical
lines there.

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They are slightly slanted but
not too much.

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Now, I want it to be unit
impulse.

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This is the force access and
this is the time access.

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Since the impulse is the area
under this curve,

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if I want that to be one,
then if this is h,

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the height to which I --
In other words,

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the magnitude of the force must
be one over h in order

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that the area be one,
in order, in other words,

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that this integral be one,
the area under that curve be

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one.
So the unit impulse looks like

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that.
The narrower it is here,

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the higher it has to be that
way.

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The bigger the force must be if
you want the end result to be a

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unit impulse.
Now, to solve a problem,

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00:05:27,000 --> 00:05:33,000
a typical problem,
then, would be a spring.

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The mass is traveling on the
track.

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Let's suppose the spring
constant is one,

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so there would be a
differential equation.

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And the right-hand side would
be this f of t.

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Well, let's give it its name,
the name I gave it before.

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Remember, I called the unit box
function the thing which was one

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between zero and h and zero
everywhere else.

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00:06:00,000 --> 00:06:06,000
The notation we used for that
was u, and then it had a double

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subscript from the starting
point and the finishing point.

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So oh-- u(oh) of t.

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00:06:10,000 --> 00:06:16,000
This much represents the thing
if it only rose to the high one.

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But if it, instead,
rises to the height one over h

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in order to make that
area one, I have to multiply it

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00:06:21,000 --> 00:06:27,000
by the factor one over h.
Now, if you want to solve this

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by the Laplace transform.
In other words,

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00:06:28,000 --> 00:06:34,000
see what the motion of that
mass is as I apply this unit

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00:06:32,000 --> 00:06:38,000
impulse to it over that time
interval.

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00:06:34,000 --> 00:06:40,000
You have to take the Laplace
transform, if that is the way we

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are doing it.
Now, the left-hand side is just

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routine and would involve the
initial conditions.

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The whole interest is taking
the Laplace transform of the

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right-hand side.
And that is what I want to do

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now.
The problem is what is the

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Laplace transform of this guy?

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00:07:02,000 --> 00:07:08,000
Well, remember,
to do everything else,

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you do everything by writing in
terms of the unit step function?

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This function that we are
talking about is one over h

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times what you get by
first stepping up to one.

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That is the unit step function,
which goes up by one and tries

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to stay at one ever after.
And then, when it gets to h,

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it has got to step down.
Well, the way you make it step

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down is by subtracting off the
function, which is the unit step

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function but where the step
takes place, not at time zero

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but at time h.
In other words,

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I translate the unit step
function of course with,

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I don't think I have to draw
that picture again.

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The unit step function looks
like zing.

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00:07:47,000 --> 00:07:53,000
And if you translate it to the
right by h it looks like zing.

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And then make it negative to
subtract it off.

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And what you will get is this
box function.

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So we want to take the Laplace
transform of this thing.

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Well, let's assume,
for the sake of argument that

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you didn't remember.
Well, you had to use the

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formula at 2:00 AM this morning
and, therefore,

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you do remember it.
[LAUGHTER] So I don't have to

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recopy the formula onto the
board.

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Maybe if there is room there.
All right, let's put it up

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there.
It says that u of t minus a

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times f,
any f, so let's call it g so

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you won't confuse it with this
particular one,

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times g translated.
If you translate a function

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from t, if you translate it to
the right by a then its Laplace

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transform is e to the minus a s
times whatever the

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old Laplace transform was,
g of s.

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Multiply by an exponential on
the right.

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On the left that corresponds to
translation.

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Except you must remember to put
in that factor u for a secret

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00:09:02,000 --> 00:09:08,000
reason which I spent half of
Wednesday explaining.

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What do we have here? The
Laplace transform of u of t,

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that is easy.
That is simply one over s.

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The Laplace transform of this

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other guy we get from the
formula.

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It is basically one over s.
No, the Laplace transform of u

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00:09:21,000 --> 00:09:27,000
of t.
But because it has been

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translated to the right by h,
I have to multiply it by that

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factor e to the minus
h times s.

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That is the answer.
And, if you want to solve

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problems, this is what you would
feed into the equation.

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And you would calculate and
calculate and calculate it.

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But that is not what I want to
do now because that was

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Wednesday and this is Friday.
You have the right to expect

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something new.
Here is what I am going to do

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new.
I am going to let h go to zero.

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As h goes to zero,
this function gets narrower and

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narrower, but it also has to get
higher and higher because its

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area has to stay one.
What I am interested in,

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first of all,
is what happens to the Laplace

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transform as h goes to zero.
In other words,

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what is the limit,
as h goes to zero of --

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Well, what is that function?
One minus e to the negative hs

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divided by hs.

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Well, this is an 18.01 problem,
an ordinary calculus problem,

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but let's do it nicely.
You see, the nice way to do it

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is to make a substitution.
We will change h s to u

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because it is occurring
as a unit in both cases.

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This is going to be the same as
the limit as u goes to zero.

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I think there are too many u's

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here already.
I cannot use u,

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you cannot use t,
v is velocity,

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w is wavefunction.
There is no letter.

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00:11:13,000 --> 00:11:19,000
All right, u.
It is one minus e to the

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00:11:17,000 --> 00:11:23,000
negative u over u.

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So what is the answer?
Well, either you know the

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answer or you replace this by,
say, the first couple of terms

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of the Taylor series.
But I think most of you would

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use L'Hopital's rule,
so let's do that.

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The derivative of the top is
zero here.

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00:11:40,000 --> 00:11:46,000
The derivative by the chain
rule of e to the negative u is e

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00:11:44,000 --> 00:11:50,000
to the negative u times minus
one.

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00:11:47,000 --> 00:11:53,000
And that minus one cancels that
minus.

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So the derivative of the top is
simply e to the negative u and

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00:11:53,000 --> 00:11:59,000
the derivative of the bottom is
one.

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So, as u goes to zero,
that limit is one.

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00:12:00,000 --> 00:12:06,000
Interesting.
Let's draw a picture this way.

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00:12:03,000 --> 00:12:09,000
I will draw it schematically.
Up here is the function one

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00:12:08,000 --> 00:12:14,000
over h times u zero h of t,
our box function,

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00:12:14,000 --> 00:12:20,000
except it has the height one
over h instead of the

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00:12:19,000 --> 00:12:25,000
height one.
We have just calculated that

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00:12:23,000 --> 00:12:29,000
its Laplace transform is that
funny thing, one minus e to the

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00:12:28,000 --> 00:12:34,000
minus hs divided by hs.

