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YEN-JIE LEE: So
welcome, everybody.
00:00:26.370 --> 00:00:27.740
My name is Yen-Jie Lee.
00:00:27.740 --> 00:00:32.210
I am a assistant professor
of physics in the physics
00:00:32.210 --> 00:00:36.500
department, and I will
be your instructor
00:00:36.500 --> 00:00:39.860
of this semester on 8.03.
00:00:39.860 --> 00:00:44.930
So of course, one first
question you have is,
00:00:44.930 --> 00:00:50.240
why do we want to learn
about vibrations and waves?
00:00:50.240 --> 00:00:52.460
Why do we learn about this?
00:00:52.460 --> 00:00:55.550
Why do we even care?
00:00:55.550 --> 00:00:58.350
The answer is really, simple.
00:00:58.350 --> 00:01:00.920
If you look at
this slide, you can
00:01:00.920 --> 00:01:05.720
see that the reason you
can follow this class
00:01:05.720 --> 00:01:11.270
is because I'm producing sound
wave by oscillating the air,
00:01:11.270 --> 00:01:14.180
and you can receive
those sound waves.
00:01:14.180 --> 00:01:17.120
And you can see me--
00:01:17.120 --> 00:01:19.790
that's really pretty
amazing by itself--
00:01:19.790 --> 00:01:22.880
because there are
a lot of photons
00:01:22.880 --> 00:01:25.580
or electromagnetic waves.
00:01:25.580 --> 00:01:27.700
They are bouncing
around in this room,
00:01:27.700 --> 00:01:34.790
and your eye actually receive
those electromagnetic waves.
00:01:34.790 --> 00:01:37.430
And that translates
into your brain waves.
00:01:37.430 --> 00:01:41.075
You obviously, start to think
about what this instructor is
00:01:41.075 --> 00:01:43.400
trying to tell you.
00:01:43.400 --> 00:01:47.030
And of course, all those
things we learned from 8.03
00:01:47.030 --> 00:01:50.870
is closely connected
to probability density
00:01:50.870 --> 00:01:57.500
waves, which you will learn
from 8.04, quantum physics.
00:01:57.500 --> 00:01:59.660
And finally, it's
also, of course,
00:01:59.660 --> 00:02:05.030
related to a recent discovery
of the gravitational waves.
00:02:05.030 --> 00:02:08.050
When we are sitting
here, maybe there
00:02:08.050 --> 00:02:12.800
are already some space-time
distortion already passing
00:02:12.800 --> 00:02:16.370
through our body and
you don't feel it.
00:02:16.370 --> 00:02:18.550
When I'm moving
around like this,
00:02:18.550 --> 00:02:22.370
I am creating also the
gravitational waves,
00:02:22.370 --> 00:02:25.430
but it's so small
to be detected.
00:02:25.430 --> 00:02:26.960
So that's actually really cool.
00:02:26.960 --> 00:02:32.570
So the take-home message is
that we cannot even recognize
00:02:32.570 --> 00:02:37.310
the universe without using
waves and the vibrations.
00:02:37.310 --> 00:02:40.810
So that's actually why we
care about this subject.
00:02:40.810 --> 00:02:42.575
And the last is actually
why this subject
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is so cool even without quantum,
without any fancy names.
00:02:49.170 --> 00:02:55.420
So what is actually the
relation of 8.03 to other class
00:02:55.420 --> 00:02:57.280
or other field of studies?
00:02:57.280 --> 00:03:00.410
It's closely related to
classical mechanics, which
00:03:00.410 --> 00:03:03.050
I will use it
immediately, and I hope
00:03:03.050 --> 00:03:05.060
you will still
remember what you have
00:03:05.060 --> 00:03:07.480
learned from 8.01 and 8.02.
00:03:07.480 --> 00:03:13.310
Electromagnetic force is
actually closely related also,
00:03:13.310 --> 00:03:16.640
and we are going to use
a technique we learned
00:03:16.640 --> 00:03:21.270
from this class to understand
optics, quantum mechanics,
00:03:21.270 --> 00:03:24.800
and also there are many
practical applications, which
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you will learn from this class.
00:03:29.200 --> 00:03:31.390
This is the concrete goal.
00:03:31.390 --> 00:03:36.100
We care about the future
of our space time.
00:03:36.100 --> 00:03:40.450
We would like to predict
what is going to happen
00:03:40.450 --> 00:03:42.790
when we set up an experiment.
00:03:42.790 --> 00:03:47.290
We would like to design
experiments which can improve
00:03:47.290 --> 00:03:50.110
our understanding of nature.
00:03:50.110 --> 00:03:53.850
But without using the
most powerful tool
00:03:53.850 --> 00:03:58.030
is very, very difficult
to make progress.
00:03:58.030 --> 00:04:03.900
So the most powerful tool
we have is mathematics.
00:04:03.900 --> 00:04:07.840
You will see that it
really works in this class.
00:04:07.840 --> 00:04:09.610
But the first thing
we have to learn
00:04:09.610 --> 00:04:15.610
is how to translate physical
situations into mathematics
00:04:15.610 --> 00:04:19.959
so that we can actually include
this really wonderful tool
00:04:19.959 --> 00:04:22.960
to help us to solve problems.
00:04:22.960 --> 00:04:25.180
Once we have done
that, we will start
00:04:25.180 --> 00:04:29.180
to look at single
harmonic oscillator,
00:04:29.180 --> 00:04:32.590
then we try to couple all
those oscillators together
00:04:32.590 --> 00:04:36.590
to see how they interact
with each other.
00:04:36.590 --> 00:04:41.860
Finally, we go to an infinite
number of oscillators.
00:04:41.860 --> 00:04:46.080
Sounds scary, but it's
actually not scary after all.
00:04:46.080 --> 00:04:49.810
And we will see waves
because waves are actually
00:04:49.810 --> 00:04:52.990
coming from an infinite number
of oscillating particles,
00:04:52.990 --> 00:04:55.360
if you think about it.
00:04:55.360 --> 00:04:57.880
Then we would do Fourier
decomposition of waves
00:04:57.880 --> 00:04:59.890
to see what we can
learn about it.
00:04:59.890 --> 00:05:04.390
We learn how to put
together physical systems.
00:05:04.390 --> 00:05:08.800
That brings us to the issue
of boundary conditions,
00:05:08.800 --> 00:05:12.520
and we will also enjoy
what we have learned
00:05:12.520 --> 00:05:14.620
by looking at the
phenomenon related
00:05:14.620 --> 00:05:18.610
to electromagnetic waves
and practical application
00:05:18.610 --> 00:05:21.530
and optics.
00:05:21.530 --> 00:05:22.240
Any questions?
00:05:22.240 --> 00:05:26.080
If you have any questions,
please stop me any time.
00:05:26.080 --> 00:05:29.040
So if you don't stop me, I'm
going to continue talking.
00:05:31.600 --> 00:05:33.950
So that gets started.
00:05:33.950 --> 00:05:37.090
So the first example,
the concrete example
00:05:37.090 --> 00:05:43.570
I'm going to talk about is a
spring block, a massive block
00:05:43.570 --> 00:05:45.520
system.
00:05:45.520 --> 00:05:50.150
So this is actually what
I have on that table.
00:05:50.150 --> 00:05:54.740
So basically, I have a
highly-idealized spring.
00:05:54.740 --> 00:05:58.480
This is ideal spring
with spring constant, k,
00:05:58.480 --> 00:06:01.480
and the natural length L0.
00:06:01.480 --> 00:06:04.150
So that is actually what I have.
00:06:04.150 --> 00:06:13.830
And at t equal to 0, what I am
going to do is I am going to--
00:06:13.830 --> 00:06:16.480
I should remove this
mass a little bit,
00:06:16.480 --> 00:06:22.900
and I hold this mass still and
release that really carefully.
00:06:22.900 --> 00:06:24.910
So that is actually
the experiment,
00:06:24.910 --> 00:06:27.070
which I am going to do.
