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PROFESSOR: So, I'm back.

10
00:00:27,020 --> 00:00:30,500
Welcome back, also, to 8.03.

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00:00:30,500 --> 00:00:33,640
So today, what we
are going to do

12
00:00:33,640 --> 00:00:36,930
is something really interesting.

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00:00:36,930 --> 00:00:40,880
It's to understand
how we use symmetry

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00:00:40,880 --> 00:00:48,700
to help us with prediction
of physical situations.

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00:00:48,700 --> 00:00:55,430
So first, I will go through two
concrete examples of symmetry,

16
00:00:55,430 --> 00:00:58,080
and see what we can
learn from there.

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00:00:58,080 --> 00:01:01,680
And also, today, we are
going to go to infinite

18
00:01:01,680 --> 00:01:03,480
number of coupled oscillators.

19
00:01:03,480 --> 00:01:04,700
OK?

20
00:01:04,700 --> 00:01:06,740
I think we are done
with finite numbers.

21
00:01:06,740 --> 00:01:08,330
OK?

22
00:01:08,330 --> 00:01:08,990
All right.

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00:01:08,990 --> 00:01:11,090
So what we have
learned last time

24
00:01:11,090 --> 00:01:14,330
when Bolek was
giving lectures, I

25
00:01:14,330 --> 00:01:17,780
hope we have learned
that driving force can

26
00:01:17,780 --> 00:01:20,420
excite a specific normal mode.

27
00:01:20,420 --> 00:01:21,060
Right?

28
00:01:21,060 --> 00:01:24,360
So if you drive the
system at the frequency,

29
00:01:24,360 --> 00:01:27,540
the system like, then
the system will respond,

30
00:01:27,540 --> 00:01:32,300
and will oscillate the driving
frequency with large amplitude.

31
00:01:32,300 --> 00:01:33,860
OK?

32
00:01:33,860 --> 00:01:38,030
And also, we have learned
that the full solution

33
00:01:38,030 --> 00:01:39,870
of a coupled
oscillator is actually

34
00:01:39,870 --> 00:01:47,860
pretty similar to the situation
we got from single oscillators.

35
00:01:47,860 --> 00:01:50,120
So that where you have
a particular solution,

36
00:01:50,120 --> 00:01:52,190
and a homogeneous solution.

37
00:01:52,190 --> 00:01:57,650
And the full solution will
be a superposition of the two

38
00:01:57,650 --> 00:02:02,490
component, and all the
unknown coefficients

39
00:02:02,490 --> 00:02:05,590
in the homogeneous
part of the solution.

40
00:02:05,590 --> 00:02:06,470
OK?

41
00:02:06,470 --> 00:02:09,830
And today, I hope I can
help you and convince you

42
00:02:09,830 --> 00:02:15,590
that symmetry actually can help
us to solve the number of modes

43
00:02:15,590 --> 00:02:20,690
without knowing the detail
of M minus one K metrics.

44
00:02:20,690 --> 00:02:22,790
So that actually
sounds really cool,

45
00:02:22,790 --> 00:02:26,400
and I would like to talk about
that in this lecture today.

46
00:02:26,400 --> 00:02:30,120
So this is actually what
we have been doing so far.

47
00:02:30,120 --> 00:02:33,560
So we tried everything
in terms of metrics.

48
00:02:33,560 --> 00:02:35,930
So we start from the
equation of motion,

49
00:02:35,930 --> 00:02:39,530
and X double dot,
you go to minus KX.

50
00:02:39,530 --> 00:02:44,760
And then we write everything
in a complex notation--

51
00:02:44,760 --> 00:02:49,240
exponential i omega
t plus phi times A--

52
00:02:49,240 --> 00:02:50,990
A is actually the vector, right?

53
00:02:50,990 --> 00:02:53,590
So it's actually A1, A2, A3.

54
00:02:53,590 --> 00:02:56,810
It's actually the amplitude
of the oscillation

55
00:02:56,810 --> 00:02:59,640
of the first, second, and
third and etc. etc. it's

56
00:02:59,640 --> 00:03:01,850
a component of the system.

57
00:03:01,850 --> 00:03:03,140
Right?

58
00:03:03,140 --> 00:03:06,960
Then, we actually found
that, in the end of the day,

59
00:03:06,960 --> 00:03:09,380
we are actually
solving this problem

60
00:03:09,380 --> 00:03:12,390
like eigenvalue problem.

61
00:03:12,390 --> 00:03:16,750
So basically, we have M
minus 1 K metrics describe

62
00:03:16,750 --> 00:03:21,730
how each component in the system
interacts with each other.

63
00:03:21,730 --> 00:03:22,580
OK?

64
00:03:22,580 --> 00:03:26,240
Then, what is actually
the angle of frequency

65
00:03:26,240 --> 00:03:27,590
of the the normal modes?

66
00:03:27,590 --> 00:03:31,290
Essentially, coming from
this eigenvalue problem ,

67
00:03:31,290 --> 00:03:35,450
M minus 1 K, A equal
to omega square A.

68
00:03:35,450 --> 00:03:39,200
Then you just go ahead and
solve the eigenvalue problem.

69
00:03:39,200 --> 00:03:41,120
Then you will be
able to figure out

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00:03:41,120 --> 00:03:44,320
why are there no more mode
frequencies, and therefore,

71
00:03:44,320 --> 00:03:47,060
what are the
relative-- the ratio

72
00:03:47,060 --> 00:03:51,720
of the amplitude in the normal
mode, which is actually the A

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00:03:51,720 --> 00:03:52,220
vector.

74
00:03:52,220 --> 00:03:52,720
OK?

75
00:03:52,720 --> 00:03:53,900
The eigenvector.

76
00:03:53,900 --> 00:03:57,440
OK, so that's actually
what we have been doing.

77
00:03:57,440 --> 00:03:58,940
OK?

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00:03:58,940 --> 00:04:02,270
And today, what
I'm going to do is

79
00:04:02,270 --> 00:04:06,170
to introduce you a very
important concept in physics.

80
00:04:06,170 --> 00:04:10,050
Not only in physics, but also
in mathematics, and also art,

81
00:04:10,050 --> 00:04:10,550
right?

82
00:04:10,550 --> 00:04:14,120
So you see symmetry
in art, for example.

83
00:04:14,120 --> 00:04:15,210
We can see here--

84
00:04:15,210 --> 00:04:17,269
there are several graphs here--

85
00:04:17,269 --> 00:04:20,120
and you can see that
their apparent symmetry,

86
00:04:20,120 --> 00:04:25,610
or rotational symmetry, they
are refraction symmetry.

87
00:04:25,610 --> 00:04:30,500
And you can see that when
we build the particle

88
00:04:30,500 --> 00:04:32,240
detector for example
lower right plot

89
00:04:32,240 --> 00:04:37,010
is a CNS detector in the
Large Hadron Collider.

90
00:04:37,010 --> 00:04:41,720
We also try to build this
detector symmetric, right?

91
00:04:41,720 --> 00:04:44,940
Because otherwise, if we
get a very complicated shape

92
00:04:44,940 --> 00:04:47,330
of detector, then the
analysis of the data

93
00:04:47,330 --> 00:04:49,760
will be really complicated.

94
00:04:49,760 --> 00:04:54,020
So therefore, everybody
like symmetry, and everybody

95
00:04:54,020 --> 00:04:56,690
don't like, really, chaos.

96
00:04:56,690 --> 00:04:57,330
Right?

97
00:04:57,330 --> 00:04:57,830
OK?

98
00:04:57,830 --> 00:04:59,840
So, that's really nice.

99
00:04:59,840 --> 00:05:04,640
The question is, how do
we speak the language

100
00:05:04,640 --> 00:05:06,950
that the nature speak?

101
00:05:06,950 --> 00:05:11,460
How do we actually
describe symmetry?

102
00:05:11,460 --> 00:05:13,490
That's actually the
question I'm asking,

103
00:05:13,490 --> 00:05:16,380
and I'm going to show you
that, OK, we can actually

104
00:05:16,380 --> 00:05:20,310
use mathematics to
describe symmetry.

105
00:05:20,310 --> 00:05:25,850
So before we go to infinite
number of oscillators,

106
00:05:25,850 --> 00:05:30,730
let me give you a concrete
example of symmetry,

107
00:05:30,730 --> 00:05:35,020
and then see if we can
understand how to use the math

108
00:05:35,020 --> 00:05:37,060
to describe symmetry.

109
00:05:37,060 --> 00:05:37,930
OK?

110
00:05:37,930 --> 00:05:42,220
So there is a
two-component system.

111
00:05:42,220 --> 00:05:46,240
Two pendulums, which we worked
together in the last few

112
00:05:46,240 --> 00:05:50,410
lectures, that they are
coupled to each other,

113
00:05:50,410 --> 00:05:53,350
and there's a parent
symmetry of this system.

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00:05:53,350 --> 00:05:56,180
Can somebody tell me
what is the symmetry,

115
00:05:56,180 --> 00:05:58,570
you can see from this system?

116
00:05:58,570 --> 00:05:59,890
Somebody?

117
00:05:59,890 --> 00:06:02,046
Anybody?

118
00:06:02,046 --> 00:06:03,010
AUDIENCE: Reflection.

119
00:06:03,010 --> 00:06:05,160
PROFESSOR: The
reflection symmetry.