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That is the top line.
All this is completely kosher,

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00:12:39,000 --> 00:12:45,000
but now I am going to let h go
to zero.

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00:12:43,000 --> 00:12:49,000
And the question is what do we
get now?

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Well, I just calculated for you
that this thing approaches one,

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has the limit one.
And now, let's fill in the

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00:12:58,000 --> 00:13:04,000
picture.
What does this thing approach?

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00:13:03,000 --> 00:13:09,000
Well, it approaches a function
which is zero everywhere.

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00:13:10,000 --> 00:13:16,000
As h approaches zero,
this green box turns into a box

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00:13:17,000 --> 00:13:23,000
which is zero everywhere except
at zero.

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And there, it is infinitely
high.

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00:13:26,000 --> 00:13:32,000
So, keep going up.

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00:13:35,000 --> 00:13:41,000
Now, of course,
that is not a function.

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00:13:38,000 --> 00:13:44,000
People call it a function but
it isn't.

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00:13:41,000 --> 00:13:47,000
Mathematicians call it a
generalized function,

208
00:13:45,000 --> 00:13:51,000
but that is not a function
either.

209
00:13:48,000 --> 00:13:54,000
It is just a way of making you
feel comfortable by talking

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00:13:53,000 --> 00:13:59,000
about something which isn't
really a function.

211
00:13:56,000 --> 00:14:02,000
It was given the name,
introduced formally into

212
00:14:00,000 --> 00:14:06,000
mathematics by a physicist,
Dirac.

213
00:14:05,000 --> 00:14:11,000
And he, looking ahead to the
future, did what many people do

214
00:14:09,000 --> 00:14:15,000
who introduce something into the
literature, a formula or a

215
00:14:13,000 --> 00:14:19,000
function or something which they
think is going to be important.

216
00:14:17,000 --> 00:14:23,000
They never name it directly
after themselves,

217
00:14:20,000 --> 00:14:26,000
but they always use as the
symbol for it the first letter

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00:14:24,000 --> 00:14:30,000
of their name.
I cannot tell you how often

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00:14:26,000 --> 00:14:32,000
that has happened.
Maybe even Euler called e for

220
00:14:31,000 --> 00:14:37,000
that reason, although he claims
it was in Latin because it has

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00:14:37,000 --> 00:14:43,000
to do with exponentials.
Well, luckily his name began

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00:14:42,000 --> 00:14:48,000
with an E, too.
That is Paul Dirac's delta

223
00:14:45,000 --> 00:14:51,000
function.
I won't dignify it by the name

224
00:14:49,000 --> 00:14:55,000
function by writing that out,
by putting the world function

225
00:14:54,000 --> 00:15:00,000
here, too, but it is called the
delta function.

226
00:15:00,000 --> 00:15:06,000
From this point on,
the entire rest of the lecture

227
00:15:03,000 --> 00:15:09,000
has a slight fictional element.
The entire rest of the lecture

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00:15:08,000 --> 00:15:14,000
is in figurative quotation
marks, so you are not entirely

229
00:15:12,000 --> 00:15:18,000
responsible for anything I say.
This is a non-function,

230
00:15:16,000 --> 00:15:22,000
but you put it in there and
call it a function.

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00:15:19,000 --> 00:15:25,000
And you naturally want to
complete, if it's a function

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00:15:23,000 --> 00:15:29,000
then it must have a Laplace
transform, even though it

233
00:15:27,000 --> 00:15:33,000
doesn't, so the diagram is
completed that way.

234
00:15:32,000 --> 00:15:38,000
And its Laplace transform is
declared to be one.

235
00:15:35,000 --> 00:15:41,000
So let's start listing the
properties of this weird thing.

236
00:15:54,000 --> 00:16:00,000
The delta function,
its Laplace transform is one.

237
00:16:05,000 --> 00:16:11,000
Now, one of the things is we
have not yet expressed the fact

238
00:16:09,000 --> 00:16:15,000
that it is a unit impulse.
In other words,

239
00:16:13,000 --> 00:16:19,000
since the areas of all of these
boxes, they all have areas one

240
00:16:17,000 --> 00:16:23,000
as they are shrunk this way they
get higher that way.

241
00:16:22,000 --> 00:16:28,000
By convention,
one says that the area under

242
00:16:25,000 --> 00:16:31,000
the orange curve also remains
one in the limit.

243
00:16:30,000 --> 00:16:36,000
Now, how am I going to express
that?

244
00:16:32,000 --> 00:16:38,000
Well, it is done by the
following formula that the

245
00:16:35,000 --> 00:16:41,000
integral, the total impulse of
the delta function should be

246
00:16:39,000 --> 00:16:45,000
one.
Now, where do I integrate?

247
00:16:41,000 --> 00:16:47,000
Well, from any place that it is
zero to any place that it is

248
00:16:45,000 --> 00:16:51,000
zero on the other side of that
vertical line.

249
00:16:48,000 --> 00:16:54,000
But, in order to avoid
controversy, people integrate

250
00:16:52,000 --> 00:16:58,000
all the way from negative
infinity to infinity since it

251
00:16:55,000 --> 00:17:01,000
doesn't hurt.
Does it?

252
00:16:57,000 --> 00:17:03,000
It is zero practically all the
time.

253
00:17:01,000 --> 00:17:07,000
This is the function whose
Laplace transform is one.

254
00:17:06,000 --> 00:17:12,000
Its integral from minus
infinity to infinity is one.

255
00:17:11,000 --> 00:17:17,000
How else can we calculate for
it?

256
00:17:15,000 --> 00:17:21,000
Well, I would like to calculate
its convolution.

257
00:17:20,000 --> 00:17:26,000
Here is f of t.
What happens if I convolute it

258
00:17:26,000 --> 00:17:32,000
with the delta function?
Well, if you go back to the

259
00:17:31,000 --> 00:17:37,000
definition of the convolution,
you know, it is that funny

260
00:17:35,000 --> 00:17:41,000
integral, you are going to do a
lot of head scratching because

261
00:17:40,000 --> 00:17:46,000
it is not really all that clear
how to integrate with the delta

262
00:17:44,000 --> 00:17:50,000
function.
Instead of doing that let's

263
00:17:46,000 --> 00:17:52,000
assume that it follows the laws
of the Laplace transform.