00:06:27.070 --> 00:06:33.390
And we were wondering what
is going to happen afterward.
00:06:33.390 --> 00:06:36.940
Well, the mass as you
move, will it stay there
00:06:36.940 --> 00:06:40.790
or it just disappear,
I don't know
00:06:40.790 --> 00:06:42.220
before I solved this question.
00:06:44.890 --> 00:06:49.690
Now I have put together a
concrete question to you,
00:06:49.690 --> 00:06:52.900
but I don't know how
to proceed because you
00:06:52.900 --> 00:06:54.040
say everything works.
00:06:54.040 --> 00:06:55.190
What I am going to do?
00:06:55.190 --> 00:06:58.540
I mean, I don't know.
00:06:58.540 --> 00:07:03.160
So as I mentioned before, there
is a pretty powerful tool,
00:07:03.160 --> 00:07:04.780
mathematics.
00:07:04.780 --> 00:07:08.140
So I'm going to use
that, even though I don't
00:07:08.140 --> 00:07:11.587
know why mathematics can work.
00:07:11.587 --> 00:07:12.670
Have you thought about it?
00:07:15.340 --> 00:07:19.180
So let's try it and see
how we can make progress.
00:07:19.180 --> 00:07:25.540
So the first thing which you
can do in order to make progress
00:07:25.540 --> 00:07:29.170
is to define a
coordinate system.
00:07:29.170 --> 00:07:32.950
So here I define a
coordinate system, which
00:07:32.950 --> 00:07:34.630
is in the horizontal direction.
00:07:34.630 --> 00:07:36.580
It's the x direction.
00:07:36.580 --> 00:07:39.640
And the x equal
to 0, the origin,
00:07:39.640 --> 00:07:44.800
is the place which the
spring is not stressed,
00:07:44.800 --> 00:07:47.330
is at its natural length.
00:07:47.330 --> 00:07:51.490
That is actually what I
define as x equal to 0.
00:07:51.490 --> 00:07:56.020
And once I define
this, I can now
00:07:56.020 --> 00:08:01.720
express what is actually the
initial position of the mass
00:08:01.720 --> 00:08:03.990
by these coordinates is x0.
00:08:03.990 --> 00:08:08.020
It can be expressed
as x initial.
00:08:08.020 --> 00:08:11.830
And also, initially, I said
that this mass is not moving.
00:08:11.830 --> 00:08:16.810
Therefore, the
velocity at 0 is 0.
00:08:16.810 --> 00:08:20.000
So now I can also formulate
my question really concretely
00:08:20.000 --> 00:08:21.460
with some mathematics.
00:08:21.460 --> 00:08:25.330
Basically, you can see
that at time equal to t,
00:08:25.330 --> 00:08:28.690
I was wondering
where is this mass.
00:08:28.690 --> 00:08:31.060
So actually, the question
is, what is actually
00:08:31.060 --> 00:08:34.210
x as a function of t?
00:08:34.210 --> 00:08:38.150
So you can see that once I have
the mathematics to help me,
00:08:38.150 --> 00:08:41.490
everything becomes
pretty simple.
00:08:41.490 --> 00:08:44.770
So once I have those
defined, I would
00:08:44.770 --> 00:08:48.460
like to predict what is going
to happen at time equal to t.
00:08:48.460 --> 00:08:54.370
Therefore, I would like to make
use physical laws to actually
00:08:54.370 --> 00:08:56.540
help me to solve this problem.
00:08:56.540 --> 00:09:01.200
So apparently what we are
going to use is Newton's law.
00:09:01.200 --> 00:09:05.700
And I am going to go
through this example
00:09:05.700 --> 00:09:10.100
really slowly so that
everybody is on the same page.
00:09:10.100 --> 00:09:12.300
So the first thing
which I usually do
00:09:12.300 --> 00:09:18.610
is now I would like to do
a force diagram analysis.
00:09:18.610 --> 00:09:19.930
So I have this mass.
00:09:19.930 --> 00:09:25.080
This setup is on Earth,
and the question is,
00:09:25.080 --> 00:09:28.240
how many forces are
acting on this mass?
00:09:28.240 --> 00:09:32.185
Can anybody answer my question.
00:09:32.185 --> 00:09:33.151
AUDIENCE: Two.
00:09:33.151 --> 00:09:35.566
We got the--
00:09:35.566 --> 00:09:38.685
So acceleration and
the spring force.
00:09:38.685 --> 00:09:41.150
YEN-JIE LEE: OK, so
your answer is two.
00:09:41.150 --> 00:09:43.210
Any different?
00:09:43.210 --> 00:09:44.850
Three.
00:09:44.850 --> 00:09:45.450
Very good.
00:09:45.450 --> 00:09:48.120
So we have two and three.
00:09:48.120 --> 00:09:52.750
And the answer
actually is three.
00:09:52.750 --> 00:09:54.260
So look at this scene.
00:09:54.260 --> 00:09:57.120
I am drawing in and
I have product here.
00:09:57.120 --> 00:10:00.105
So this is actually the most
difficult part of the question,
00:10:00.105 --> 00:10:01.220
actually.
00:10:01.220 --> 00:10:05.140
So once you pass this step,
everything is straightforward.
00:10:05.140 --> 00:10:06.660
It's just mathematics.
00:10:06.660 --> 00:10:09.150
It's not my problem any more,
but the math department,
00:10:09.150 --> 00:10:12.300
they have problem, OK?
00:10:12.300 --> 00:10:13.320
All right.
00:10:13.320 --> 00:10:16.952
So now let's look at this mass.
00:10:16.952 --> 00:10:17.910
There are three forces.
00:10:17.910 --> 00:10:21.915
The first one as you mentioned
correctly is F spring.
00:10:21.915 --> 00:10:26.190
It's pulling the mass.
00:10:26.190 --> 00:10:31.020
And since we are
working on Earth,
00:10:31.020 --> 00:10:33.240
we have not yet
moved the whole class
00:10:33.240 --> 00:10:36.810
to the moon or somewhere
else, but there
00:10:36.810 --> 00:10:39.945
would be gravitational
force pointing downward.
00:10:42.720 --> 00:10:49.440
But this whole setup is on a
table of friction, this table.
00:10:49.440 --> 00:10:51.555
Therefore, there will
be no more force.
00:10:54.708 --> 00:10:56.369
So don't forget this one.
00:10:56.369 --> 00:10:57.535
There will be no more force.
00:11:00.080 --> 00:11:02.995
So the answer is that
we have three forces.
00:11:05.740 --> 00:11:08.265
The normal force is, actually,
a complicated subject,
00:11:08.265 --> 00:11:09.640
which you will
need to understand
00:11:09.640 --> 00:11:11.440
that will quantum physics.
00:11:15.160 --> 00:11:18.790
So now I have three force,
and now I can actually
00:11:18.790 --> 00:11:24.670
calculate the total
force, the total force,
00:11:24.670 --> 00:11:35.510
F. F is equal to
Fs plus Fn plus Fg.
00:11:38.830 --> 00:11:43.000
So since we know
that the mass is
00:11:43.000 --> 00:11:45.310
moving in the
horizontal direction,
00:11:45.310 --> 00:11:49.900
the mass didn't suddenly
jump and disappear.
00:11:49.900 --> 00:11:51.760
So it is there.
00:11:51.760 --> 00:11:59.020
Therefore, we know that the
normal force is actually
00:11:59.020 --> 00:12:07.690
equal to minus Fg, which is
actually Ng in the y direction.
00:12:07.690 --> 00:12:13.480
And here I define y is
actually pointing up,
00:12:13.480 --> 00:12:16.660
and the x is pointing
to the right-hand side.
00:12:16.660 --> 00:12:18.760
Therefore, what
is going to happen
00:12:18.760 --> 00:12:22.730
is that the total force
is actually just Fs.