120
00:06:05,160 --> 00:06:11,530
So if you reflect this system,
as I show you in the slide,

121
00:06:11,530 --> 00:06:15,670
you can see that if you
reflect this picture,

122
00:06:15,670 --> 00:06:17,630
it looks identical.

123
00:06:17,630 --> 00:06:18,310
Right?

124
00:06:18,310 --> 00:06:20,980
So that is actually
really, really good news.

125
00:06:20,980 --> 00:06:24,310
That means if I do
this reflection, XY,

126
00:06:24,310 --> 00:06:26,284
and go to minus X2--

127
00:06:26,284 --> 00:06:27,700
you have a minus
sign, because you

128
00:06:27,700 --> 00:06:30,910
can see that after reflection--
the amplitude changes sign.

129
00:06:30,910 --> 00:06:31,750
Right?

130
00:06:31,750 --> 00:06:37,120
X2 go to minus X1, the
system looks identical,

131
00:06:37,120 --> 00:06:39,260
and the physics
should not change.

132
00:06:39,260 --> 00:06:39,760
OK?

133
00:06:39,760 --> 00:06:43,570
So that's actually what
we can learn from there.

134
00:06:43,570 --> 00:06:47,250
So that means if I have--

135
00:06:47,250 --> 00:06:53,708
I do this reflection, then I
can actually define X tilde--

136
00:06:53,708 --> 00:07:01,550
T-- this is equal to
minus X2 minus X1.

137
00:07:01,550 --> 00:07:02,050
OK?

138
00:07:02,050 --> 00:07:05,091
To become paired with X. OK?

139
00:07:05,091 --> 00:07:11,830
And this is also going to be the
solution of the equation motion

140
00:07:11,830 --> 00:07:14,990
if the original X is
already a solution.

141
00:07:14,990 --> 00:07:15,590
OK?

142
00:07:15,590 --> 00:07:19,190
So that's the power of
reflection symmetry.

143
00:07:19,190 --> 00:07:19,690
OK?

144
00:07:19,690 --> 00:07:25,810
If X is a solution, then
I do this reflection,

145
00:07:25,810 --> 00:07:32,900
and I can figure out that
X tilde is also a solution.

146
00:07:32,900 --> 00:07:34,390
OK?

147
00:07:34,390 --> 00:07:39,760
So how do I actually
describe the symmetry

148
00:07:39,760 --> 00:07:41,800
in the form of mathematics?

149
00:07:41,800 --> 00:07:48,550
What we actually do is to define
S matrix, symmetry matrix.

150
00:07:48,550 --> 00:07:53,140
And in this case, when we talk
about reflection symmetry,

151
00:07:53,140 --> 00:07:57,040
it's actually defined as
zero minus 1 minus 1, 0.

152
00:07:57,040 --> 00:07:59,800
This is actually a
two by two matrix.

153
00:07:59,800 --> 00:08:06,430
And if I do this operation,
S operate on this X matrix,

154
00:08:06,430 --> 00:08:10,130
then that is actually is
going to give you the X tilde.

155
00:08:10,130 --> 00:08:10,630
OK?

156
00:08:10,630 --> 00:08:15,596
So that's the nature of the
role of the symmetry matrix.

157
00:08:15,596 --> 00:08:17,220
OK?

158
00:08:17,220 --> 00:08:19,620
Any questions?

159
00:08:19,620 --> 00:08:21,060
OK.

160
00:08:21,060 --> 00:08:25,560
So now we have defined
a symmetry matrix.

161
00:08:25,560 --> 00:08:30,210
And then you can ask,
why do we actually care,

162
00:08:30,210 --> 00:08:32,970
and why do we actually
introduce symmetry matrix?

163
00:08:32,970 --> 00:08:33,809
Right?

164
00:08:33,809 --> 00:08:37,530
Because I can always write
down the X tilde in that way.

165
00:08:37,530 --> 00:08:42,340
That is because I think by
the end of this lecture,

166
00:08:42,340 --> 00:08:47,820
you will find that
if S matrix describes

167
00:08:47,820 --> 00:08:51,600
the symmetry of the
system, OK, that

168
00:08:51,600 --> 00:08:57,930
would mean S matrix will commute
with M minus 1 K matrix--

169
00:08:57,930 --> 00:09:01,020
which, we don't know commute
yet, but I will introduce you--

170
00:09:01,020 --> 00:09:06,770
that means M minus 1 K matrix
and S can actually swap freely.

171
00:09:06,770 --> 00:09:08,070
OK?

172
00:09:08,070 --> 00:09:12,240
If that happens,
then S matrix will

173
00:09:12,240 --> 00:09:17,240
share the same sets
of eigenvectors

174
00:09:17,240 --> 00:09:19,650
as the M minus 1 K matrix.

175
00:09:19,650 --> 00:09:21,300
What does that mean?

176
00:09:21,300 --> 00:09:22,920
That means-- OK.

177
00:09:22,920 --> 00:09:24,840
Before we are doing
this solution,

178
00:09:24,840 --> 00:09:29,060
right, we are solving M minus
1 K matrix eigenvalue problem,

179
00:09:29,060 --> 00:09:30,630
right?

180
00:09:30,630 --> 00:09:32,550
And then, we get
the eigenvector,

181
00:09:32,550 --> 00:09:36,940
which is the amplitude
ratio of normal modes.

182
00:09:36,940 --> 00:09:40,950
And that means you
have an alternative way

183
00:09:40,950 --> 00:09:43,170
to get the normal mode.

184
00:09:43,170 --> 00:09:47,480
You can solve the eigenvalue
problem of S matrix,

185
00:09:47,480 --> 00:09:51,665
then you can get the same
set of amplitude ratios

186
00:09:51,665 --> 00:09:55,550
as M minus 1 K matrix
eigenvalue problem.

187
00:09:55,550 --> 00:09:56,190
OK?

188
00:09:56,190 --> 00:10:01,170
And then usually, the
eigenvalue problem of S matrix

189
00:10:01,170 --> 00:10:05,710
is far much easier than
M minus 1 K matrix.

190
00:10:05,710 --> 00:10:06,210
OK?

191
00:10:06,210 --> 00:10:08,740
So that's actually
why we're doing this.

192
00:10:08,740 --> 00:10:09,400
OK?

193
00:10:09,400 --> 00:10:13,260
So now, I would
like to convince you

194
00:10:13,260 --> 00:10:24,870
that S matrix and M minus 1 K
matrix will share eigenvectors.

195
00:10:31,390 --> 00:10:32,870
OK?

196
00:10:32,870 --> 00:10:33,950
So.

197
00:10:33,950 --> 00:10:41,480
Let's go ahead and prove this,
or demonstrate this idea.

198
00:10:41,480 --> 00:10:42,110
OK?

199
00:10:42,110 --> 00:10:45,230
So the original equation
of motion looks like this.

200
00:10:45,230 --> 00:10:52,580
X double dot equal to
M minus 1 K X. Right?

201
00:10:52,580 --> 00:10:57,530
So now, this is actually the
original equation of motion.

202
00:10:57,530 --> 00:11:03,980
And if this system satisfy
the reflection symmetry,

203
00:11:03,980 --> 00:11:09,620
that means X tilde is
also a solution, right?

204
00:11:09,620 --> 00:11:11,780
Therefore, what does that mean?

205
00:11:11,780 --> 00:11:16,510
That means X tilde
double dot will be also

206
00:11:16,510 --> 00:11:23,650
equal to M minus 1 K X tilde.

207
00:11:23,650 --> 00:11:27,010
Because it's also a solution to
the equation of motion, right?

208
00:11:27,010 --> 00:11:28,450
That's pretty natural.

209
00:11:28,450 --> 00:11:29,530
OK?

210
00:11:29,530 --> 00:11:31,900
Now.

211
00:11:31,900 --> 00:11:34,420
I can actually use
this expression,

212
00:11:34,420 --> 00:11:38,890
X tilde is equal to
S times X. Right?

213
00:11:38,890 --> 00:11:41,160
All of those things
are matrix, OK?

214
00:11:41,160 --> 00:11:43,220
Just to be careful.

215
00:11:43,220 --> 00:11:46,870
That means I can
write this like this--

216
00:11:46,870 --> 00:12:00,430
S X double dot equal
to M minus 1 K S X. OK?

217
00:12:00,430 --> 00:12:03,460
There's no matrix,
and I also replace--

218
00:12:03,460 --> 00:12:08,332
I'm just replacing
X tilde by S X. OK?

219
00:12:08,332 --> 00:12:12,830
And also, I call this, actually,
1; I call this actually 2.

220
00:12:12,830 --> 00:12:13,600
OK?

221
00:12:13,600 --> 00:12:20,130
I can multiply X from
the left-hand side of 1.

222
00:12:20,130 --> 00:12:20,910
OK?

223
00:12:20,910 --> 00:12:22,320
And see what will happen.

224
00:12:22,320 --> 00:12:24,670
So if I do that, then
what I am going to get

225
00:12:24,670 --> 00:12:27,540
is S X double dot--

226
00:12:27,540 --> 00:12:28,270
OK?

227
00:12:28,270 --> 00:12:39,430
That will be equal to
S M minus 1 K X. OK?

228
00:12:39,430 --> 00:12:43,090
If you compare this equation,
and the equation number

229
00:12:43,090 --> 00:12:45,670
three, these two
equations, you will see

230
00:12:45,670 --> 00:12:48,530
that let-hand side is the same.

231
00:12:48,530 --> 00:12:49,840
Right?