264
00:17:50,000 --> 00:17:56,000
In that case,
its Laplace transform would be

265
00:17:53,000 --> 00:17:59,000
what?
Well, the whole thing of a

266
00:17:55,000 --> 00:18:01,000
convolution is that the Laplace
transform of the convolution is

267
00:18:00,000 --> 00:18:06,000
the product of the two separate
Laplace transforms.

268
00:18:05,000 --> 00:18:11,000
So that is going to be F of s
times the Laplace

269
00:18:09,000 --> 00:18:15,000
transform of the delta function,
which is one.

270
00:18:13,000 --> 00:18:19,000
Now, what must this thing be?
Well, there is some ambiguity

271
00:18:18,000 --> 00:18:24,000
as to what it is for negative
values of t.

272
00:18:21,000 --> 00:18:27,000
But if we, by brute force,
decide for negative values of t

273
00:18:26,000 --> 00:18:32,000
it is going to have the value
zero, that is the way we make

274
00:18:31,000 --> 00:18:37,000
things unique.
In fact, why don't we make f of

275
00:18:35,000 --> 00:18:41,000
unique that way to start with?

276
00:18:38,000 --> 00:18:44,000
This is a function now that is
allowed to do anything it wants

277
00:18:41,000 --> 00:18:47,000
on the right-hand side of zero
starting at zero,

278
00:18:44,000 --> 00:18:50,000
but on the left-hand side of
zero it is wiped away and must

279
00:18:48,000 --> 00:18:54,000
be zero.
This is a definite thing now.

280
00:18:50,000 --> 00:18:56,000
Its convolution is this.
And the inverse Laplace

281
00:18:53,000 --> 00:18:59,000
transform is --
The answer, in other words,

282
00:18:57,000 --> 00:19:03,000
is the same thing as what u of
t f of t would be.

283
00:19:02,000 --> 00:19:08,000
It's the same thing,
F of s.

284
00:19:04,000 --> 00:19:10,000
And so, the conclusion is that
these are equal,

285
00:19:08,000 --> 00:19:14,000
since they must be unique.
They have been made unique by

286
00:19:12,000 --> 00:19:18,000
making them zero for t negative.
In other words,

287
00:19:16,000 --> 00:19:22,000
apply to a function,
well, I won't recopy it.

288
00:19:19,000 --> 00:19:25,000
But the point is that delta t,
for the convolution operation,

289
00:19:24,000 --> 00:19:30,000
is acting like an identity.
If I multiply,

290
00:19:29,000 --> 00:19:35,000
in the sense of convolution,
it is a peculiar operation.

291
00:19:33,000 --> 00:19:39,000
But algebraically,
it has a lot of the properties

292
00:19:36,000 --> 00:19:42,000
of multiplication.
It is communitive.

293
00:19:39,000 --> 00:19:45,000
It is linear in both factors.
In other words,

294
00:19:42,000 --> 00:19:48,000
it is almost anything you would
want with multiplication.

295
00:19:46,000 --> 00:19:52,000
It has an identity element,
identity function.

296
00:19:49,000 --> 00:19:55,000
And the identity function is
the Dirac delta function.

297
00:19:53,000 --> 00:19:59,000
Anything else here?
Yeah, I will throw in one more

298
00:19:57,000 --> 00:20:03,000
thing.
It would just require one more

299
00:20:01,000 --> 00:20:07,000
phony argument,
which I won't bother giving

300
00:20:04,000 --> 00:20:10,000
you, but it is not totally
implausible.

301
00:20:06,000 --> 00:20:12,000
After all, u of t,
the unit step function is not

302
00:20:11,000 --> 00:20:17,000
differentiable,
is not a differentiable

303
00:20:13,000 --> 00:20:19,000
function.
It looks like this.

304
00:20:15,000 --> 00:20:21,000
Here its derivative is zero,
here its derivative is zero,

305
00:20:19,000 --> 00:20:25,000
and in this class it is not
even defined in between.

306
00:20:23,000 --> 00:20:29,000
But, I don't care,
I will make it go straight up.

307
00:20:27,000 --> 00:20:33,000
The question is what's its
derivative?

308
00:20:31,000 --> 00:20:37,000
Well, zero here,
zero there and infinity at

309
00:20:34,000 --> 00:20:40,000
zero, so it must be the delta
function.

310
00:20:37,000 --> 00:20:43,000
That has exactly the right
properties.

311
00:20:40,000 --> 00:20:46,000
So the same people who will
tell you this will tell you that

312
00:20:44,000 --> 00:20:50,000
also.
And, in fact,

313
00:20:45,000 --> 00:20:51,000
when you use it to solve
differential equations it acts

314
00:20:50,000 --> 00:20:56,000
as if that is true.
I think I have given you an

315
00:20:53,000 --> 00:20:59,000
example on your homework.
Let me now show you a typical

316
00:20:57,000 --> 00:21:03,000
example of the way the Dirac
delta function would be used to

317
00:21:02,000 --> 00:21:08,000
solve a problem.
Let's go back to our little

318
00:21:06,000 --> 00:21:12,000
spring, since it is the easiest
thing.

319
00:21:09,000 --> 00:21:15,000
You are familiar with it from a
physical point of view,

320
00:21:13,000 --> 00:21:19,000
and it is the easiest thing to
illustrate on.

321
00:21:16,000 --> 00:21:22,000
We have our spring mass system.
Where is it?

322
00:21:20,000 --> 00:21:26,000
Is it on the board?
Up there.

323
00:21:22,000 --> 00:21:28,000
That one.
And the differential equation

324
00:21:25,000 --> 00:21:31,000
we are going to solve is y
double prime plus y

325
00:21:29,000 --> 00:21:35,000
equals --
And now, I am going to assume

326
00:21:33,000 --> 00:21:39,000
that the spring is kicked with
impulse a.

327
00:21:37,000 --> 00:21:43,000
I am not going to kick it at
time t equals zero,

328
00:21:42,000 --> 00:21:48,000
since that would get us into
slight technical difficulties.

329
00:21:47,000 --> 00:21:53,000
Anyway, it is more fun to kick
it at time pi over two.