00:12:27.740 --> 00:12:31.660
And this is equal
to minus k, which
00:12:31.660 --> 00:12:35.290
is the spring
constant and x, which
00:12:35.290 --> 00:12:43.230
is the position of the little
mass at time equal to t.
00:12:46.350 --> 00:12:55.670
So once we have those
forces and the total force,
00:12:55.670 --> 00:12:57.770
actually, we can
use Newton's law.
00:13:03.674 --> 00:13:09.240
So F is equal to m times a.
00:13:09.240 --> 00:13:16.760
And this is actually equal
to m d squared xt dt squared
00:13:16.760 --> 00:13:20.060
in the x direction,
and that is actually
00:13:20.060 --> 00:13:26.880
equal to mx double dot t x.
00:13:26.880 --> 00:13:28.190
So here is my notation.
00:13:28.190 --> 00:13:32.810
I'm going to use each of the dot
is actually the differentiation
00:13:32.810 --> 00:13:34.630
with respect to t.
00:13:37.760 --> 00:13:45.600
So this is actually equal to
minus kxt in the x direction.
00:13:45.600 --> 00:13:49.730
So you can see that here is
actually what you already
00:13:49.730 --> 00:13:51.330
know about Newton's law.
00:13:51.330 --> 00:13:55.250
And that is actually coming
from the force analysis.
00:13:55.250 --> 00:13:58.550
So in this example,
it's simple enough such
00:13:58.550 --> 00:14:00.830
that you can write
it down immediately,
00:14:00.830 --> 00:14:03.110
but in the later
examples, things
00:14:03.110 --> 00:14:06.740
will become very complicated
and things will be slightly more
00:14:06.740 --> 00:14:08.960
difficult. Therefore,
you will really need help
00:14:08.960 --> 00:14:12.080
from the force diagram.
00:14:12.080 --> 00:14:15.320
So now we have everything
in the x direction,
00:14:15.320 --> 00:14:18.140
therefore, I can drop the x hat.
00:14:18.140 --> 00:14:25.010
Therefore, finally, my equation
of motion is x double dot t.
00:14:25.010 --> 00:14:32.720
And this is equal to
minus k over n x of t.
00:14:32.720 --> 00:14:37.280
To make my life
easier, I am going
00:14:37.280 --> 00:14:43.460
to define omega equal to
square root of k over n.
00:14:43.460 --> 00:14:45.390
You will see why afterward.
00:14:45.390 --> 00:14:48.810
It looks really weird why
professor Lee wants to do this,
00:14:48.810 --> 00:14:53.680
but afterward, you will see that
omega really have a meaning,
00:14:53.680 --> 00:14:58.000
and that is equal to
minus omega squared x.
00:15:00.710 --> 00:15:06.110
So we have solved this problem,
actually, as a physicist.
00:15:06.110 --> 00:15:08.420
Now the problem is
what is actually
00:15:08.420 --> 00:15:11.180
the solution to
this differential
00:15:11.180 --> 00:15:13.440
second-order
differential equation.
00:15:13.440 --> 00:15:15.570
And as I mentioned,
this is actually
00:15:15.570 --> 00:15:17.530
not the content
of 8.03, actually,
00:15:17.530 --> 00:15:20.930
it's a content of 18.03, maybe.
00:15:20.930 --> 00:15:24.650
How many of you actually
have taken 18.03?
00:15:24.650 --> 00:15:27.380
Everybody knows the
solution, so very good.
00:15:27.380 --> 00:15:30.475
I am safe.
00:15:30.475 --> 00:15:31.475
So what is the solution?
00:15:35.730 --> 00:15:46.940
The solution is x of t equal
to a cosine of omega t plus b
00:15:46.940 --> 00:15:49.820
sine omega t.
00:15:49.820 --> 00:15:53.120
So my friends from
the math department
00:15:53.120 --> 00:15:58.160
tell me secretly that this
is actually the solution.
00:15:58.160 --> 00:16:00.470
And I trust him or her.
00:16:04.340 --> 00:16:07.130
So that's very nice.
00:16:07.130 --> 00:16:09.950
Now I have the
solution, and how do I
00:16:09.950 --> 00:16:12.450
know this is the only solution?
00:16:12.450 --> 00:16:14.850
How do I know?
00:16:14.850 --> 00:16:17.330
Actually, there
are two unknowns,
00:16:17.330 --> 00:16:20.964
just to remind you
what you have learned.
00:16:20.964 --> 00:16:23.810
There are two unknowns.
00:16:23.810 --> 00:16:29.860
And if you plug this
thing into this equation,
00:16:29.860 --> 00:16:32.000
you satisfy that equation.
00:16:32.000 --> 00:16:34.880
If you don't trust me,
you can do it offline.
00:16:34.880 --> 00:16:37.610
It's always good
to check to make
00:16:37.610 --> 00:16:39.290
sure I didn't make a mistake.
00:16:39.290 --> 00:16:40.980
But that's very good news.
00:16:40.980 --> 00:16:46.250
So that means we will
have two unknowns,
00:16:46.250 --> 00:16:49.760
and those will
satisfy the equation.
00:16:49.760 --> 00:16:54.900
So by uniqueness
theorem, this is actually
00:16:54.900 --> 00:17:01.080
the one and the only one
solution in my universe,
00:17:01.080 --> 00:17:07.800
also yours, which satisfy
the equation because
00:17:07.800 --> 00:17:09.700
of the uniqueness theorem.
00:17:09.700 --> 00:17:12.690
So I hope I have
convinced you that we
00:17:12.690 --> 00:17:16.810
have solved this equation.
00:17:16.810 --> 00:17:22.349
So now I take my
physicist hat back and now
00:17:22.349 --> 00:17:24.400
it is actually my job again.
00:17:24.400 --> 00:17:27.030
So now we have the
solution, and we
00:17:27.030 --> 00:17:29.290
need to determine
what is actually
00:17:29.290 --> 00:17:32.440
these two unknown coefficients.
00:17:32.440 --> 00:17:37.770
So what I'm going to use is to
use the two initial conditions.
00:17:44.840 --> 00:17:49.180
The first initial condition
is x of 0 equal to x initial.
00:17:52.770 --> 00:17:57.540
The second one is that since
I released this mass really
00:17:57.540 --> 00:18:03.180
carefully and the initial
velocity is 0, therefore,
00:18:03.180 --> 00:18:09.160
I have x dot 0 equal to 0.
00:18:09.160 --> 00:18:11.610
From this, you can solve.
00:18:11.610 --> 00:18:16.000
Plug these two conditions
into this equation.
00:18:16.000 --> 00:18:20.370
You can actually figure out
that a is equal to x initial.
00:18:26.400 --> 00:18:29.690
And b is equal to 0.
00:18:33.000 --> 00:18:34.110
Any questions so far?
00:18:39.830 --> 00:18:41.460
Very good.
00:18:41.460 --> 00:18:43.530
So now we have the solution.
00:18:43.530 --> 00:18:46.640
So finally, what is
actually the solution?
00:18:46.640 --> 00:19:03.660
The solution we get is x of t
equal to x initial cosine omega
00:19:03.660 --> 00:19:04.160
t.
00:19:08.120 --> 00:19:12.620
So this is actually the
amplitude of the oscillation,
00:19:12.620 --> 00:19:17.240
and this is actually
the angular velocity.
00:19:17.240 --> 00:19:20.420
So you may be
asking why angular?
00:19:20.420 --> 00:19:22.970
Where is the
angular coming from?
00:19:22.970 --> 00:19:25.910
Because this is actually
a one-dimensional motion.
00:19:25.910 --> 00:19:28.490
Where is the angular
velocity coming from?
00:19:28.490 --> 00:19:34.250
And I will explain that
in the later lecture.
00:19:34.250 --> 00:19:38.220
And also this is actually
a harmonic oscillation.
00:19:38.220 --> 00:19:40.700
So what we are
actually predicting
00:19:40.700 --> 00:19:46.490
is that this mass is going
to do this, have a fixed
00:19:46.490 --> 00:19:48.830
amplitude and it's
actually going
00:19:48.830 --> 00:19:54.920
to go back and forth with the
angular frequency of omega.