232
00:12:49,840 --> 00:12:52,570
Right-hand side-- huh!

233
00:12:52,570 --> 00:12:54,920
Something interesting
is happening.

234
00:12:54,920 --> 00:12:59,020
M minus 1 K S must
be equal to S M

235
00:12:59,020 --> 00:13:02,870
minus 1 K. What does that mean?

236
00:13:02,870 --> 00:13:04,820
This means that
they are the same.

237
00:13:04,820 --> 00:13:11,980
M minus 1 K S is actually
equal to S M minus 1 K.

238
00:13:11,980 --> 00:13:17,330
So if I say, this distance
satisfy a symmetry

239
00:13:17,330 --> 00:13:21,650
described by S
matrix, that means

240
00:13:21,650 --> 00:13:26,350
X tilde, which is actually
the transformed amplitude,

241
00:13:26,350 --> 00:13:32,230
will be also a solution
to the equation of motion.

242
00:13:32,230 --> 00:13:38,800
And therefore, an inevitable
consequence is that M minus 1

243
00:13:38,800 --> 00:13:43,060
K S will be equal
to S M minus 1 K.

244
00:13:43,060 --> 00:13:46,150
Usually, when you
started physics,

245
00:13:46,150 --> 00:13:48,940
we write this in
terms of commutator.

246
00:13:55,320 --> 00:13:55,920
OK?

247
00:13:55,920 --> 00:13:59,250
So we call this, these two
things actually commute.

248
00:13:59,250 --> 00:14:00,270
OK?

249
00:14:00,270 --> 00:14:02,460
So commutator is
actually defined

250
00:14:02,460 --> 00:14:06,990
as A bracket of A and B.
This is actually equal--

251
00:14:06,990 --> 00:14:13,230
defined as A B minus B A. OK?

252
00:14:13,230 --> 00:14:16,330
If A and B commute--

253
00:14:16,330 --> 00:14:16,830
OK?

254
00:14:16,830 --> 00:14:20,050
It's this new word,
probably, for most of you--

255
00:14:20,050 --> 00:14:26,090
if they commute, that
means A B in the bracket

256
00:14:26,090 --> 00:14:28,950
is equal to zero.

257
00:14:28,950 --> 00:14:30,720
OK?

258
00:14:30,720 --> 00:14:34,890
So this expression,
I can actually

259
00:14:34,890 --> 00:14:36,440
write it down like this.

260
00:14:36,440 --> 00:14:43,890
Commutator of S M minus 1
K, that is equal to zero.

261
00:14:43,890 --> 00:14:47,010
And you will see
this really a lot

262
00:14:47,010 --> 00:14:49,180
when you study quantum physics.

263
00:14:49,180 --> 00:14:50,010
OK?

264
00:14:50,010 --> 00:14:51,540
So I hope this
actually gives you

265
00:14:51,540 --> 00:14:54,870
some flavor about commutator.

266
00:14:54,870 --> 00:14:56,100
OK?

267
00:14:56,100 --> 00:14:59,100
So now, that's
actually pretty nice.

268
00:14:59,100 --> 00:15:01,950
This means that
they commute, OK?

269
00:15:01,950 --> 00:15:16,120
If I take X of t this is
equal to A 1 cosine omega 1 t.

270
00:15:16,120 --> 00:15:18,810
OK?

271
00:15:18,810 --> 00:15:24,378
So, this means that
A is actually--

272
00:15:24,378 --> 00:15:26,670
sorry, X is actually
a solution, which

273
00:15:26,670 --> 00:15:28,570
is a normal mode, a solution.

274
00:15:28,570 --> 00:15:29,070
Right?

275
00:15:29,070 --> 00:15:34,680
And A is actually amplitude the
vector, the amplitude vector

276
00:15:34,680 --> 00:15:37,620
of the first normal
mode, and omega 1

277
00:15:37,620 --> 00:15:43,091
is actually the first
normal mode frequency.

278
00:15:43,091 --> 00:15:43,590
OK?

279
00:15:43,590 --> 00:15:58,110
If this is the case, then
I will have X tilde of t

280
00:15:58,110 --> 00:16:07,490
will be also oppositional
to A 1 cosine omega 1 t.

281
00:16:07,490 --> 00:16:11,300
Because if I actually
exchange X1 and X2,

282
00:16:11,300 --> 00:16:15,004
the oscillation frequency
is not going to change.

283
00:16:15,004 --> 00:16:16,860
Right?

284
00:16:16,860 --> 00:16:21,860
Therefore, since this system
is in the same normal mode

285
00:16:21,860 --> 00:16:27,100
with angular frequency
omega 1, therefore

286
00:16:27,100 --> 00:16:32,420
the amplitude ratio of the
first and second oscillator

287
00:16:32,420 --> 00:16:34,531
will stay constant.

288
00:16:34,531 --> 00:16:35,030
Right?

289
00:16:35,030 --> 00:16:37,271
Because you are in one
of the normal modes.

290
00:16:37,271 --> 00:16:37,770
Right?

291
00:16:37,770 --> 00:16:42,095
Therefore, I can
conclude that X tilde

292
00:16:42,095 --> 00:16:46,130
is going to be proportional
to this expression.

293
00:16:46,130 --> 00:16:47,890
Because they are in
the same normal mode,

294
00:16:47,890 --> 00:16:50,240
oscillating at the
same frequency.

295
00:16:50,240 --> 00:16:50,780
OK?

296
00:16:50,780 --> 00:16:52,730
Is that too fast?

297
00:16:52,730 --> 00:16:54,770
Everybody is following?

298
00:16:54,770 --> 00:16:55,760
OK.

299
00:16:55,760 --> 00:16:57,570
So that's nice.

300
00:16:57,570 --> 00:17:12,710
So this means that S X of t will
be equal to S A 1 cosine omega

301
00:17:12,710 --> 00:17:14,720
1t, OK?

302
00:17:14,720 --> 00:17:16,609
So this is actually
coming from here, right?

303
00:17:16,609 --> 00:17:24,270
I am replacing X tilde by S
X based on this definition.

304
00:17:24,270 --> 00:17:26,119
OK?

305
00:17:26,119 --> 00:17:29,980
Then again, I replace, I
write, X explicitly which

306
00:17:29,980 --> 00:17:32,990
is actually A cosine omega 1 t.

307
00:17:32,990 --> 00:17:33,890
OK?

308
00:17:33,890 --> 00:17:35,330
Then you get this expression.

309
00:17:35,330 --> 00:17:39,090
And from this
expression above, you

310
00:17:39,090 --> 00:17:44,090
see that you conclude that
this is proportional to A 1

311
00:17:44,090 --> 00:17:48,000
cosine omega 1 t.

312
00:17:48,000 --> 00:17:49,070
That's very nice.

313
00:17:49,070 --> 00:17:57,240
That means S A cosine omega 1
t is proportional to A 1 cosine

314
00:17:57,240 --> 00:17:58,620
omega t.

315
00:17:58,620 --> 00:18:00,865
And you can actually
cancel this.

316
00:18:00,865 --> 00:18:08,060
And you see that S A 1
is proportional to A 1.

317
00:18:08,060 --> 00:18:17,760
Or I can write it as S A
1 is equal to beta A 1.

318
00:18:17,760 --> 00:18:19,880
What does that mean?

319
00:18:19,880 --> 00:18:26,060
This means that A 1 originally--

320
00:18:26,060 --> 00:18:27,850
where's A 1 coming from?

321
00:18:27,850 --> 00:18:32,830
A 1 is the amplitude
of all the components

322
00:18:32,830 --> 00:18:34,340
in the first normal mode.

323
00:18:34,340 --> 00:18:36,400
Right?

324
00:18:36,400 --> 00:18:39,770
That's coming from the
eigenvalue problem, which

325
00:18:39,770 --> 00:18:42,700
it actually does in this light.

326
00:18:42,700 --> 00:18:45,970
Eigenvalue problem M
minus 1 K A equal to omega

327
00:18:45,970 --> 00:18:49,360
square A will give you the
solution of normal mode

328
00:18:49,360 --> 00:18:56,987
and their eigenvectors, which
is amplitude ratios of all

329
00:18:56,987 --> 00:18:58,195
the components in the system.

330
00:18:58,195 --> 00:18:58,810
Right?

331
00:18:58,810 --> 00:19:06,350
So that means A 1 is not only M
minus 1 K matrix eigenvectors,

332
00:19:06,350 --> 00:19:11,970
it's also eigenvector
of S matrix.

333
00:19:11,970 --> 00:19:13,350
OK?

334
00:19:13,350 --> 00:19:15,670
So that is actually
very good news.

335
00:19:15,670 --> 00:19:18,640
And I can also do the
same thing for A 2,

336
00:19:18,640 --> 00:19:21,540
to prove that it
also works for A 2--

337
00:19:21,540 --> 00:19:23,920
the derivation is
identical, so I am not

338
00:19:23,920 --> 00:19:25,490
going to do that again.

339
00:19:25,490 --> 00:19:33,220
So that means, actually,
starting from here, OK--

340
00:19:33,220 --> 00:19:37,960
if X and X tilde
are both solutions

341
00:19:37,960 --> 00:19:40,800
to the equation of motion.

342
00:19:40,800 --> 00:19:46,390
I will conclude that S matrix
and M minus 1 K matrix,

343
00:19:46,390 --> 00:19:47,800
they commute.

344
00:19:47,800 --> 00:19:49,690
OK?