330
00:21:52,000 --> 00:21:58,000
The thing is,

331
00:21:54,000 --> 00:22:00,000
what is happening?
Well, we have got to have

332
00:21:58,000 --> 00:22:04,000
initial conditions.
The initial conditions are

333
00:22:02,000 --> 00:22:08,000
going to be, let's start at time
zero.

334
00:22:05,000 --> 00:22:11,000
We will start it at the
position one.

335
00:22:08,000 --> 00:22:14,000
So I take my spring,
I drag it to the position one,

336
00:22:11,000 --> 00:22:17,000
I take the little mass there
and then let it go.

337
00:22:15,000 --> 00:22:21,000
And so it starts going birr.
But right when it gets to the

338
00:22:19,000 --> 00:22:25,000
equilibrium point I give it a,
"cha!" with unit impulse.

339
00:22:23,000 --> 00:22:29,000
I started it from rest.
Those will be the initial

340
00:22:27,000 --> 00:22:33,000
conditions.
And I want to say that I kicked

341
00:22:31,000 --> 00:22:37,000
it, not with unit impulse,
but with the impulse a.

342
00:22:35,000 --> 00:22:41,000
Bigger.
And I did that at time pi over

343
00:22:38,000 --> 00:22:44,000
two.
So how are we going to say

344
00:22:41,000 --> 00:22:47,000
that?
Well, kick it means delivered

345
00:22:43,000 --> 00:22:49,000
that impulse over an extremely
short time interval,

346
00:22:47,000 --> 00:22:53,000
but in such a way kicked it
sufficiently hard that the total

347
00:22:52,000 --> 00:22:58,000
impulse was a.
The way to say that is kick it

348
00:22:56,000 --> 00:23:02,000
with the Dirac delta function.
Translate it to the point time

349
00:23:02,000 --> 00:23:08,000
pi over two.
Not at zero any longer.

350
00:23:06,000 --> 00:23:12,000
t minus pi over two.

351
00:23:09,000 --> 00:23:15,000
But that would kick it with a
unit impulse.

352
00:23:12,000 --> 00:23:18,000
I want it to kick it with the
impulse a, so I will just

353
00:23:16,000 --> 00:23:22,000
multiply that by the constant
factor a.

354
00:23:20,000 --> 00:23:26,000
Let's put this over here.
y of zero equals one,

355
00:23:24,000 --> 00:23:30,000
that's the starting value.
Now we have a problem.

356
00:23:30,000 --> 00:23:36,000
The only thing new in solving
this with the Laplace transform

357
00:23:33,000 --> 00:23:39,000
is I have this funny right-hand
side.

358
00:23:36,000 --> 00:23:42,000
But it corresponds to a
physical situation.

359
00:23:38,000 --> 00:23:44,000
Let's do it.
You take the Laplace transform

360
00:23:41,000 --> 00:23:47,000
of both sides of the equation.
Remember how to do that?

361
00:23:44,000 --> 00:23:50,000
You have to take account of the
initial conditions.

362
00:23:48,000 --> 00:23:54,000
The Laplace transform of the
second derivative is you

363
00:23:51,000 --> 00:23:57,000
multiply by s squared,
and then you have to subtract.

364
00:23:55,000 --> 00:24:01,000
You have to use these initial
conditions.

365
00:23:59,000 --> 00:24:05,000
This one won't give you
anything, but the first one

366
00:24:02,000 --> 00:24:08,000
means I have to subtract one
times s.

367
00:24:05,000 --> 00:24:11,000
That is the Laplace transform
of y double prime.

368
00:24:09,000 --> 00:24:15,000
The Laplace transform of y,
of course, is just capital Y.

369
00:24:13,000 --> 00:24:19,000
And how about the Laplace

370
00:24:16,000 --> 00:24:22,000
transform of the right-hand
side.

371
00:24:18,000 --> 00:24:24,000
Well, we will have the constant
factor a because the Laplace

372
00:24:22,000 --> 00:24:28,000
transform is linear.
And now, the delta function

373
00:24:25,000 --> 00:24:31,000
would have the transform one.
But when I translate it,

374
00:24:30,000 --> 00:24:36,000
pi over two,
that means I have to use that

375
00:24:33,000 --> 00:24:39,000
formula.
Translate it by pi over two

376
00:24:35,000 --> 00:24:41,000
means take the one that it would
have been otherwise and multiply

377
00:24:40,000 --> 00:24:46,000
it by e, that exponential
factor.

378
00:24:42,000 --> 00:24:48,000
It would be e to the minus pi
over two,

379
00:24:46,000 --> 00:24:52,000
that is the A times s times
one, which would be the g of s,

380
00:24:49,000 --> 00:24:55,000
the Laplace transform
or the delta function before it

381
00:24:54,000 --> 00:25:00,000
had been translated.
But I don't have to put that in

382
00:24:57,000 --> 00:25:03,000
because it's one.
I am multiplying by one.

383
00:25:01,000 --> 00:25:07,000
And to do everything now is
routine.

384
00:25:03,000 --> 00:25:09,000
Solve for the Laplace
transform.

385
00:25:05,000 --> 00:25:11,000
Well, what is it?
It is y is equal to.

386
00:25:08,000 --> 00:25:14,000
I put the s on the other side.
That makes the right-hand side

387
00:25:12,000 --> 00:25:18,000
the sum of two terms.
And I divide by the coefficient

388
00:25:15,000 --> 00:25:21,000
of y, which is s squared plus
one.

389
00:25:18,000 --> 00:25:24,000
The s is over on the right-hand
side and it is divided by s

390
00:25:22,000 --> 00:25:28,000
squared plus one.
And the other factor is there,

391
00:25:25,000 --> 00:25:31,000
too.
And it, too,

392
00:25:26,000 --> 00:25:32,000
is divided by s squared plus
one.

393
00:25:35,000 --> 00:25:41,000
Now, we take the inverse
Laplace transform of those two

394
00:25:38,000 --> 00:25:44,000
terms and add them up.

395
00:26:00,000 --> 00:26:06,000
What will we get?
Well, y is equal to,

396
00:26:02,000 --> 00:26:08,000
the inverse Laplace transform
of s over s squared plus one is

397
00:26:07,000 --> 00:26:13,000
cosine t.

398
00:26:11,000 --> 00:26:17,000
Now, for this thing we will
have to use our formula.