00:19:54.920 --> 00:20:00.230
So we can now do an experiment
to verify if this is actually
00:20:00.230 --> 00:20:01.940
really the case.
00:20:01.940 --> 00:20:04.400
So there's a small difference.
00:20:04.400 --> 00:20:08.870
There's another spring here,
but essentially, the solution
00:20:08.870 --> 00:20:10.290
will be very similar.
00:20:10.290 --> 00:20:15.860
You may get this
in a p-set or exam.
00:20:15.860 --> 00:20:20.670
So now I can turn on the air
so that I make this surface
00:20:20.670 --> 00:20:23.880
frictionless.
00:20:23.880 --> 00:20:29.120
And you can see that now
I actually move this thing
00:20:29.120 --> 00:20:33.390
slightly away from the
equilibrium position,
00:20:33.390 --> 00:20:36.410
and I release that carefully.
00:20:36.410 --> 00:20:42.030
So you can see that really it's
actually going back and forth
00:20:42.030 --> 00:20:43.080
harmonically.
00:20:46.290 --> 00:20:50.880
I can change the amplitude
and see what will happen.
00:20:50.880 --> 00:20:53.540
The amplitude is
becoming bigger,
00:20:53.540 --> 00:20:59.120
and you can see that the
oscillation amplitude really
00:20:59.120 --> 00:21:03.365
depends on where you put
that initially with respect
00:21:03.365 --> 00:21:04.750
to the equilibrium position.
00:21:04.750 --> 00:21:08.330
I can actually make a small
amplitude oscillation also.
00:21:08.330 --> 00:21:11.570
Now you can see that now the
amplitude is small but still
00:21:11.570 --> 00:21:14.660
oscillating back and forth.
00:21:14.660 --> 00:21:16.401
So that's very encouraging.
00:21:20.730 --> 00:21:25.340
Let's take another example,
which I actually rotate
00:21:25.340 --> 00:21:28.520
the whole thing by 90 degrees.
00:21:28.520 --> 00:21:32.150
You are going to get a
question about this system
00:21:32.150 --> 00:21:33.510
in your p-set.
00:21:33.510 --> 00:21:38.500
The amazing thing is that
the solution is the same.
00:21:38.500 --> 00:21:41.330
What is that?
00:21:41.330 --> 00:21:45.470
And you don't believe me,
let me do the experiment.
00:21:45.470 --> 00:21:48.680
I actually shifted the position.
00:21:48.680 --> 00:21:53.220
I changed the position, and I
release that really carefully.
00:21:53.220 --> 00:21:56.460
You see that this mass is
oscillating up and down.
00:21:56.460 --> 00:22:00.410
The amplitude did not change.
00:22:00.410 --> 00:22:03.890
The frequency did not change
as a function of time.
00:22:03.890 --> 00:22:08.960
It really matched with the
solution we found here.
00:22:11.690 --> 00:22:14.060
It's truly amazing.
00:22:14.060 --> 00:22:16.040
No?
00:22:16.040 --> 00:22:19.940
The problem is that we are
so used to this already.
00:22:19.940 --> 00:22:25.250
You have seen this maybe
100 times before my lecture,
00:22:25.250 --> 00:22:28.040
so therefore, you
got so used to this.
00:22:28.040 --> 00:22:30.680
Therefore, when I say,
OK, I make a prediction.
00:22:30.680 --> 00:22:34.120
This is what happened, you
are just so used to this
00:22:34.120 --> 00:22:36.740
or you don't feel
the excitement.
00:22:36.740 --> 00:22:41.510
But for me, after I teach
this class so many times,
00:22:41.510 --> 00:22:46.030
I still find this
thing really amazing.
00:22:46.030 --> 00:22:48.110
Why is that?
00:22:48.110 --> 00:22:53.120
This means that
actually, mathematics
00:22:53.120 --> 00:22:56.520
really works, first of all.
00:22:56.520 --> 00:23:03.560
That means we can use the same
tool for the understanding
00:23:03.560 --> 00:23:11.300
of gravitational waves, for the
prediction of the Higgs boson,
00:23:11.300 --> 00:23:13.920
for the calculation
of the property
00:23:13.920 --> 00:23:17.810
of the quark-gluon plasma
in the early universe,
00:23:17.810 --> 00:23:22.280
and also at the
same time the motion
00:23:22.280 --> 00:23:25.400
of this spring-mass system.
00:23:25.400 --> 00:23:30.350
We actually use always the
same tool, the mathematics,
00:23:30.350 --> 00:23:32.840
to understand this system.
00:23:32.840 --> 00:23:35.360
And nobody will understands why.
00:23:35.360 --> 00:23:37.845
If you understand
why, please tell me.
00:23:37.845 --> 00:23:38.720
I would like to know.
00:23:38.720 --> 00:23:39.886
I will be very proud of you.
00:23:43.780 --> 00:23:49.400
Rene Descartes said
once, "But in my opinion,
00:23:49.400 --> 00:23:54.190
all things in nature
occur mathematically."
00:23:54.190 --> 00:23:55.340
Apparently, he's right.
00:23:58.570 --> 00:24:03.670
Albert Einstein also once said,
"The most incomprehensible
00:24:03.670 --> 00:24:08.810
thing about the universe is
that it is comprehensible."
00:24:08.810 --> 00:24:13.870
So I would say this
is really something
00:24:13.870 --> 00:24:17.470
we need to appreciate
the need to think
00:24:17.470 --> 00:24:21.250
about why this is the case.
00:24:21.250 --> 00:24:22.325
Any questions?
00:24:26.790 --> 00:24:29.990
So you may say, oh, come on.
00:24:29.990 --> 00:24:33.990
We just solved the problem
of an ideal spring.
00:24:33.990 --> 00:24:35.380
Who cares?
00:24:35.380 --> 00:24:39.120
It's so simple, so easy,
and you are making really
00:24:39.120 --> 00:24:41.760
a big thing out of this.
00:24:41.760 --> 00:24:44.490
But actually, what
we have been solving
00:24:44.490 --> 00:24:47.110
is really much more than that.
00:24:47.110 --> 00:24:53.850
This equation is much more
than just a spring-mass system.
00:24:53.850 --> 00:24:58.800
Actually, if you think about
this question carefully,
00:24:58.800 --> 00:25:02.130
there's really no
Hooke's law forever.
00:25:02.130 --> 00:25:05.820
Hooke's law will give you
a potential proportional
00:25:05.820 --> 00:25:07.800
to x squared.
00:25:07.800 --> 00:25:15.970
And if you are so far away, you
pull the spring so really hard,
00:25:15.970 --> 00:25:19.080
you can store the energy
of the whole universe.
00:25:19.080 --> 00:25:21.730
Does that make sense?
00:25:21.730 --> 00:25:22.980
No.
00:25:22.980 --> 00:25:25.620
At some point, it
should break down.
00:25:25.620 --> 00:25:27.990
So there's really no Hook's law.
00:25:27.990 --> 00:25:31.860
But there's also
Hook's law everywhere.
00:25:31.860 --> 00:25:37.230
If you look at this system,
it follows the harmonic
00:25:37.230 --> 00:25:38.280
oscillation.
00:25:38.280 --> 00:25:40.950
If you look at this
system I perturb this,
00:25:40.950 --> 00:25:42.150
it goes back and forth.
00:25:44.940 --> 00:25:46.560
It's almost like everywhere.
00:25:46.560 --> 00:25:48.540
Why is this the case?
00:25:48.540 --> 00:25:51.580
I'm going to answer this
question immediately.
00:25:54.280 --> 00:25:58.860
So let's take a
look at an example.
00:25:58.860 --> 00:26:01.060
So if I consider
a potential, this
00:26:01.060 --> 00:26:03.870
is an artificial
potential, which
00:26:03.870 --> 00:26:07.110
you can find in
Georgi's book, so v
00:26:07.110 --> 00:26:11.780
is equal to E times L
over x plus x over L.