345
00:19:49,690 --> 00:19:55,720
How to tell if a system satisfy
a specific symmetry defined

346
00:19:55,720 --> 00:19:57,550
by my symmetry matrix?

347
00:19:57,550 --> 00:20:02,940
Is by this way, you can check
if M minus 1 K and S commute.

348
00:20:02,940 --> 00:20:06,700
If they commute, that
means the system actually

349
00:20:06,700 --> 00:20:09,550
satisfy this symmetry.

350
00:20:09,550 --> 00:20:13,570
And also, the consequence
is that from there,

351
00:20:13,570 --> 00:20:17,090
you will conclude
that if you have

352
00:20:17,090 --> 00:20:20,640
also a set of
eigenvectors from M

353
00:20:20,640 --> 00:20:23,690
minus 1 K matrix
eigenvalue problem, then

354
00:20:23,690 --> 00:20:31,360
that is going to be also
the eigenvector of S. OK?

355
00:20:31,360 --> 00:20:32,725
Any questions?

356
00:20:36,290 --> 00:20:38,030
OK.

357
00:20:38,030 --> 00:20:43,400
So M minus 1 K eigenvectors.

358
00:20:48,610 --> 00:20:52,648
Also S eigenvector.

359
00:20:55,641 --> 00:20:56,140
OK?

360
00:20:56,140 --> 00:20:57,670
That's actually
what we have learned

361
00:20:57,670 --> 00:21:00,370
from this small exercise.

362
00:21:00,370 --> 00:21:03,550
Now, you can say,
wait, wait, wait, wait.

363
00:21:03,550 --> 00:21:05,650
This is actually not
what we need, right?

364
00:21:05,650 --> 00:21:09,910
I would like-- we would like
to argue that S matrix--

365
00:21:09,910 --> 00:21:13,720
I can solve S matrix
eigenvalue problem,

366
00:21:13,720 --> 00:21:16,760
and I can learn about the
solution of M minus 1 K matrix,

367
00:21:16,760 --> 00:21:17,260
right?

368
00:21:17,260 --> 00:21:19,770
This logic is actually in the
opposite direction, right?

369
00:21:19,770 --> 00:21:22,420
You said, OK, you
solved things already,

370
00:21:22,420 --> 00:21:25,360
then, actually, it's also S
matrix eigenvalue problem.

371
00:21:25,360 --> 00:21:30,280
So now what I am going to
do is to reverse the logic,

372
00:21:30,280 --> 00:21:31,940
and see if it works.

373
00:21:31,940 --> 00:21:32,440
OK?

374
00:21:32,440 --> 00:21:34,790
Again, to see what will happen.

375
00:21:34,790 --> 00:21:35,830
OK?

376
00:21:35,830 --> 00:21:41,350
So now, I would like to prove
that if I solve S matrix

377
00:21:41,350 --> 00:21:47,660
eigenvalue problem, I have also
solved the eigenvectors for M

378
00:21:47,660 --> 00:21:48,810
minus 1 K matrix.

379
00:21:48,810 --> 00:21:51,400
Run the logic in the
opposite direction.

380
00:21:51,400 --> 00:21:52,300
OK?

381
00:21:52,300 --> 00:21:55,810
So, if I were given two things--

382
00:21:55,810 --> 00:22:05,870
one, S A is equal to beta A.
Number two, S matrix and M

383
00:22:05,870 --> 00:22:10,160
minus 1 K matrix commute.

384
00:22:10,160 --> 00:22:10,660
OK?

385
00:22:10,660 --> 00:22:13,870
If those are the
given conditions,

386
00:22:13,870 --> 00:22:20,010
then I can actually conclude
that S M minus 1 K--

387
00:22:20,010 --> 00:22:20,510
OK?

388
00:22:20,510 --> 00:22:24,280
I can actually contract
this expression--

389
00:22:24,280 --> 00:22:27,730
I write that S M
minus 1 K A, OK?

390
00:22:27,730 --> 00:22:30,220
Because they commute, right?

391
00:22:30,220 --> 00:22:35,140
They can actually
swap M minus 1 K and S

392
00:22:35,140 --> 00:22:38,920
safely without actually
introducing any more terms.

393
00:22:38,920 --> 00:22:47,500
This will be equal to
M minus 1 K S A. OK?

394
00:22:47,500 --> 00:22:50,800
And S A, from the
first expression,

395
00:22:50,800 --> 00:22:53,420
S A is equal to beta A. Right?

396
00:22:53,420 --> 00:22:56,890
Beta is a number, OK?

397
00:22:56,890 --> 00:23:04,160
Therefore this expression will
become beta M minus 1 K A.

398
00:23:04,160 --> 00:23:06,690
So, beta can penetrate
through matrix,

399
00:23:06,690 --> 00:23:10,070
because beta is just a
number, is eigenvalue.

400
00:23:10,070 --> 00:23:12,110
It's eigenvalue of S matrix.

401
00:23:12,110 --> 00:23:14,070
OK?

402
00:23:14,070 --> 00:23:15,200
So what does this mean?

403
00:23:18,570 --> 00:23:20,320
OK.

404
00:23:20,320 --> 00:23:22,100
So what does it mean?

405
00:23:22,100 --> 00:23:30,310
So this means that if you look
at this part and that part--

406
00:23:30,310 --> 00:23:33,760
you look at the beginning and
the end of the expression-- you

407
00:23:33,760 --> 00:23:41,470
immediately conclude that M
minus 1 K A, this expression

408
00:23:41,470 --> 00:23:46,371
is also an eigenvector
of S matrix.

409
00:23:46,371 --> 00:23:46,870
Right?

410
00:23:46,870 --> 00:23:52,756
So you have S matrix
acting M minus 1 K A.

411
00:23:52,756 --> 00:23:56,750
And that will give you
something proportional

412
00:23:56,750 --> 00:24:00,370
to M minus 1 K A. You see?

413
00:24:00,370 --> 00:24:01,810
It's magic, right?

414
00:24:01,810 --> 00:24:03,810
It's actually not magic,
but it's actually just,

415
00:24:03,810 --> 00:24:06,730
you know, really
logical extension.

416
00:24:06,730 --> 00:24:07,370
Right?

417
00:24:07,370 --> 00:24:08,700
OK?

418
00:24:08,700 --> 00:24:09,640
Very cool!

419
00:24:09,640 --> 00:24:14,140
So that means this is also
an eigenvector of S. Right?

420
00:24:14,140 --> 00:24:16,570
And also, another thing
which is interesting

421
00:24:16,570 --> 00:24:22,203
is that they share the
same eigenvalue, beta.

422
00:24:22,203 --> 00:24:23,030
Right?

423
00:24:23,030 --> 00:24:24,435
They have the same eigenvalue.

424
00:24:28,440 --> 00:24:30,960
OK?

425
00:24:30,960 --> 00:24:34,230
So, if eigenvalues of S--

426
00:24:34,230 --> 00:24:36,390
so you can get several
eigenvalues, right?

427
00:24:36,390 --> 00:24:39,300
In this case, two by two
matrix, you will get--

428
00:24:39,300 --> 00:24:40,050
how many?

429
00:24:40,050 --> 00:24:41,280
Two, right?

430
00:24:41,280 --> 00:24:43,080
Two eigenvalues.

431
00:24:43,080 --> 00:24:45,900
If those two eigenvalues
are different,

432
00:24:45,900 --> 00:24:49,350
then I can conclude
that M minus 1

433
00:24:49,350 --> 00:24:57,580
K A must be proportional
to A. Right?

434
00:24:57,580 --> 00:25:01,750
Because this is actually
the same eigenvalue problem,

435
00:25:01,750 --> 00:25:04,480
and the same eigenvalue, beta.

436
00:25:04,480 --> 00:25:07,490
Since all the eigenvalues
from the solution

437
00:25:07,490 --> 00:25:12,500
of eigenvalue problem
of S A equal to beta A,

438
00:25:12,500 --> 00:25:14,590
those eigenvalues
are all different,

439
00:25:14,590 --> 00:25:16,450
therefore I can
argue that M minus 1

440
00:25:16,450 --> 00:25:20,770
K A is proportional to A. OK?

441
00:25:20,770 --> 00:25:31,740
Therefore, M minus 1 K A is
equal to omega square A. Omega

442
00:25:31,740 --> 00:25:34,276
square is actually
some constant.

443
00:25:37,030 --> 00:25:38,370
OK?

444
00:25:38,370 --> 00:25:41,370
This is actually amazing,
because that means given

445
00:25:41,370 --> 00:25:44,040
the two conditions--
the first one,

446
00:25:44,040 --> 00:25:48,780
I can figure out the eigenvalue
and the eigenvectors of S

447
00:25:48,780 --> 00:25:54,170
matrix; second, if S matrix and
M minus 1 K matrix interaction

448
00:25:54,170 --> 00:25:57,300
matrix, they commute--

449
00:25:57,300 --> 00:26:01,170
then I can actually
already figure out

450
00:26:01,170 --> 00:26:07,340
what are the eigenvectors
of M minus 1 K matrix.

451
00:26:07,340 --> 00:26:09,950
OK?

452
00:26:09,950 --> 00:26:13,480
And another thing which
we've learned from here

453
00:26:13,480 --> 00:26:16,550
is that, wow, that's good!

454
00:26:16,550 --> 00:26:19,760
Because the eigenvectors
are already solved.