399
00:26:15,000 --> 00:26:21,000
If this weren't here,
the inverse Laplace transform

400
00:26:19,000 --> 00:26:25,000
of a over s squared plus one
would

401
00:26:24,000 --> 00:26:30,000
be what?
Well, it would be a times the

402
00:26:27,000 --> 00:26:33,000
sine of t.

403
00:26:35,000 --> 00:26:41,000
In other words,
if this is the g of s

404
00:26:37,000 --> 00:26:43,000
then the function on the left
would be basically A sine t.

405
00:26:41,000 --> 00:26:47,000
But because it has been

406
00:26:43,000 --> 00:26:49,000
multiplied by that exponential
factor, e to the minus as

407
00:26:47,000 --> 00:26:53,000
where a is pi over two,

408
00:26:50,000 --> 00:26:56,000
the left-hand side has to be
changed from A sine t

409
00:26:53,000 --> 00:26:59,000
to what it would be
with the translated form.

410
00:26:58,000 --> 00:27:04,000
So the rest of it is u of t
minus pi over two,

411
00:27:01,000 --> 00:27:07,000
because a is pi over
two, times what it would have

412
00:27:05,000 --> 00:27:11,000
been just from the factor g of s
itself.

413
00:27:09,000 --> 00:27:15,000
In other words,
A times the sine of,

414
00:27:11,000 --> 00:27:17,000
again, t minus pi over two.

415
00:27:15,000 --> 00:27:21,000
I am applying that formula,
but I am applying it in that

416
00:27:19,000 --> 00:27:25,000
direction.
I started with this,

417
00:27:21,000 --> 00:27:27,000
and I want to recover the
left-hand side.

418
00:27:24,000 --> 00:27:30,000
And that is what it must look
like.

419
00:27:26,000 --> 00:27:32,000
The A, of course,
just gets dragged along for the

420
00:27:29,000 --> 00:27:35,000
free ride.
Now, as I emphasized to you

421
00:27:34,000 --> 00:27:40,000
last time, and I hope you did on
your homework that you handed

422
00:27:38,000 --> 00:27:44,000
in, you mustn't leave it in that
form.

423
00:27:41,000 --> 00:27:47,000
You have to make cases because
people will expect you to tell

424
00:27:46,000 --> 00:27:52,000
them what the meaning of this
is.

425
00:27:49,000 --> 00:27:55,000
Now, if t is less than pi over
two, this is zero.

426
00:27:52,000 --> 00:27:58,000
And, therefore,
that term does not exist.

427
00:27:56,000 --> 00:28:02,000
So the first part of it is just
the cosine t term if

428
00:28:00,000 --> 00:28:06,000
t lies between zero and pi over
two.

429
00:28:05,000 --> 00:28:11,000
If t is bigger than pi over two
then this factor is

430
00:28:10,000 --> 00:28:16,000
one.
It's the unit step function.

431
00:28:12,000 --> 00:28:18,000
And I, therefore,
must add in this term.

432
00:28:16,000 --> 00:28:22,000
Now, what is that term?
What is the sine of t minus pi

433
00:28:20,000 --> 00:28:26,000
over two?
The sine of t looks

434
00:28:25,000 --> 00:28:31,000
like that.
The sine of t,

435
00:28:27,000 --> 00:28:33,000
if I translate it,
looks like this.

436
00:28:31,000 --> 00:28:37,000
If I translate it by pi over
two.

437
00:28:33,000 --> 00:28:39,000
And let's finish it up,
the pi that was over here moved

438
00:28:38,000 --> 00:28:44,000
into position.
That curve is the curve

439
00:28:41,000 --> 00:28:47,000
negative cosine t.

440
00:28:51,000 --> 00:28:57,000
And so the answer is if t is
bigger than pi over two,

441
00:28:55,000 --> 00:29:01,000
it is cosine t minus A
times cosine t.

442
00:29:00,000 --> 00:29:06,000
Or, in other words,

443
00:29:02,000 --> 00:29:08,000
it is one minus A times cosine
t.

444
00:29:11,000 --> 00:29:17,000
Now, do those match up?
They have always got to match

445
00:29:13,000 --> 00:29:19,000
up, or you have made a mistake.
You always have to get a

446
00:29:17,000 --> 00:29:23,000
continuous function when you
have just discontinuities.

447
00:29:20,000 --> 00:29:26,000
Do we get a continuous
function?

448
00:29:22,000 --> 00:29:28,000
Yeah, when t is pi over two
the value here is

449
00:29:25,000 --> 00:29:31,000
zero.
The value of this is also zero

450
00:29:27,000 --> 00:29:33,000
at pi over two.
There is no conflict in the

451
00:29:31,000 --> 00:29:37,000
values.
Values doesn't suddenly jump.

452
00:29:33,000 --> 00:29:39,000
The function is continuous.
It is not differential but it

453
00:29:38,000 --> 00:29:44,000
is continuous.
Well, what function does that

454
00:29:41,000 --> 00:29:47,000
look like?
There are cases.

455
00:29:43,000 --> 00:29:49,000
It starts out life as the
function cosine t.

456
00:29:47,000 --> 00:29:53,000
So it gets to here.
And at t equals pi over two,

457
00:29:51,000 --> 00:29:57,000
the mass gets kicked
and that changes the function.

458
00:29:55,000 --> 00:30:01,000
Now, what are the values?
Well, if A is bigger than one

459
00:30:01,000 --> 00:30:07,000
this is a negative
number and it therefore becomes

460
00:30:07,000 --> 00:30:13,000
the function negative
cosine t.

461
00:30:11,000 --> 00:30:17,000
Now, negative cosine t looks
like this, the blue guy.

462
00:30:16,000 --> 00:30:22,000
Negative cosine t is a function
that looks like this.

463
00:30:21,000 --> 00:30:27,000
So it goes from here,
it reverses direction,

464
00:30:26,000 --> 00:30:32,000
the mass reverses direction
from what you thought it was

465
00:30:31,000 --> 00:30:37,000
going to do.
And it does that because A is

466
00:30:36,000 --> 00:30:42,000
so large that that impulse was
enough to make it reverse

467
00:30:40,000 --> 00:30:46,000
direction.
Of course it might only do

468
00:30:42,000 --> 00:30:48,000
this, but this is what will
happen if A is bigger than one.