00:26:11.780 --> 00:26:14.640
And if you practice as
a function of x, then
00:26:14.640 --> 00:26:16.800
basically you get
this funny shape.
00:26:16.800 --> 00:26:20.300
It's not proportional
to x squared.
00:26:20.300 --> 00:26:23.130
Therefore, you
will see that, OK,
00:26:23.130 --> 00:26:28.710
the resulting motion for the
particle in this potential,
00:26:28.710 --> 00:26:31.950
it's not going to
be harmonic motion.
00:26:31.950 --> 00:26:35.370
But if I zoom in,
zoom in, and zoom in
00:26:35.370 --> 00:26:38.550
and basically, you will see
that if I am patient enough,
00:26:38.550 --> 00:26:44.440
I zoom in enough, you'll
see that this is a parabola.
00:26:44.440 --> 00:26:50.100
Again, you follow Hooke's law.
00:26:50.100 --> 00:26:52.590
So that is actually really cool.
00:26:52.590 --> 00:26:59.100
So if I consider an
arbitrary v of x,
00:26:59.100 --> 00:27:05.740
we can do a Taylor
expansion to this potential.
00:27:05.740 --> 00:27:10.920
So basically v of x
will be equal to v of 0
00:27:10.920 --> 00:27:16.890
plus v prime 0 divided
by 1 factorial times
00:27:16.890 --> 00:27:24.900
x plus v double prime
0 over 2 factorial
00:27:24.900 --> 00:27:30.670
x squared plus v
triple prime 0 divided
00:27:30.670 --> 00:27:37.110
by 3 factorial x to the third
plus infinite number of terms.
00:27:37.110 --> 00:27:47.010
v 0 is the position of where
you have minimum potential.
00:27:47.010 --> 00:27:50.550
So that's actually where
the equilibrium position
00:27:50.550 --> 00:27:54.020
is in my coordinate system.
00:27:54.020 --> 00:27:55.770
It's the standard,
the coordinate system
00:27:55.770 --> 00:28:02.070
I used for the solving
the spring-mass question.
00:28:02.070 --> 00:28:06.900
So if I calculate the
force, the force, f of x,
00:28:06.900 --> 00:28:12.690
will be equal to
minus d dx v of x.
00:28:12.690 --> 00:28:15.150
And that will be
equal to minus v
00:28:15.150 --> 00:28:24.000
prime 0 minus v double
prime 0 x minus 1
00:28:24.000 --> 00:28:34.320
over 2 v triple prime 0 x
squared plus many other terms.
00:28:34.320 --> 00:28:39.980
Since I have mentioned
that v of 0--
00:28:46.108 --> 00:28:50.160
this will be x.
00:28:50.160 --> 00:28:55.260
v of 0 is actually the
position of the minima.
00:28:55.260 --> 00:29:01.002
Therefore, v prime of
0 will be equal to 0.
00:29:03.540 --> 00:29:05.100
Therefore.
00:29:05.100 --> 00:29:07.610
This term is gone.
00:29:11.660 --> 00:29:16.490
So what essentially is left over
is the remaining terms here.
00:29:19.910 --> 00:29:26.450
Now, if I assume that x is very
small, what is going to happen?
00:29:26.450 --> 00:29:31.090
Anybody know when x is very
small, what is going to happen?
00:29:31.090 --> 00:29:32.140
Anybody have the answer?
00:29:32.140 --> 00:29:34.720
AUDIENCE: [INAUDIBLE].
00:29:34.720 --> 00:29:36.110
YEN-JIE LEE: Exactly.
00:29:36.110 --> 00:29:41.630
So when x is very small, he said
that the higher order terms all
00:29:41.630 --> 00:29:43.095
become negligible.
00:29:43.095 --> 00:29:43.910
OK?
00:29:43.910 --> 00:29:45.290
So that is essentially correct.
00:29:45.290 --> 00:29:50.060
So when x is very
small, then I only
00:29:50.060 --> 00:29:55.130
need to consider the
leading order term.
00:29:55.130 --> 00:29:59.030
But how small is the question.
00:29:59.030 --> 00:30:00.900
How small is small?
00:30:03.760 --> 00:30:07.630
Actually, what you can
do is to take the ratio
00:30:07.630 --> 00:30:12.517
between these two terms.
00:30:12.517 --> 00:30:14.350
So if you take the
ratio, then basically you
00:30:14.350 --> 00:30:19.300
would get a condition
xv triple dot
00:30:19.300 --> 00:30:29.940
0, which will be much smaller
than v double prime 0.
00:30:29.940 --> 00:30:32.050
So that is essentially
the condition
00:30:32.050 --> 00:30:37.630
which is required to satisfy it
so that we actually can ignore
00:30:37.630 --> 00:30:39.340
all the higher-order terms.
00:30:39.340 --> 00:30:42.160
Then the whole
question becomes f
00:30:42.160 --> 00:30:48.910
of x equal to minus
v double prime 0 x.
00:30:48.910 --> 00:30:52.100
And that essentially,
Hooke's law.
00:30:52.100 --> 00:30:54.760
So you can see that
first of all, there's
00:30:54.760 --> 00:30:57.190
no Hooke's law in general.
00:30:57.190 --> 00:31:01.240
Secondly, Hook's law
essentially applicable
00:31:01.240 --> 00:31:06.490
almost everywhere when you
have a well-behaved potential
00:31:06.490 --> 00:31:10.860
and if you only perturb
the system really slightly
00:31:10.860 --> 00:31:14.960
with very small amplitude,
then it always works.
00:31:14.960 --> 00:31:19.120
So what I would like
to say is that after we
00:31:19.120 --> 00:31:23.620
have done this exercise, you
will see that, actually, we
00:31:23.620 --> 00:31:28.960
have solved all the
possible systems, which
00:31:28.960 --> 00:31:33.630
have a well-behaved potential.
00:31:33.630 --> 00:31:39.830
It has a minima, and if I have
the amplitude small enough,
00:31:39.830 --> 00:31:46.400
then the system is going to do
simple harmonic oscillation.
00:31:46.400 --> 00:31:47.860
Any questions?
00:31:54.540 --> 00:31:56.065
No question, then
we'll continue.
00:32:02.060 --> 00:32:07.040
So let's come back to
this equation of motion.
00:32:07.040 --> 00:32:13.360
x double dot plus omega
squared x, this is equal to 0.
00:32:16.380 --> 00:32:19.600
There are two
important properties
00:32:19.600 --> 00:32:22.930
of this linear
equation of motion.
00:32:26.950 --> 00:32:48.130
The first one is that if x1 of t
and x2 of t are both solutions,
00:32:48.130 --> 00:32:54.280
then x12, which is
the superposition
00:32:54.280 --> 00:33:01.340
of the first and second
solution, is also a solution.
00:33:13.930 --> 00:33:17.690
The second thing, which is very
interesting about this equation
00:33:17.690 --> 00:33:27.235
of motion, is that there's a
time translation invariance.
00:33:31.990 --> 00:33:38.830
So this means that if
x of t is a solution,
00:33:38.830 --> 00:33:49.285
then xt prime equal to xt
plus a is also a solution.
00:33:52.190 --> 00:33:55.520
So that is really
cool, because that
00:33:55.520 --> 00:34:04.370
means if I change t equal to
0, so I shift the 0-th time,
00:34:04.370 --> 00:34:08.090
the whole physics
did not change.
00:34:10.989 --> 00:34:13.650
So this is actually
because of the chain law.
00:34:13.650 --> 00:34:19.929
So if you have chain
law dx t plus a dt, that
00:34:19.929 --> 00:34:35.690
is equal to d t plus a
dt, dx t prime dt prime
00:34:35.690 --> 00:34:39.949
evaluated at t prime
equal to t plus a.
00:34:39.949 --> 00:34:52.969
And that is equal to dx t prime
dt, t prime equal to t plus a.