455
00:26:19,760 --> 00:26:23,690
Therefore, I just have
to calculate this.

456
00:26:23,690 --> 00:26:25,420
It's just a normal operation.

457
00:26:25,420 --> 00:26:27,320
It's not the eigenvalue
problem anymore.

458
00:26:27,320 --> 00:26:30,530
I just multiply M
minus 1 K times A,

459
00:26:30,530 --> 00:26:34,350
then I can actually get
the value omega square.

460
00:26:34,350 --> 00:26:34,850
You see?

461
00:26:34,850 --> 00:26:38,260
That's actually much easier than
solving the eigenvalue problem

462
00:26:38,260 --> 00:26:40,505
of M minus 1 K matrix.

463
00:26:40,505 --> 00:26:41,005
OK?

464
00:26:43,710 --> 00:26:47,980
That's actually very good news.

465
00:26:47,980 --> 00:26:53,370
Finally, I think the most
important consequence

466
00:26:53,370 --> 00:27:04,870
is that once we solve
this system, which

467
00:27:04,870 --> 00:27:10,860
satisfy the symmetry
described by this S matrix,

468
00:27:10,860 --> 00:27:16,290
we have solved all the
possible systems which

469
00:27:16,290 --> 00:27:18,390
satisfy the same symmetry.

470
00:27:18,390 --> 00:27:23,520
For example, in this case,
I solve a coupled pendulum

471
00:27:23,520 --> 00:27:24,670
problem, OK?

472
00:27:24,670 --> 00:27:26,290
They look symmetric.

473
00:27:26,290 --> 00:27:27,420
Right?

474
00:27:27,420 --> 00:27:31,260
And I can, of course,
I can draw another one,

475
00:27:31,260 --> 00:27:32,495
which is like this.

476
00:27:32,495 --> 00:27:34,140
It's more circular.

477
00:27:34,140 --> 00:27:36,730
And there are two walls,
which is actually--

478
00:27:36,730 --> 00:27:40,510
there are three springs
connected to the wall.

479
00:27:40,510 --> 00:27:42,720
This problem is already
also solved, right?

480
00:27:42,720 --> 00:27:46,330
Because it also satisfy
the same symmetry.

481
00:27:46,330 --> 00:27:50,070
And of course-- like,
you know, like this,

482
00:27:50,070 --> 00:27:53,510
go crazy, and even more.

483
00:27:53,510 --> 00:27:55,180
This is also solved!

484
00:27:55,180 --> 00:27:55,680
Right?

485
00:27:55,680 --> 00:27:57,000
Because this is also symmetric.

486
00:27:57,000 --> 00:27:57,530
Right?

487
00:27:57,530 --> 00:27:59,420
I can add more.

488
00:27:59,420 --> 00:28:00,190
Right?

489
00:28:00,190 --> 00:28:01,240
Like this.

490
00:28:01,240 --> 00:28:02,750
This is also symmetric.

491
00:28:02,750 --> 00:28:03,720
Right?

492
00:28:03,720 --> 00:28:06,150
And this-- let's think.

493
00:28:06,150 --> 00:28:11,430
The eigenvector of this M minus
1 K matrix eigenvalue problem

494
00:28:11,430 --> 00:28:15,420
will be identical to what
we have already solved here.

495
00:28:15,420 --> 00:28:16,290
OK?

496
00:28:16,290 --> 00:28:18,810
So, that is actually
really amazing.

497
00:28:18,810 --> 00:28:22,890
If you speak the right language,
and cut into the problem

498
00:28:22,890 --> 00:28:25,150
in the right angle,
you actually find

499
00:28:25,150 --> 00:28:31,010
that actually, you can solve
multiple problems at one time.

500
00:28:31,010 --> 00:28:32,970
OK?

501
00:28:32,970 --> 00:28:33,740
Any questions?

502
00:28:36,710 --> 00:28:37,910
OK.

503
00:28:37,910 --> 00:28:40,950
So now this is
actually very nice,

504
00:28:40,950 --> 00:28:44,550
and this is actually a
very important preparation

505
00:28:44,550 --> 00:28:48,690
to the next step, actually.

506
00:28:48,690 --> 00:28:52,500
So now, we have understood
coupled oscillator,

507
00:28:52,500 --> 00:28:55,860
and we have learned a
little bit about symmetry.

508
00:28:55,860 --> 00:28:59,460
Therefore, I would
like to go to infinite

509
00:28:59,460 --> 00:29:01,981
number of coupled oscillator.

510
00:29:01,981 --> 00:29:02,480
OK?

511
00:29:02,480 --> 00:29:04,770
So that is actually
the next step, which

512
00:29:04,770 --> 00:29:10,440
we are going to move on in 8.03

513
00:29:10,440 --> 00:29:13,920
So this is actually one
example infinite system.

514
00:29:13,920 --> 00:29:15,660
OK?

515
00:29:15,660 --> 00:29:17,501
I cannot write the
whole universe.

516
00:29:17,501 --> 00:29:18,000
Why?

517
00:29:18,000 --> 00:29:21,480
Because it's infinite, so I
couldn't include everything

518
00:29:21,480 --> 00:29:22,585
in the slide.

519
00:29:22,585 --> 00:29:24,210
But this is actually
an example system.

520
00:29:24,210 --> 00:29:27,520
Done OK?

521
00:29:27,520 --> 00:29:30,870
Looks hopeless, right?

522
00:29:30,870 --> 00:29:34,060
In general, we don't
know how to solve

523
00:29:34,060 --> 00:29:37,030
infinite system, because if you
have infinite number of things

524
00:29:37,030 --> 00:29:41,700
that are connected to
each other in random ways,

525
00:29:41,700 --> 00:29:46,350
then the problem becomes
really, really complicated.

526
00:29:46,350 --> 00:29:46,920
OK?

527
00:29:46,920 --> 00:29:52,500
In general, I don't know
how to solve this problem.

528
00:29:52,500 --> 00:29:56,930
And if you are a EE major,
the first thing, maybe, you

529
00:29:56,930 --> 00:29:59,760
like to do is, ah, now
I have this picture,

530
00:29:59,760 --> 00:30:01,770
and I can put everything
in my computer,

531
00:30:01,770 --> 00:30:04,710
and see how things evolve
as a function of time!

532
00:30:04,710 --> 00:30:06,810
Right?

533
00:30:06,810 --> 00:30:08,730
Of course we can rely
on the computers,

534
00:30:08,730 --> 00:30:10,810
and see what we
can learn from it.

535
00:30:10,810 --> 00:30:14,100
And if you made your
major of mathematics,

536
00:30:14,100 --> 00:30:16,344
you will say, no, this
is not the problem

537
00:30:16,344 --> 00:30:17,260
I am going to work on.

538
00:30:19,900 --> 00:30:20,400
OK?

539
00:30:20,400 --> 00:30:23,280
I don't care.

540
00:30:23,280 --> 00:30:26,760
But as a physicist, what we
are going to do is that, huh--

541
00:30:26,760 --> 00:30:30,366
we look at this
infinite system, OK?

542
00:30:30,366 --> 00:30:31,960
It's kind of interesting, right?

543
00:30:31,960 --> 00:30:35,040
It's a lot of things, a lot
of small balls connected

544
00:30:35,040 --> 00:30:36,600
to big balls, right?

545
00:30:36,600 --> 00:30:39,250
Super big ones, and plotting
things in log scale.

546
00:30:39,250 --> 00:30:41,580
So those balls are
really, really large

547
00:30:41,580 --> 00:30:45,210
compared to all the other
balls connected to this system.

548
00:30:45,210 --> 00:30:46,860
Therefore, as a
physicist, I'm going

549
00:30:46,860 --> 00:30:49,440
to ignore all the other balls.

550
00:30:52,710 --> 00:30:56,760
Oh, if I do that,
then it becomes--

551
00:30:56,760 --> 00:30:59,800
there is some kind of
symmetry you can actually

552
00:30:59,800 --> 00:31:01,440
see from here, right?

553
00:31:01,440 --> 00:31:04,960
What is actually the
symmetry? you see?

554
00:31:04,960 --> 00:31:07,960
There are three balls that
connected to each other.

555
00:31:07,960 --> 00:31:10,080
They are equally spaced.

556
00:31:10,080 --> 00:31:13,500
We have a translation symmetry.

557
00:31:13,500 --> 00:31:14,670
You see?

558
00:31:14,670 --> 00:31:16,320
So you can see that,
actually, that's

559
00:31:16,320 --> 00:31:19,490
how we think about a problem.

560
00:31:19,490 --> 00:31:22,950
Of course, different field have
different kind of thinking,

561
00:31:22,950 --> 00:31:26,220
and different kind of problem
they would like to focus on.

562
00:31:26,220 --> 00:31:28,410
But as a physicist,
I would like to know

563
00:31:28,410 --> 00:31:30,570
how the system will work,
and that is actually

564
00:31:30,570 --> 00:31:32,220
what I'm going to do.

565
00:31:32,220 --> 00:31:33,880
OK?

566
00:31:33,880 --> 00:31:36,400
So that's very nice.

567
00:31:36,400 --> 00:31:41,030
We are going to discuss
infinite system.

568
00:31:41,030 --> 00:31:42,870
So what is actually
the infinite system

569
00:31:42,870 --> 00:31:45,626
I am going to talk about?

570
00:31:45,626 --> 00:31:52,630
It's actually there is infinite
system with space translation

571
00:31:52,630 --> 00:31:54,580
symmetry.