469
00:30:47,000 --> 00:30:53,000
This will be A,
which is a lot bigger than one.

470
00:30:50,000 --> 00:30:56,000
If it's not so much bigger than
one it might look like that.

471
00:30:55,000 --> 00:31:01,000
So A is just bigger than one.
How's that?

472
00:30:59,000 --> 00:31:05,000
Well, what if A is less than
one?

473
00:31:01,000 --> 00:31:07,000
Well, in that case it stays
positive.

474
00:31:04,000 --> 00:31:10,000
If A is less than one,
this is now still a positive

475
00:31:07,000 --> 00:31:13,000
number.
And, therefore,

476
00:31:09,000 --> 00:31:15,000
the cosine continues on its
merry way.

477
00:31:12,000 --> 00:31:18,000
The only thing is it might be a
little more sluggish or it might

478
00:31:17,000 --> 00:31:23,000
be very peppy and do that.
Let's just go that far.

479
00:31:20,000 --> 00:31:26,000
This will be the case A less
than one.

480
00:31:23,000 --> 00:31:29,000
Well, of course,
the most interesting case is

481
00:31:26,000 --> 00:31:32,000
what happens if A is exactly
equal to one?

482
00:31:32,000 --> 00:31:38,000
The porridge is exactly just
right, I think that's the

483
00:31:37,000 --> 00:31:43,000
phrase.
Too hot.

484
00:31:38,000 --> 00:31:44,000
Too cold.
Just right.

485
00:31:40,000 --> 00:31:46,000
When A is equal to one,
it is zero.

486
00:31:44,000 --> 00:31:50,000
It starts out as cosine t.

487
00:31:48,000 --> 00:31:54,000
When it gets to t,
it continues on ever after as

488
00:31:52,000 --> 00:31:58,000
the function zero.
I have a visual aid for the

489
00:31:57,000 --> 00:32:03,000
only time this term.
It didn't work at all.

490
00:32:02,000 --> 00:32:08,000
I mean, on the other hand,
the last hour,

491
00:32:06,000 --> 00:32:12,000
the people who worked it were
not intrinsically baseball

492
00:32:11,000 --> 00:32:17,000
players, so we will use the
equation of the pendulum

493
00:32:15,000 --> 00:32:21,000
instead.
That is a lot easier than mass

494
00:32:18,000 --> 00:32:24,000
spring.
This is a pendulum.

495
00:32:20,000 --> 00:32:26,000
It is undamped because I
declare it to be and it swings

496
00:32:25,000 --> 00:32:31,000
back and forth.
And here I am releasing it.

497
00:32:30,000 --> 00:32:36,000
The variable is not x or y but
theta, the angle through.

498
00:32:34,000 --> 00:32:40,000
Here theta is one,
let's say.

499
00:32:37,000 --> 00:32:43,000
That's about one radian.
It starts there and swings back

500
00:32:41,000 --> 00:32:47,000
and forth.
It is not damped,

501
00:32:44,000 --> 00:32:50,000
so it never loses amplitude,
particularly if I swish it,

502
00:32:48,000 --> 00:32:54,000
if I move my hand a little bit.
I want someone who knows how to

503
00:32:53,000 --> 00:32:59,000
bat a baseball.
That was the problem last hour.

504
00:32:57,000 --> 00:33:03,000
Two people.
One to release it.

505
00:33:01,000 --> 00:33:07,000
I will stand up and try to hold
it here.

506
00:33:04,000 --> 00:33:10,000
Somebody releases it.
And then somebody who has to be

507
00:33:08,000 --> 00:33:14,000
very skillful should apply a
unit impulse of exactly one when

508
00:33:13,000 --> 00:33:19,000
it gets to the equilibrium
point.

509
00:33:16,000 --> 00:33:22,000
So who can do that?
Who can play baseball here?

510
00:33:20,000 --> 00:33:26,000
Come on.
Somebody elected?

511
00:33:30,000 --> 00:33:36,000
All right. Come on. [APPLAUSE]

512
00:33:41,000 --> 00:33:47,000
Somebody release it,
too.

513
00:33:44,000 --> 00:33:50,000
Somebody tall to handle it all.
I think that will be me.

514
00:33:52,000 --> 00:33:58,000
Just hold it at what you would
take to be one radian.

515
00:34:00,000 --> 00:34:06,000
He releases it.
When it gets to the bottom,

516
00:34:04,000 --> 00:34:10,000
you will have to get way down,
and maybe on this side.

517
00:34:11,000 --> 00:34:17,000
Are you a lefty or a righty?
Rightly.

518
00:34:15,000 --> 00:34:21,000
Okay.
Bat it what part.

519
00:34:17,000 --> 00:34:23,000
Give it a good swat.
I will stand up higher.

520
00:34:22,000 --> 00:34:28,000
Help.
I'm not very stable.

521
00:34:25,000 --> 00:34:31,000
[APPLAUSE] A trial run.
Again.

522
00:34:30,000 --> 00:34:36,000
Okay.
A little further out.

523
00:34:32,000 --> 00:34:38,000
First of all,
you have to see where it's

524
00:34:36,000 --> 00:34:42,000
going.
Why don't you stand,

525
00:34:38,000 --> 00:34:44,000
oh, you bat rightly.
That's right.

526
00:34:41,000 --> 00:34:47,000
Okay.
Let's try it again.

527
00:34:59,000 --> 00:35:05,000
Strike one.
It's okay.
It's the beginning of the
baseball season.
One more.
The Red Sox are having trouble,
too.
Not bad. [APPLAUSE]

528
00:35:13,000 --> 00:35:19,000
If he had hit even harder it
would have reversed direction

529
00:35:16,000 --> 00:35:22,000
and gone that way.
If you hadn't hit it quite as

530
00:35:20,000 --> 00:35:26,000
hard it would have continued on,
still at cosine t,

531
00:35:24,000 --> 00:35:30,000
but with less amplitude.
But if you hit it exactly right

532
00:35:28,000 --> 00:35:34,000
--
It is fun to try to do.

533
00:35:31,000 --> 00:35:37,000
Toomre in our department is a
master at this,

534
00:35:36,000 --> 00:35:42,000
but he has been practicing for
years.

535
00:35:40,000 --> 00:35:46,000
He can take a little mallet and
go blunk, and it stops

536
00:35:45,000 --> 00:35:51,000
absolutely dead.
It is unbelievable.