00:34:52.969 --> 00:35:00.020
So that means if I have
changed the t equal to 0
00:35:00.020 --> 00:35:04.670
to other place, the
whole equation of motion
00:35:04.670 --> 00:35:06.860
is still the same.
00:35:06.860 --> 00:35:10.880
On the other hand, if the
k, or say the potential,
00:35:10.880 --> 00:35:14.640
is time dependent, then that
may break this symmetry.
00:35:17.970 --> 00:35:18.932
Any questions?
00:35:22.150 --> 00:35:27.670
So before we take a
five minute break,
00:35:27.670 --> 00:35:33.640
I would like to discuss further
about this point, this linear
00:35:33.640 --> 00:35:34.990
and nonlinear event.
00:35:34.990 --> 00:35:38.230
So you can see that
the force is actually
00:35:38.230 --> 00:35:42.760
linearly dependent on x.
00:35:42.760 --> 00:35:49.420
But what will happen
if I increase x more?
00:35:49.420 --> 00:35:50.350
Something will happen.
00:35:50.350 --> 00:35:54.850
That means the higher-ordered
term should also be
00:35:54.850 --> 00:35:57.370
taken into account carefully.
00:35:57.370 --> 00:36:01.210
So that means the
solution of this kind, x
00:36:01.210 --> 00:36:07.070
initial cosine omega t,
will not work perfectly.
00:36:07.070 --> 00:36:12.690
In 8.03, we only consider the
linear term most of the time.
00:36:12.690 --> 00:36:15.700
But actually, I would
like to make sure
00:36:15.700 --> 00:36:17.450
that everybody
can at this point,
00:36:17.450 --> 00:36:21.220
the higher-order
contribution is actually
00:36:21.220 --> 00:36:23.290
visible in our daily life.
00:36:23.290 --> 00:36:28.710
So let me actually give
you a concrete example.
00:36:28.710 --> 00:36:33.520
So here I have two pendulums.
00:36:33.520 --> 00:36:36.550
So I can now perturb
this pendulum slightly.
00:36:36.550 --> 00:36:42.970
And you you'll see that it goes
back and forth and following
00:36:42.970 --> 00:36:45.610
simple harmonic emotion.
00:36:45.610 --> 00:36:48.670
So if I have both
things slightly
00:36:48.670 --> 00:36:52.810
oscillating with
small amplitude, what
00:36:52.810 --> 00:36:58.140
is going to happen is that
both pendulums reach maxima
00:36:58.140 --> 00:37:00.800
amplitude at the same time.
00:37:00.800 --> 00:37:05.360
You can see that very clearly.
00:37:05.360 --> 00:37:08.916
I don't need to
do this carefully.
00:37:08.916 --> 00:37:14.150
You see that they always
reach maxima at the same time
00:37:14.150 --> 00:37:16.040
when the amplitude is small.
00:37:16.040 --> 00:37:17.250
Why?
00:37:17.250 --> 00:37:24.000
That is because the higher-order
terms are not important.
00:37:24.000 --> 00:37:26.350
So now let's do a experiment.
00:37:26.350 --> 00:37:29.010
And now I go crazy.
00:37:29.010 --> 00:37:33.840
I make the amplitude
very large so that I
00:37:33.840 --> 00:37:37.800
break that approximation.
00:37:37.800 --> 00:37:39.300
So let's see what will happen.
00:37:39.300 --> 00:37:41.805
So now I do this then.
00:37:41.805 --> 00:37:47.170
I release at the same time
and see what will happen.
00:37:47.170 --> 00:37:49.410
You see that originally
they are in phase.
00:37:49.410 --> 00:37:53.830
They are reaching
maxima at the same time.
00:37:53.830 --> 00:37:57.110
But if we are patient
enough, you see that now?
00:37:57.110 --> 00:38:01.020
They are is
oscillating, actually,
00:38:01.020 --> 00:38:02.920
at different frequencies.
00:38:02.920 --> 00:38:06.150
Originally, the
solution, the omega,
00:38:06.150 --> 00:38:09.810
is really independent
of the amplitude.
00:38:09.810 --> 00:38:11.970
So they should,
actually, be isolating
00:38:11.970 --> 00:38:13.810
at the same frequency.
00:38:13.810 --> 00:38:16.260
But clearly you
can see here, when
00:38:16.260 --> 00:38:19.380
you increase the
amplitude, then you
00:38:19.380 --> 00:38:24.600
need to consider also
the nonlinear effects.
00:38:24.600 --> 00:38:30.067
So any questions before we
take a five-minute break.
00:38:30.067 --> 00:38:32.150
So if not, then we would
take a five-minute break,
00:38:32.150 --> 00:38:34.850
and we come back at 25.
00:38:40.590 --> 00:38:43.290
So welcome back, everybody.
00:38:43.290 --> 00:38:45.630
So we will continue
the discussion
00:38:45.630 --> 00:38:51.170
of this equation of motion, x
double dot plus omega square x
00:38:51.170 --> 00:38:54.120
equal to 0.
00:38:54.120 --> 00:38:58.750
So there are three
possible way to like
00:38:58.750 --> 00:39:01.935
the solution to this equation.
00:39:01.935 --> 00:39:04.860
So the first one as
I mentioned before,
00:39:04.860 --> 00:39:15.960
x of t equal to a cosine
omega t plus b sine omega t.
00:39:15.960 --> 00:39:17.800
So this is actually
the functional form
00:39:17.800 --> 00:39:21.010
we have been using before.
00:39:21.010 --> 00:39:26.520
And we can actually also
rewrite it in a different way.
00:39:26.520 --> 00:39:39.600
So x or t equal to capital
A cosine omega t plus phi.
00:39:39.600 --> 00:39:41.640
You may say, wait a second.
00:39:41.640 --> 00:39:44.190
You just promised me that
this is the first one,
00:39:44.190 --> 00:39:47.190
the one is the one
and only one solution
00:39:47.190 --> 00:39:51.750
in the universe, which actually
satisfy the equation of motion.
00:39:51.750 --> 00:39:53.960
Now you write another one.
00:39:53.960 --> 00:39:56.210
What is going on?
00:39:56.210 --> 00:39:57.680
Why?
00:39:57.680 --> 00:39:59.060
But actually, they are the same.
00:40:01.770 --> 00:40:09.740
This is actually A
cosine phi cosine omega t
00:40:09.740 --> 00:40:16.250
minus A sine phi sine omega t.
00:40:19.250 --> 00:40:23.390
So the good thing
is that A and phi
00:40:23.390 --> 00:40:27.770
are arbitrary constant
so that it should be you
00:40:27.770 --> 00:40:31.040
can use two initial
conditions to determine
00:40:31.040 --> 00:40:32.570
the arbitrary constant.
00:40:32.570 --> 00:40:39.050
So you can see that one and
two are completely equivalent.
00:40:39.050 --> 00:40:44.390
So I hope that solves
some of the questions
00:40:44.390 --> 00:40:48.320
because you really
find it confusing
00:40:48.320 --> 00:40:53.960
why we have different
presentations of the solution.
00:40:53.960 --> 00:40:59.600
So there's a third one, which
is actually much more fancier.
00:40:59.600 --> 00:41:04.270
The third one is
that I have x of t.
00:41:04.270 --> 00:41:10.575
This is actually
a real part of A--
00:41:10.575 --> 00:41:16.650
again, the amplitude--
exponential i omega t
00:41:16.650 --> 00:41:24.030
plus phi, where i is equal to
the square root of minus 1.
00:41:27.800 --> 00:41:28.460
Wait a second.
00:41:28.460 --> 00:41:32.530
We will say, well, professor,
why are you writing
00:41:32.530 --> 00:41:34.975
such a horrible solution?
00:41:38.820 --> 00:41:40.150
Right?
00:41:40.150 --> 00:41:41.150
Really strange.
00:41:41.150 --> 00:41:42.400
But that will explain you why.