572
00:31:54,580 --> 00:32:04,270
So, to save some time, I have
already written down the matrix

573
00:32:04,270 --> 00:32:08,290
involving this system here.

574
00:32:08,290 --> 00:32:13,790
What I am interested is
mass sprint system, OK?

575
00:32:13,790 --> 00:32:16,810
Infinite number of
mass and spring.

576
00:32:16,810 --> 00:32:22,590
And they actually satisfy
space translation symmetry.

577
00:32:22,590 --> 00:32:25,170
OK?

578
00:32:25,170 --> 00:32:29,860
They are connected to each other
by springs, with natural length

579
00:32:29,860 --> 00:32:34,780
A and spring constant K. OK?

580
00:32:34,780 --> 00:32:36,520
And there are infinite
number of them,

581
00:32:36,520 --> 00:32:39,520
actually, lined up
from the left-hand side

582
00:32:39,520 --> 00:32:42,220
of the edge of the universe
to the right-hand side

583
00:32:42,220 --> 00:32:43,520
edge of the universe.

584
00:32:43,520 --> 00:32:44,020
OK?

585
00:32:44,020 --> 00:32:45,280
I've prepared this system.

586
00:32:45,280 --> 00:32:45,780
OK?

587
00:32:45,780 --> 00:32:47,170
It took me a long time.

588
00:32:47,170 --> 00:32:49,030
OK?

589
00:32:49,030 --> 00:32:50,340
All right?

590
00:32:50,340 --> 00:32:53,420
But it's very difficult to
describe this kind of system,

591
00:32:53,420 --> 00:32:53,920
right?

592
00:32:53,920 --> 00:32:57,130
So the first thing we
have learned from 8.03

593
00:32:57,130 --> 00:32:59,630
is that in order to
describe this system,

594
00:32:59,630 --> 00:33:03,210
I need to define a
coordinate system, right?

595
00:33:03,210 --> 00:33:07,100
And also have everything
properly labeled.

596
00:33:07,100 --> 00:33:08,830
So I introduce a label--

597
00:33:08,830 --> 00:33:13,480
j minus 1 j, j plus
one, j plus two--

598
00:33:13,480 --> 00:33:17,170
just to name each little
mass I'm talking about.

599
00:33:17,170 --> 00:33:17,740
OK?

600
00:33:17,740 --> 00:33:19,300
No other purpose.

601
00:33:19,300 --> 00:33:24,930
Then, once I have the label, I
can actually write everything,

602
00:33:24,930 --> 00:33:29,680
express the displacement of
little mass, as X j minus 1,

603
00:33:29,680 --> 00:33:34,650
X j X j plus one, X j plus two.

604
00:33:34,650 --> 00:33:39,410
That's just the displacement
from the equilibrium position

605
00:33:39,410 --> 00:33:40,310
of the mass.

606
00:33:40,310 --> 00:33:42,690
OK?

607
00:33:42,690 --> 00:33:44,675
And this system
will have equation

608
00:33:44,675 --> 00:33:46,720
of motion looks like this.

609
00:33:46,720 --> 00:33:53,920
So if now I focus on
the little mass, Z. OK?

610
00:33:53,920 --> 00:33:57,810
Then I can actually write
down the equation of motion.

611
00:33:57,810 --> 00:34:05,560
There are two springs
connected to these mass.

612
00:34:05,560 --> 00:34:06,220
Right?

613
00:34:06,220 --> 00:34:11,230
Therefore, you are going
to have two spring force.

614
00:34:11,230 --> 00:34:12,040
Right?

615
00:34:12,040 --> 00:34:15,100
Since this is actually idealize
the springs with spring

616
00:34:15,100 --> 00:34:18,580
constant capital K,
therefore, I can write down

617
00:34:18,580 --> 00:34:21,144
immediately the
equation of motion

618
00:34:21,144 --> 00:34:27,790
is actually equal to M X
double dot j is equal to minus

619
00:34:27,790 --> 00:34:32,784
K X j minus X j minus 1 minus--

620
00:34:32,784 --> 00:34:37,300
this is actually the
right-hand side spring force--

621
00:34:37,300 --> 00:34:41,050
minus K X j minus X j plus 1.

622
00:34:41,050 --> 00:34:43,139
We have done this
exercise before, right,

623
00:34:43,139 --> 00:34:46,560
with a simpler problem.

624
00:34:46,560 --> 00:34:47,380
OK?

625
00:34:47,380 --> 00:34:50,650
As usual, I can
collect all the parents

626
00:34:50,650 --> 00:34:58,000
associated with X j minus 1,
X j, and X j plus 1, together.

627
00:34:58,000 --> 00:35:01,900
Then I get this expression,
which actually looks nice.

628
00:35:01,900 --> 00:35:03,840
OK?

629
00:35:03,840 --> 00:35:08,550
And I assume that this
system is actually undergoing

630
00:35:08,550 --> 00:35:12,810
some kind of oscillation.

631
00:35:12,810 --> 00:35:13,440
OK?

632
00:35:13,440 --> 00:35:17,010
Therefore, I assume
that this solution, X j

633
00:35:17,010 --> 00:35:22,730
will be equal to A j is
the amplitude of j's mass.

634
00:35:22,730 --> 00:35:23,610
OK?

635
00:35:23,610 --> 00:35:28,380
Cosine omega t plus phi, omega
is actually the oscillation

636
00:35:28,380 --> 00:35:31,020
frequency, and phi is
actually the phase,

637
00:35:31,020 --> 00:35:34,841
and I don't know why this is
actually omega and A j yet.

638
00:35:34,841 --> 00:35:35,340
OK?

639
00:35:35,340 --> 00:35:37,930
We would like to
figure that out.

640
00:35:37,930 --> 00:35:45,240
And as usual, you can actually
write down the M matrix, OK?

641
00:35:45,240 --> 00:35:50,520
M matrix is actually really
simple, in the diagonal terms--

642
00:35:50,520 --> 00:35:53,760
diagonal terms are all m,
and the off diagonal terms

643
00:35:53,760 --> 00:35:55,462
are all zero.

644
00:35:55,462 --> 00:35:57,096
Right?

645
00:35:57,096 --> 00:35:58,720
And you don't really
need to copy them,

646
00:35:58,720 --> 00:36:01,650
because they're all derived
in the lecture notes.

647
00:36:01,650 --> 00:36:03,411
M minus 1 K matrix--

648
00:36:03,411 --> 00:36:03,910
ha!

649
00:36:03,910 --> 00:36:06,940
I have already
arranged my terms here;

650
00:36:06,940 --> 00:36:08,480
therefore it looks like this.

651
00:36:08,480 --> 00:36:12,070
It have a strange structure,
you have three terms,

652
00:36:12,070 --> 00:36:17,790
kind of in the diagonal
terms, and this actually

653
00:36:17,790 --> 00:36:23,740
is shifting as a function
of number of rows,

654
00:36:23,740 --> 00:36:29,380
and all the other parts of
the matrix actually zero.

655
00:36:29,380 --> 00:36:29,880
OK?

656
00:36:29,880 --> 00:36:34,440
It's an infinite times
infinite dimension matrix.

657
00:36:34,440 --> 00:36:37,100
Finally, I would
like to also write my

658
00:36:37,100 --> 00:36:41,080
A matrix is the vector
of amplitude, right?

659
00:36:41,080 --> 00:36:42,880
So you have many, many numbers--

660
00:36:42,880 --> 00:36:45,940
A j, A j plus 1, A j plus 2.

661
00:36:45,940 --> 00:36:47,940
OK?

662
00:36:47,940 --> 00:36:49,970
And et cetera, et cetera.

663
00:36:49,970 --> 00:36:50,910
OK?

664
00:36:50,910 --> 00:36:55,100
Now, very easy, right?

665
00:36:55,100 --> 00:36:58,220
The question is actually
can be solved, right?

666
00:36:58,220 --> 00:37:02,730
You just have to solve the
M minus 1 K matrix, right?

667
00:37:02,730 --> 00:37:03,690
That's easy, right?

668
00:37:03,690 --> 00:37:07,950
It's an infinite number times
infinite number matrix, right?

669
00:37:07,950 --> 00:37:09,090
Super easy!

670
00:37:09,090 --> 00:37:09,930
No, actually not.

671
00:37:09,930 --> 00:37:10,708
Right?

672
00:37:10,708 --> 00:37:14,790
[LAUGHTER] So we are in trouble.

673
00:37:14,790 --> 00:37:16,960
I don't know how to
solve this problem.

674
00:37:16,960 --> 00:37:19,250
OK?

675
00:37:19,250 --> 00:37:22,000
What can we do?

676
00:37:22,000 --> 00:37:24,698
Anybody have any
suggestion to me?

677
00:37:24,698 --> 00:37:26,547
AUDIENCE: Ask the
math department?

678
00:37:26,547 --> 00:37:27,380
PROFESSOR: Ah, yeah!

679
00:37:27,380 --> 00:37:29,730
Math department is
coming in to help.

680
00:37:29,730 --> 00:37:31,360
Yes.

681
00:37:31,360 --> 00:37:33,650
But actually,
before asking them,

682
00:37:33,650 --> 00:37:37,320
we learn some concept, which
we just learned, right?

683
00:37:37,320 --> 00:37:40,070
This-- what kind of
property of this system?

684
00:37:40,070 --> 00:37:41,480
AUDIENCE: Symmetry.