537
00:35:49,000 --> 00:35:55,000
I should have had him give the
lecture.

538
00:35:53,000 --> 00:35:59,000
Now, I would like to do
something truly serious.

539
00:36:00,000 --> 00:36:06,000
Here, I guess.
Because there is a certain

540
00:36:03,000 --> 00:36:09,000
amount of engineering lingo you
have to learn.

541
00:36:07,000 --> 00:36:13,000
It is used by almost everybody.
Not architects and biologists

542
00:36:12,000 --> 00:36:18,000
probably quite yet,
but anybody that uses the

543
00:36:16,000 --> 00:36:22,000
Laplace transform will use these
words in connection with it.

544
00:36:21,000 --> 00:36:27,000
I really think,
since it is such a widespread

545
00:36:25,000 --> 00:36:31,000
technique, that these are things
you should know.

546
00:36:31,000 --> 00:36:37,000
Anyway, it will be easy.
It is just the enrichment of

547
00:36:34,000 --> 00:36:40,000
your vocabulary.
It is always fun to learn new

548
00:36:37,000 --> 00:36:43,000
vocabulary words.
So, let's just consider a

549
00:36:40,000 --> 00:36:46,000
general second order equation.
By the way, all this applies to

550
00:36:45,000 --> 00:36:51,000
higher order equations,
too.

551
00:36:47,000 --> 00:36:53,000
It applies to systems.
The same words are used,

552
00:36:50,000 --> 00:36:56,000
but let's use something that
you know.

553
00:36:52,000 --> 00:36:58,000
Here is a system.
It could be a spring mass

554
00:36:55,000 --> 00:37:01,000
dashpot system.
It could be an RLC circuit.

555
00:37:00,000 --> 00:37:06,000
Or that pendulum,
a damped pendulum,

556
00:37:02,000 --> 00:37:08,000
anything that is modeled by
that differential equation with

557
00:37:06,000 --> 00:37:12,000
constant coefficients,
second-order.

558
00:37:08,000 --> 00:37:14,000
This is the input.
The input can be any kind of a

559
00:37:11,000 --> 00:37:17,000
function.
Exponential functions,

560
00:37:13,000 --> 00:37:19,000
sine, cosine.
It could be a Dirac delta

561
00:37:16,000 --> 00:37:22,000
function.
It could be a sum of these

562
00:37:18,000 --> 00:37:24,000
things.
It could be a Fourier series.

563
00:37:21,000 --> 00:37:27,000
Anything of the sort of stuff
we have been talking about

564
00:37:24,000 --> 00:37:30,000
throughout the last few weeks.
And let's have simple initial

565
00:37:30,000 --> 00:37:36,000
conditions so that doesn't louse
things up, the simplest possible

566
00:37:34,000 --> 00:37:40,000
ones.
The mass starts at the

567
00:37:36,000 --> 00:37:42,000
equilibrium point from rest.
Of course, it doesn't stay that

568
00:37:40,000 --> 00:37:46,000
way because there is an input
that is asking it to move along.

569
00:37:45,000 --> 00:37:51,000
Now all I want to do is solve
this in general with a Laplace

570
00:37:49,000 --> 00:37:55,000
transform.
If I do it in general,

571
00:37:51,000 --> 00:37:57,000
that is always easier than
doing it in particular since you

572
00:37:56,000 --> 00:38:02,000
don't ever have to do any
calculations.

573
00:38:00,000 --> 00:38:06,000
It is s squared Y.
There are no other terms here

574
00:38:05,000 --> 00:38:11,000
because the initial conditions
are zero.

575
00:38:08,000 --> 00:38:14,000
This part will be a times s Y.

576
00:38:12,000 --> 00:38:18,000
Again, no other terms because
the initial conditions are zero.

577
00:38:17,000 --> 00:38:23,000
Plus b times Y.
And all that is equal to

578
00:38:21,000 --> 00:38:27,000
whatever the Laplace transform
is of the right-hand side.

579
00:38:26,000 --> 00:38:32,000
So it is F of s.
Next step.

580
00:38:31,000 --> 00:38:37,000
Boy, this is an easy problem.
You solve for Y.

581
00:38:35,000 --> 00:38:41,000
Well, Y is F of s times one
over s squared plus as plus b.

582
00:38:45,000 --> 00:38:51,000
Now, what is that?
The next step now is to figure

583
00:38:50,000 --> 00:38:56,000
out what the answer to the
problem is, what's the Y of t?

584
00:38:55,000 --> 00:39:01,000
Well, you do that by taking the

585
00:39:00,000 --> 00:39:06,000
inverse Laplace transform.
But because these are general

586
00:39:04,000 --> 00:39:10,000
functions, I don't have to write
down any specific answer.

587
00:39:09,000 --> 00:39:15,000
The only thing is to use the
convolution because this is the

588
00:39:13,000 --> 00:39:19,000
product of two functions of s.
The inverse transform will be

589
00:39:18,000 --> 00:39:24,000
the convolution of their
respective things.

590
00:39:21,000 --> 00:39:27,000
The answer is going to be the
convolution of F of t,

591
00:39:26,000 --> 00:39:32,000
the input function in other
words, convoluted with the

592
00:39:30,000 --> 00:39:36,000
inverse Laplace transform of
that thing.

593
00:39:35,000 --> 00:39:41,000
Now, we have to have a name for
that, and those are the two

594
00:39:39,000 --> 00:39:45,000
words I want to introduce you to
because they are used

595
00:39:43,000 --> 00:39:49,000
everywhere.
The function,

596
00:39:44,000 --> 00:39:50,000
on the right-hand side,
this function one over s

597
00:39:48,000 --> 00:39:54,000
squared plus as plus b,

598
00:39:51,000 --> 00:39:57,000
notice it only depends upon the
left-hand side of the

599
00:39:55,000 --> 00:40:01,000
differential equation,
on the damping constant.

600
00:40:00,000 --> 00:40:06,000
The spring constant if you are
thinking of a mass spring

601
00:40:03,000 --> 00:40:09,000
dashpot system.
So this depends only on the

602
00:40:06,000 --> 00:40:12,000
system, not on what input is
going into it.

603
00:40:08,000 --> 00:40:14,000
And it is called the transfer
function.