00:41:45.030 --> 00:41:48.120
So three is actually
a mathematical trick.
00:41:48.120 --> 00:41:53.220
I'm not going to prove anything
here because I'm a physicist,
00:41:53.220 --> 00:41:58.600
but I would like to share with
you what I think is going on.
00:41:58.600 --> 00:42:01.410
I think three is really
just a mathematical trick
00:42:01.410 --> 00:42:05.310
from the math department.
00:42:05.310 --> 00:42:10.780
In principle, I can drive it
an even more horrible way.
00:42:10.780 --> 00:42:24.486
x of t equal to a real part of
A cosine omega t plus phi plus i
00:42:24.486 --> 00:42:29.501
f of t.
00:42:29.501 --> 00:42:36.910
And f of t is a real function.
00:42:36.910 --> 00:42:38.730
In principle, I can do that.
00:42:38.730 --> 00:42:41.870
It's even more horrible.
00:42:41.870 --> 00:42:43.500
Why is that?
00:42:43.500 --> 00:42:46.470
Because I now have
this function.
00:42:46.470 --> 00:42:48.870
I take the real
part, and I actually
00:42:48.870 --> 00:42:56.200
take the two out
of this operation.
00:42:56.200 --> 00:42:59.900
So f of t is actually
the real function.
00:42:59.900 --> 00:43:02.170
It can be something arbitrary.
00:43:02.170 --> 00:43:11.690
And i can now plot the locus
of this function, the solution
00:43:11.690 --> 00:43:15.280
on the complex print.
00:43:15.280 --> 00:43:19.800
Now I'm plotting this solution
on this complex print.
00:43:19.800 --> 00:43:23.780
What is going to happen is
that you're going to have--
00:43:31.250 --> 00:43:33.310
That is what you
are going to get.
00:43:33.310 --> 00:43:36.520
If I am lucky, if this
f of t is confined
00:43:36.520 --> 00:43:39.250
in some specific
region, if I not lucky,
00:43:39.250 --> 00:43:41.160
then it goes out
of the print there.
00:43:41.160 --> 00:43:42.490
I couldn't see it.
00:43:42.490 --> 00:43:44.490
Maybe it go to the
moon or something.
00:43:47.150 --> 00:43:50.090
But if you are smart
enough, and I'm
00:43:50.090 --> 00:44:00.080
sure you are, if I choose f
of t equal to A sine omega
00:44:00.080 --> 00:44:06.410
t plus phi, can anybody tell
me what is going to happen?
00:44:09.314 --> 00:44:10.766
AUDIENCE: [INAUDIBLE].
00:44:15.122 --> 00:44:16.950
YEN-JIE LEE: Would
you count a circle?
00:44:19.920 --> 00:44:22.770
Very good.
00:44:22.770 --> 00:44:27.700
If I plot the locus
again of this function,
00:44:27.700 --> 00:44:33.970
the real axis, imaginary axis,
then you should get a circle.
00:44:33.970 --> 00:44:37.030
Some miracle happened.
00:44:37.030 --> 00:44:41.530
If you choose the
f of t correctly,
00:44:41.530 --> 00:44:50.400
wisely, then you can actually
turn all this mess into order.
00:44:50.400 --> 00:44:51.649
Any questions?
00:44:56.440 --> 00:44:59.590
So I can now follow
up about this.
00:45:09.380 --> 00:45:20.110
So now I have x of t is equal to
the real part of A cosine omega
00:45:20.110 --> 00:45:28.260
t plus phi plus iA
sine omega t plus phi.
00:45:31.390 --> 00:45:36.120
And just a reminder,
exponential i theta
00:45:36.120 --> 00:45:43.830
is equal to cosine
theta plus i sine theta.
00:45:43.830 --> 00:45:45.920
Therefore, I arrive this.
00:45:45.920 --> 00:45:54.020
This is a real part of A
exponential i omega t plus phi.
00:45:58.840 --> 00:46:01.090
So if I do this
really carefully,
00:46:01.090 --> 00:46:08.680
I look at this the position of
the point at a specific time.
00:46:08.680 --> 00:46:13.500
So now time is equal to t.
00:46:13.500 --> 00:46:16.915
And this is the real axis, and
this is the imaginary axis.
00:46:16.915 --> 00:46:21.100
So I have this circle here.
00:46:21.100 --> 00:46:24.520
So at time equal to t,
what you are getting
00:46:24.520 --> 00:46:27.400
is that x is actually--
00:46:27.400 --> 00:46:31.430
before taking the real part, A,
exponential i omega t plus phi,
00:46:31.430 --> 00:46:33.730
it's actually here.
00:46:33.730 --> 00:46:39.380
And this vector actually
shows the amplitude.
00:46:39.380 --> 00:46:46.540
Amplitude is A. And the angle
between this vector pointing
00:46:46.540 --> 00:46:54.460
to the position of this
function is omega t plus phi.
00:46:54.460 --> 00:47:00.100
So this is actually the
angle between this vector
00:47:00.100 --> 00:47:03.500
and the real axis.
00:47:03.500 --> 00:47:06.060
So that's pretty cool.
00:47:06.060 --> 00:47:06.770
Why?
00:47:06.770 --> 00:47:12.410
Because now I understand why
I call this omega angular
00:47:12.410 --> 00:47:15.830
velocity or angular frequency.
00:47:15.830 --> 00:47:22.140
Because the solution to
the equation of motion,
00:47:22.140 --> 00:47:26.325
which we have actually
derived before,
00:47:26.325 --> 00:47:35.610
is actually the real part of
rotation in a complex print.
00:47:35.610 --> 00:47:39.540
If you think about
it, that means now
00:47:39.540 --> 00:47:45.910
I see this particle
going up and down.
00:47:45.910 --> 00:47:49.160
I see this particle
going up and down.
00:47:49.160 --> 00:47:51.950
You can think about
that, this is Earth.
00:47:51.950 --> 00:47:55.370
If there is an extra
dimension mention,
00:47:55.370 --> 00:47:58.620
which you couldn't see.
00:47:58.620 --> 00:48:02.620
Actually, this particle
in the dimension
00:48:02.620 --> 00:48:06.200
where we can see into the extra
dimension, which is hidden
00:48:06.200 --> 00:48:09.180
is actually rotating.
00:48:09.180 --> 00:48:12.470
And while we see
that reality, it's
00:48:12.470 --> 00:48:16.040
a projection to the real axis.
00:48:16.040 --> 00:48:16.650
You see?
00:48:16.650 --> 00:48:25.210
So in reality, this particle
is actually rotating,
00:48:25.210 --> 00:48:29.450
if you add the image
and the extra dimension.
00:48:29.450 --> 00:48:33.620
So that is actually pretty
cool, but the purity artificial.
00:48:33.620 --> 00:48:37.580
So you can see that
I can choose f of t
00:48:37.580 --> 00:48:42.530
to be a different function,
and then this whole picture
00:48:42.530 --> 00:48:43.970
is different.
00:48:43.970 --> 00:48:46.490
But I also would
create a lot of trouble
00:48:46.490 --> 00:48:49.250
because then the mathematics
become complicated.
00:48:49.250 --> 00:48:50.880
I didn't gain anything.
00:48:50.880 --> 00:48:58.240
But by choosing this
functional form,
00:48:58.240 --> 00:49:02.280
you actually write a
very beautiful picture.
00:49:02.280 --> 00:49:05.920
Another thing, which
is very cool about this
00:49:05.920 --> 00:49:12.480
is that if I write this thing
in the exponential functional
00:49:12.480 --> 00:49:18.340
form, since we are dealing
with differential equations,
00:49:18.340 --> 00:49:22.170
there is a very good property
about exponential function.
00:49:22.170 --> 00:49:26.856
That is it is essentially
a phoenix function.
00:49:26.856 --> 00:49:30.480
Do you know what is a phoenix?