685
00:37:41,480 --> 00:37:42,650
PROFESSOR: Symmetry!

686
00:37:42,650 --> 00:37:43,290
Right?

687
00:37:43,290 --> 00:37:44,590
We have symmetry.

688
00:37:44,590 --> 00:37:45,350
OK?

689
00:37:45,350 --> 00:37:50,300
So this M minus 1 K
matrix looks horrible.

690
00:37:50,300 --> 00:37:55,770
But if I write down
the symmetry matrix,

691
00:37:55,770 --> 00:37:58,150
actually, it looks
slightly better.

692
00:37:58,150 --> 00:37:59,570
OK?

693
00:37:59,570 --> 00:38:03,470
So what is actually
the symmetry matrix?

694
00:38:03,470 --> 00:38:06,090
So one observation we
can make from this system

695
00:38:06,090 --> 00:38:12,380
is that if I shift this
system, A, to the left, OK?

696
00:38:12,380 --> 00:38:15,980
I shift these two mass
to the left-hand side,

697
00:38:15,980 --> 00:38:17,180
I shift all the mass.

698
00:38:17,180 --> 00:38:19,220
I have to hire
many, many students

699
00:38:19,220 --> 00:38:22,180
to move all the mass from
left-hand side of the universe

700
00:38:22,180 --> 00:38:23,780
to right-hand side
of the universe.

701
00:38:23,780 --> 00:38:24,500
OK?

702
00:38:24,500 --> 00:38:28,776
And after they have done that,
the system looks the same.

703
00:38:28,776 --> 00:38:29,650
Right?

704
00:38:29,650 --> 00:38:31,310
That's very good, OK?

705
00:38:31,310 --> 00:38:34,640
After all the hard work, right?

706
00:38:34,640 --> 00:38:39,160
So what is actually going
to be the symmetry matrix?

707
00:38:39,160 --> 00:38:39,710
OK.

708
00:38:39,710 --> 00:38:42,050
Now, I would like
to achieve something

709
00:38:42,050 --> 00:38:46,290
which is A prime
equal to S A. And then

710
00:38:46,290 --> 00:38:53,030
this S actually shift the mass
by a distance of A. Right?

711
00:38:53,030 --> 00:38:56,320
So what would be the functional
formula for this S matrix?

712
00:38:56,320 --> 00:38:58,620
It would look like this.

713
00:38:58,620 --> 00:39:02,505
It's going to be 0, 1, 0, 0, 0--

714
00:39:02,505 --> 00:39:04,940
0, 0, 1, 0, 0, 0--

715
00:39:12,130 --> 00:39:14,060
looks like this.

716
00:39:14,060 --> 00:39:15,300
OK?

717
00:39:15,300 --> 00:39:20,740
So the next two diagonal
term is all one.

718
00:39:20,740 --> 00:39:24,130
All the rest of the
component is zero.

719
00:39:24,130 --> 00:39:24,940
OK?

720
00:39:24,940 --> 00:39:29,650
And this looks a lot
more friendly compared

721
00:39:29,650 --> 00:39:31,210
to M minus 1 K matrix, right?

722
00:39:31,210 --> 00:39:34,540
Still, this is
horrible thing to do,

723
00:39:34,540 --> 00:39:39,630
because this is infinite number
times infinite number dimension

724
00:39:39,630 --> 00:39:40,281
matrix.

725
00:39:40,281 --> 00:39:40,780
OK?

726
00:39:44,200 --> 00:39:45,460
So.

727
00:39:45,460 --> 00:39:51,010
We would like to find the
eigenvectors of S matrix.

728
00:39:51,010 --> 00:39:52,300
OK?

729
00:39:52,300 --> 00:39:55,540
So this means that
if I manage to solve

730
00:39:55,540 --> 00:39:57,360
the eigenvalue problem,
assuming that--

731
00:39:57,360 --> 00:40:01,700
OK, I haven't solved it, but
assuming that I can solve it,

732
00:40:01,700 --> 00:40:03,850
then what I'm going
to do is going

733
00:40:03,850 --> 00:40:10,450
to get this S A will be equal
to beta A, where A is actually

734
00:40:10,450 --> 00:40:16,220
a eigenvector of S matrix OK?

735
00:40:16,220 --> 00:40:20,415
And S A, we just
learned from here,

736
00:40:20,415 --> 00:40:22,018
is actually equal to A prime.

737
00:40:24,826 --> 00:40:32,950
So beta is the eigenvalue, and
A is actually the eigenvector.

738
00:40:36,160 --> 00:40:38,690
So that means,
originally, I have

739
00:40:38,690 --> 00:40:46,360
A, which is something something
A j, A j plus 1, A j plus 2,

740
00:40:46,360 --> 00:40:47,260
blah blah blah.

741
00:40:47,260 --> 00:40:48,400
OK?

742
00:40:48,400 --> 00:40:55,900
And A prime, after I actually
multiply A by S matrix,

743
00:40:55,900 --> 00:41:00,730
I get A prime, which
looks like this--

744
00:41:00,730 --> 00:41:08,350
A j plus 1, A j
plus 2, A j plus 3.

745
00:41:08,350 --> 00:41:10,320
OK?

746
00:41:10,320 --> 00:41:12,980
So what I am going to do is--

747
00:41:12,980 --> 00:41:15,210
what, actually,
this S matrix does

748
00:41:15,210 --> 00:41:23,350
is to shift the A
component one row, right?

749
00:41:23,350 --> 00:41:24,150
OK?

750
00:41:24,150 --> 00:41:27,850
So then, we basically
get this expression.

751
00:41:27,850 --> 00:41:31,530
And of course, A 1 is
equal to beta, which

752
00:41:31,530 --> 00:41:33,780
is a constant, times A. Right?

753
00:41:33,780 --> 00:41:42,220
So if you compare, for example,
here, you can get that--

754
00:41:42,220 --> 00:41:49,630
A j prime will be equal to
beta A j, which is actually

755
00:41:49,630 --> 00:41:52,091
equal to A j plus 1.

756
00:41:52,091 --> 00:41:52,590
Right?

757
00:41:52,590 --> 00:41:56,420
A j prime is actually
equal to A j plus 1, right?

758
00:41:56,420 --> 00:42:01,400
It's just shifting
one unique label.

759
00:42:01,400 --> 00:42:02,470
Right?

760
00:42:02,470 --> 00:42:04,740
OK.

761
00:42:04,740 --> 00:42:07,980
So this is actually the
expression I'm looking for.

762
00:42:07,980 --> 00:42:08,970
OK?

763
00:42:08,970 --> 00:42:12,060
We don't know yet why
this is actually beta.

764
00:42:12,060 --> 00:42:13,680
Beta is a number.

765
00:42:13,680 --> 00:42:16,590
Assuming that I can solve
the eigenvalue problem.

766
00:42:16,590 --> 00:42:17,900
OK?

767
00:42:17,900 --> 00:42:25,720
But I do know, if I have
A 0, if A 0 is equal to 0,

768
00:42:25,720 --> 00:42:31,790
from this expression,
that means A 1--

769
00:42:31,790 --> 00:42:33,810
sorry, A 0 is equal to 1.

770
00:42:33,810 --> 00:42:36,510
If A 0 is 0, then
everything's 0, right?

771
00:42:36,510 --> 00:42:38,530
And it's not fun, right?

772
00:42:38,530 --> 00:42:39,030
OK.

773
00:42:39,030 --> 00:42:41,915
A 0 is equal to 1, then
something will happen.

774
00:42:41,915 --> 00:42:44,020
A 1 will be equal
to beta, right?

775
00:42:44,020 --> 00:42:45,900
From this expression, right?

776
00:42:45,900 --> 00:42:50,360
Because beta A j is
equal to A j plus 1,

777
00:42:50,360 --> 00:42:54,690
A 2 will be equal to beta
square, et cetera, et cetera.

778
00:42:54,690 --> 00:42:58,380
And then I can say
that A j, if I assume

779
00:42:58,380 --> 00:43:09,070
A 0, if A 0 is equal to 1,
then A j will be equal to beta

780
00:43:09,070 --> 00:43:11,220
to the j.

781
00:43:11,220 --> 00:43:12,290
OK?

782
00:43:12,290 --> 00:43:15,130
Am I going too fast, here?

783
00:43:15,130 --> 00:43:18,860
Everybody is following?

784
00:43:18,860 --> 00:43:19,870
No questions?

785
00:43:19,870 --> 00:43:21,280
No?

786
00:43:21,280 --> 00:43:22,930
Good.

787
00:43:22,930 --> 00:43:26,670
Actually, we found
that we have already

788
00:43:26,670 --> 00:43:29,140
solved the eigenvalue problem.

789
00:43:29,140 --> 00:43:30,000
Right?

790
00:43:30,000 --> 00:43:33,840
Because I have already the
expression for the A j,

791
00:43:33,840 --> 00:43:37,410
which is actually in the
form of beta to the j, right?

792
00:43:37,410 --> 00:43:43,800
So beta is some kind of number,
and the infinite number of beta

793
00:43:43,800 --> 00:43:49,200
actually can satisfy
this eigenvalue problem.

794
00:43:49,200 --> 00:43:51,790
No matter what kind
of beta I choose--

795
00:43:51,790 --> 00:43:58,980
it can be 1, it can be
2, 3.14, it can be pi--

796
00:43:58,980 --> 00:44:02,640
and what am I going to get
is the corresponding A j,

797
00:44:02,640 --> 00:44:06,670
corresponding A vector,
which you have satisfied

798
00:44:06,670 --> 00:44:08,230
this expression.