604
00:40:11,000 --> 00:40:17,000
Is usually called capital W of,
sometimes it is

605
00:40:15,000 --> 00:40:21,000
capital H of s,
there are different things,

606
00:40:18,000 --> 00:40:24,000
but it is always called the
transfer function.

607
00:40:27,000 --> 00:40:33,000
What we are interested in
putting here its inverse Laplace

608
00:40:31,000 --> 00:40:37,000
transform.
Well, I will call that W of t

609
00:40:34,000 --> 00:40:40,000
to go with the capital
W of s by the usual

610
00:40:39,000 --> 00:40:45,000
notation.
Its inverse Laplace transform,

611
00:40:42,000 --> 00:40:48,000
well, I cannot calculate that.
I will just give it a name,

612
00:40:46,000 --> 00:40:52,000
W of t.
And that is called the weight

613
00:40:49,000 --> 00:40:55,000
function of the system.
This is the transfer function

614
00:40:53,000 --> 00:40:59,000
of the system,
so put in "of the system" if

615
00:40:57,000 --> 00:41:03,000
you are taking notes.
And so the answer is that

616
00:41:02,000 --> 00:41:08,000
always the solution is the
convolution to this differential

617
00:41:06,000 --> 00:41:12,000
equation that we have been
solving for the last three or

618
00:41:11,000 --> 00:41:17,000
four weeks.
It is the convolution of that.

619
00:41:14,000 --> 00:41:20,000
And, therefore,
the solution is expressed as a

620
00:41:18,000 --> 00:41:24,000
definite integral of the
function of the input on the

621
00:41:22,000 --> 00:41:28,000
right-hand side,
what is forcing the equation,

622
00:41:26,000 --> 00:41:32,000
times this magic function but
flipped and translated by t.

623
00:41:32,000 --> 00:41:38,000
That says du for you guys over
there.

624
00:41:34,000 --> 00:41:40,000
In other words,
the solution to the

625
00:41:37,000 --> 00:41:43,000
differential equation is
presented as a definite

626
00:41:41,000 --> 00:41:47,000
integral.
Marvelous.

627
00:41:42,000 --> 00:41:48,000
And the only thing is the
definite integral involves this

628
00:41:47,000 --> 00:41:53,000
funny function W of t.
To understand why that is the

629
00:41:52,000 --> 00:41:58,000
solution, you have to understand
what W of t is.

630
00:41:55,000 --> 00:42:01,000
Well, formally,
of course, it's that.

631
00:42:00,000 --> 00:42:06,000
But what does it really mean?
The problem is what is W of t

632
00:42:05,000 --> 00:42:11,000
really? Not just formally,

633
00:42:08,000 --> 00:42:14,000
but what does it really mean?
I mean, is it real?

634
00:42:12,000 --> 00:42:18,000
I think the simplest way of
thinking of it,

635
00:42:16,000 --> 00:42:22,000
once you know about the delta
function is just to think of

636
00:42:21,000 --> 00:42:27,000
this differential equation y
double prime plus a y prime plus

637
00:42:27,000 --> 00:42:33,000
b. Except use as the input the

638
00:42:32,000 --> 00:42:38,000
Dirac delta function.
In other words,

639
00:42:35,000 --> 00:42:41,000
we are kicking the mass.
The mass starts at rest,

640
00:42:39,000 --> 00:42:45,000
so the initial conditions are
going to be what they were

641
00:42:43,000 --> 00:42:49,000
before. y of zero,

642
00:42:46,000 --> 00:42:52,000
y prime of zero. Both zero.

643
00:42:49,000 --> 00:42:55,000
The mass starts at rest from
the equilibrium position,

644
00:42:53,000 --> 00:42:59,000
and it is kicked in the
positive direction,

645
00:42:57,000 --> 00:43:03,000
I guess that's this way,
with unit impulse.

646
00:43:02,000 --> 00:43:08,000
At time zero with unit impulse.
In other words,

647
00:43:05,000 --> 00:43:11,000
kick it just hard enough so you
impart a unit impulse.

648
00:43:10,000 --> 00:43:16,000
So that situation is modeled by
this differential equation.

649
00:43:15,000 --> 00:43:21,000
The kick at time zero is
modeled by this input,

650
00:43:19,000 --> 00:43:25,000
the Dirac delta function.
And now, what happens if I

651
00:43:23,000 --> 00:43:29,000
solve it?
Well, you see,

652
00:43:25,000 --> 00:43:31,000
everything in the solution is
the same.

653
00:43:30,000 --> 00:43:36,000
The left stays the same,
but on the right-hand side I

654
00:43:34,000 --> 00:43:40,000
should have not f of s here.

655
00:43:37,000 --> 00:43:43,000
Since this is the delta
function, I should have one.

656
00:43:42,000 --> 00:43:48,000
What I get is,
on the left-hand side,

657
00:43:45,000 --> 00:43:51,000
s squared Y plus as Y plus bY
equals,

658
00:43:50,000 --> 00:43:56,000
for the Laplace transform of
the right-hand side is simply

659
00:43:56,000 --> 00:44:02,000
one.
And, therefore,

660
00:43:57,000 --> 00:44:03,000
Y is what?
Y is one over exactly the

661
00:44:02,000 --> 00:44:08,000
transform function.
And therefore its inverse

662
00:44:05,000 --> 00:44:11,000
Laplace transform is that weight
function.

663
00:44:09,000 --> 00:44:15,000
That is the simplest
interpretation I know of what

664
00:44:13,000 --> 00:44:19,000
this magic weight function is,
which gives the solution to all

665
00:44:18,000 --> 00:44:24,000
the differential equations,
no matter what the input is.

666
00:44:23,000 --> 00:44:29,000
The weight function is the
response of the system at rest

667
00:44:27,000 --> 00:44:33,000
to a sharp kick at time zero
with unit impulse.

668
00:44:33,000 --> 00:44:39,000
And read the notes because they
will explain to you why this

669
00:44:37,000 --> 00:44:43,000
could be thought of as the
superposition of a lot of sharp

670
00:44:42,000 --> 00:44:48,000
kicks times zero a little later.
Kick, kick, kick,

671
00:44:46,000 --> 00:44:52,000
kick.
And that's what makes the

672
00:44:49,000 --> 00:44:55,000
solution.
Next time we start systems.