00:49:30.480 --> 00:49:36.060
Phoenix is actually some kind
of animal, a long-beaked bird,
00:49:36.060 --> 00:49:41.100
which is cyclically called
the regenerated or reborn.
00:49:41.100 --> 00:49:44.100
So basically, when
this phoenix die,
00:49:44.100 --> 00:49:48.820
you will lay the eggs in the
fire and you were reborn.
00:49:48.820 --> 00:49:50.670
This is actually the
same as this function.
00:49:54.240 --> 00:49:59.650
I can do differentiation,
still an exponential function,
00:49:59.650 --> 00:50:02.460
and differentiate,
differentiate, differentiate.
00:50:02.460 --> 00:50:05.460
Still exponential function.
00:50:05.460 --> 00:50:07.620
So that is very
nice because when
00:50:07.620 --> 00:50:13.350
we deal with
differential equation,
00:50:13.350 --> 00:50:16.590
then you can actually
remove all those dots
00:50:16.590 --> 00:50:20.940
and make them become just
exponential function.
00:50:20.940 --> 00:50:25.770
So essentially, a
very nice property.
00:50:25.770 --> 00:50:32.910
So the first property, which is
very nice is that it cannot be
00:50:32.910 --> 00:50:39.480
killed by differentiation.
00:50:39.480 --> 00:50:46.980
You will see how useful this
is in the following lectures.
00:50:49.620 --> 00:50:52.230
The second thing,
which is really nice
00:50:52.230 --> 00:50:55.170
is that it has a
very nice property.
00:50:55.170 --> 00:51:02.090
So basically the exponential
i theta 1 times exponential i
00:51:02.090 --> 00:51:07.410
theta 2, and that will give
you exponential i theta
00:51:07.410 --> 00:51:08.980
1 plus theta 2.
00:51:11.610 --> 00:51:13.420
So what does that mean?
00:51:13.420 --> 00:51:22.110
That means if I have a solution
in this form, A exponential i
00:51:22.110 --> 00:51:25.400
omega t plus phi.
00:51:29.040 --> 00:51:40.200
And I do a times translation,
t become t plus A. Then
00:51:40.200 --> 00:51:47.650
this become A exponential
i omega t plus A plus phi.
00:51:51.350 --> 00:51:59.430
So this means that times
translation in this rotation
00:51:59.430 --> 00:52:04.880
is just a rotation
in complex print.
00:52:04.880 --> 00:52:05.730
You see?
00:52:05.730 --> 00:52:08.900
So now t becomes t plus
A. Then you are actually
00:52:08.900 --> 00:52:16.190
just changing the angle between
this vector and the x-axis.
00:52:16.190 --> 00:52:18.140
So as time goes
on, what is going
00:52:18.140 --> 00:52:21.300
to happen is that this thing
will go around and around
00:52:21.300 --> 00:52:28.100
and around and the physics is
always the set, no matter when
00:52:28.100 --> 00:52:32.340
you start counting, and
the translation is just
00:52:32.340 --> 00:52:35.790
the rotation in this print.
00:52:35.790 --> 00:52:37.165
Any questions?
00:52:44.940 --> 00:52:50.160
So I think this is actually a
basic slide just to remind you
00:52:50.160 --> 00:52:52.710
about Euler's formula.
00:52:52.710 --> 00:52:55.460
So basically, the
explanation i phi
00:52:55.460 --> 00:52:58.920
is equal to cosine
phi plus i sine phi.
00:52:58.920 --> 00:53:03.300
And I think it will be useful if
you are not familiar with this.
00:53:03.300 --> 00:53:06.180
It is useful to actually
review a little bit
00:53:06.180 --> 00:53:10.230
about exponential
function, which will
00:53:10.230 --> 00:53:12.050
be very useful for this class.
00:53:17.350 --> 00:53:20.260
So I'm running a
bit faster today.
00:53:20.260 --> 00:53:25.930
So let's take a look at
what we have learned today.
00:53:25.930 --> 00:53:31.060
We have analyzed the physics
of a harmonic oscillator.
00:53:31.060 --> 00:53:37.830
So basically, we start by asking
really just a verbal question,
00:53:37.830 --> 00:53:40.650
what is going to
happen to this mass
00:53:40.650 --> 00:53:44.310
on the table
attached to a spring.
00:53:44.310 --> 00:53:49.620
And what we have learned is that
we actually use mathematics.
00:53:49.620 --> 00:53:57.750
Basically, we translate all what
we have learned about this mass
00:53:57.750 --> 00:54:03.480
into mathematics by first
define a coordinate system.
00:54:03.480 --> 00:54:09.130
Then I'd write everything
using that coordinate system.
00:54:09.130 --> 00:54:13.125
Then I use Newton's law to
help us to solve this question.
00:54:15.870 --> 00:54:19.710
And we have analyzed the physics
of this harmonic oscillator.
00:54:19.710 --> 00:54:23.700
And Hooke's law, we found
that he actually, not only
00:54:23.700 --> 00:54:30.380
works for this
spring-mass system,
00:54:30.380 --> 00:54:37.260
it also works for all kinds of
different small oscillations
00:54:37.260 --> 00:54:40.530
about a point of equilibrium.
00:54:40.530 --> 00:54:44.700
So basically, it's actually
a universal solution
00:54:44.700 --> 00:54:47.070
what we have been doing.
00:54:47.070 --> 00:54:51.990
And we have found out a
complex exponential function
00:54:51.990 --> 00:54:55.450
is actually a beautiful
way to present
00:54:55.450 --> 00:54:59.910
the solution to the equation of
motion we have been studying.
00:54:59.910 --> 00:55:02.240
So everything is nice and good.
00:55:02.240 --> 00:55:06.960
However, life is
hard because there
00:55:06.960 --> 00:55:13.010
are many things which actually,
we ignored in this example.
00:55:13.010 --> 00:55:16.620
One apparent thing,
which we actually ignore,
00:55:16.620 --> 00:55:18.840
is the direct force.
00:55:18.840 --> 00:55:24.420
So you can see that before I was
actually making this pendulum
00:55:24.420 --> 00:55:27.480
oscillate back and forth.
00:55:27.480 --> 00:55:29.430
What is happening now?
00:55:29.430 --> 00:55:31.930
There are not
oscillating anymore.
00:55:31.930 --> 00:55:33.660
Why?
00:55:33.660 --> 00:55:37.020
Well, they stopped being.
00:55:37.020 --> 00:55:40.440
Apparently,
something is missing.
00:55:40.440 --> 00:55:46.930
When I actually
moved this system,
00:55:46.930 --> 00:55:51.490
if I turn off the air so
that there's friction,
00:55:51.490 --> 00:55:53.590
then it doesn't really move.
00:55:53.590 --> 00:55:58.820
If I increase a bit, the
air so that the slide have
00:55:58.820 --> 00:56:02.330
some slight freedom,
then actually, you
00:56:02.330 --> 00:56:05.990
can see that you move
a bit then you stop.
00:56:05.990 --> 00:56:12.625
If I increase this
some more, you
00:56:12.625 --> 00:56:18.660
can see that the amplitude
becomes smaller and smaller.
00:56:18.660 --> 00:56:22.400
So in the following lecture,
what we are going to do
00:56:22.400 --> 00:56:27.110
is to study how to actually
include a direct force into it
00:56:27.110 --> 00:56:31.210
again and of course, using the
same machinery which we have
00:56:31.210 --> 00:56:34.160
learned from here and
see if we can actually
00:56:34.160 --> 00:56:35.890
solve this problem.
00:56:35.890 --> 00:56:36.950
Thank you very much.
00:56:36.950 --> 00:56:39.360
We actually end
up earlier today.
00:56:39.360 --> 00:56:40.520
Sorry for that.
00:56:40.520 --> 00:56:43.280
And maybe I will make the
lecture longer next time.
00:56:45.890 --> 00:56:48.340
And if you have any
questions about what
00:56:48.340 --> 00:56:55.480
we have covered today, I'm
here available to help you.