799
00:44:08,230 --> 00:44:08,970
OK?

800
00:44:08,970 --> 00:44:12,270
So that means some magic happen.

801
00:44:12,270 --> 00:44:15,690
We have already solved
the eigenvalue problem

802
00:44:15,690 --> 00:44:20,660
without really deriving, you
know, a lot of deviation.

803
00:44:20,660 --> 00:44:21,630
Right?

804
00:44:21,630 --> 00:44:23,880
Secondly, another
thing which we learned

805
00:44:23,880 --> 00:44:30,030
is that there are infinite
number of eigenvalue which

806
00:44:30,030 --> 00:44:33,660
satisfy this eigenvalue problem.

807
00:44:33,660 --> 00:44:38,020
The question is, does
that make sense, or not?

808
00:44:38,020 --> 00:44:41,320
Infinite number of
eigenvalues can actually

809
00:44:41,320 --> 00:44:44,600
satisfy this
infinity long system.

810
00:44:44,600 --> 00:44:47,140
It's kind of making
sense, right?

811
00:44:47,140 --> 00:44:50,170
Because we have worked
on one oscillator,

812
00:44:50,170 --> 00:44:53,590
you had one normal
mode; two oscillator,

813
00:44:53,590 --> 00:44:56,050
you have two normal mode;
three oscillator, you

814
00:44:56,050 --> 00:44:57,400
have three normal mode--

815
00:44:57,400 --> 00:44:59,410
infinite number
of oscillator, you

816
00:44:59,410 --> 00:45:02,872
should have infinite
number of normal modes.

817
00:45:02,872 --> 00:45:04,120
Right?

818
00:45:04,120 --> 00:45:08,620
OK, so that is actually
a very, very good news,

819
00:45:08,620 --> 00:45:13,490
because we have already
solved the problem,

820
00:45:13,490 --> 00:45:19,041
and we also know the function
of four of eigenvectors.

821
00:45:19,041 --> 00:45:19,540
OK?

822
00:45:19,540 --> 00:45:22,150
So let's take a look
at those example

823
00:45:22,150 --> 00:45:26,390
system, which are actually
close to infinity long.

824
00:45:26,390 --> 00:45:31,180
So here, you have
a Bell Lab machine,

825
00:45:31,180 --> 00:45:34,040
which actually can
have, actually,

826
00:45:34,040 --> 00:45:37,450
multiple coupled oscillators.

827
00:45:37,450 --> 00:45:40,640
Each one of them can
oscillate up and down,

828
00:45:40,640 --> 00:45:44,260
and you can see that,
huh, if I actually

829
00:45:44,260 --> 00:45:50,760
tried to move them up and
down, that a complicated kind

830
00:45:50,760 --> 00:45:54,550
of motion can occur
from this system.

831
00:45:54,550 --> 00:45:58,450
Actually, if I do this, you see
that, ah, they are something

832
00:45:58,450 --> 00:46:00,890
similar to wave is happening.

833
00:46:00,890 --> 00:46:02,665
And if I do this continuously--

834
00:46:05,460 --> 00:46:08,940
oh, some kind of,
like, a standing wave

835
00:46:08,940 --> 00:46:11,100
is produced, right?

836
00:46:11,100 --> 00:46:14,160
And this system is actually
really, really hard

837
00:46:14,160 --> 00:46:15,820
to describe, right?

838
00:46:15,820 --> 00:46:21,180
If you look at how many things
this system can actually do.

839
00:46:21,180 --> 00:46:22,290
OK?

840
00:46:22,290 --> 00:46:25,080
Another example is actually--

841
00:46:25,080 --> 00:46:27,360
OK, so you can say, come
on, this is actually not

842
00:46:27,360 --> 00:46:29,520
infinitely long system, right?

843
00:46:29,520 --> 00:46:31,690
You have some final
number, right?

844
00:46:31,690 --> 00:46:36,580
So how about I use this
system as a demonstration.

845
00:46:36,580 --> 00:46:41,010
This is actually a much
nicer, or much better,

846
00:46:41,010 --> 00:46:47,280
approximation, OK, to
infinitely long system.

847
00:46:47,280 --> 00:46:52,310
You can see that,
OK, each mass, each--

848
00:46:52,310 --> 00:46:56,060
OK, I can say, for example, each
small component of the spring,

849
00:46:56,060 --> 00:47:01,490
essentially, can become seeded
as a small m in my graph,

850
00:47:01,490 --> 00:47:02,050
right?

851
00:47:02,050 --> 00:47:06,050
And actually, I can,
instead of oscillating them

852
00:47:06,050 --> 00:47:09,940
back and forth, I
oscillate them upside down.

853
00:47:09,940 --> 00:47:10,440
OK?

854
00:47:10,440 --> 00:47:13,550
And you can see that, huh, they
are interesting kind of motion.

855
00:47:13,550 --> 00:47:15,320
I can have--

856
00:47:15,320 --> 00:47:23,040
I can have this, which is like
a standing wave; I can do this;

857
00:47:23,040 --> 00:47:26,775
I can stop this
system, and I produce--

858
00:47:26,775 --> 00:47:27,590
woo!

859
00:47:27,590 --> 00:47:29,580
I can produce a wave.

860
00:47:29,580 --> 00:47:31,490
And then it goes back and forth.

861
00:47:31,490 --> 00:47:34,310
And I can, whoa, do
this crazy, and then you

862
00:47:34,310 --> 00:47:36,140
see that, how exciting--

863
00:47:36,140 --> 00:47:39,680
a much higher frequency
normal mode, right?

864
00:47:39,680 --> 00:47:41,570
And that's really complicated.

865
00:47:41,570 --> 00:47:47,720
And the question is, how
can we actually understand

866
00:47:47,720 --> 00:47:49,040
this kind of system?

867
00:47:49,040 --> 00:47:53,000
The thing is that this system
is so much, so complicated,

868
00:47:53,000 --> 00:47:56,330
and have infinite
amount of possibilities.

869
00:47:56,330 --> 00:47:57,920
Right?

870
00:47:57,920 --> 00:48:02,000
So how are we going
to understand this?

871
00:48:02,000 --> 00:48:06,380
Very good news is that we
have solved the normal modes

872
00:48:06,380 --> 00:48:08,270
of this kind of system, right?

873
00:48:08,270 --> 00:48:09,940
So the normal mode
looks like this--

874
00:48:09,940 --> 00:48:13,130
A j equal to beta j.

875
00:48:13,130 --> 00:48:14,150
OK?

876
00:48:14,150 --> 00:48:18,710
And the following lecture,
the rest of the lecture,

877
00:48:18,710 --> 00:48:22,430
is to understand
what does that mean,

878
00:48:22,430 --> 00:48:25,670
and also make predictions.

879
00:48:25,670 --> 00:48:26,170
OK?

880
00:48:28,700 --> 00:48:32,930
So now we have, actually,
the eigenvectors, OK?

881
00:48:32,930 --> 00:48:34,380
That's really nice.

882
00:48:34,380 --> 00:48:39,230
So from our previous discussion,
if this system actually

883
00:48:39,230 --> 00:48:43,020
satisfy the symmetry,
have the symmetry

884
00:48:43,020 --> 00:48:47,180
that is acquired by the S
matrix, which I have here, that

885
00:48:47,180 --> 00:48:50,780
means M minus 1 K
matrix will share

886
00:48:50,780 --> 00:48:55,580
the same set of
eigenvectors as S matrix.

887
00:48:55,580 --> 00:49:01,070
So what is actually part of
the work is to evaluate this.

888
00:49:01,070 --> 00:49:07,330
M minus 1 K multiplied by
A, and that will give you

889
00:49:07,330 --> 00:49:11,780
omega squared A. OK?

890
00:49:11,780 --> 00:49:16,920
So I just need to multiply
M minus 1 K matrix by A.

891
00:49:16,920 --> 00:49:17,650
What is A?

892
00:49:17,650 --> 00:49:20,050
A is actually here.

893
00:49:20,050 --> 00:49:22,950
Now what is actually
M minus 1 K matrix?

894
00:49:22,950 --> 00:49:27,170
M minus 1 K matrix
is here, have a kind

895
00:49:27,170 --> 00:49:29,380
of complicated structure.

896
00:49:29,380 --> 00:49:30,230
OK?

897
00:49:30,230 --> 00:49:32,660
On the other hand,
if I only focused

898
00:49:32,660 --> 00:49:37,040
on the jth object, the
object which is named j,

899
00:49:37,040 --> 00:49:43,070
have a label j, then actually
I can write down, OK,

900
00:49:43,070 --> 00:49:49,660
the right-hand side is actually
just omega square A j, right?

901
00:49:49,660 --> 00:49:51,650
Because this is actually--

902
00:49:51,650 --> 00:49:54,470
if I only focus on
the j component,

903
00:49:54,470 --> 00:49:57,920
OK, left-hand side is
actually just M minus 1 K A

904
00:49:57,920 --> 00:50:02,200
multiplied by A, right?

905
00:50:02,200 --> 00:50:05,450
OK, so basically, there
are only these three

906
00:50:05,450 --> 00:50:07,750
terms coming into play, right?

907
00:50:07,750 --> 00:50:11,300
If this is A j minus
1, so anything minus 1,

908
00:50:11,300 --> 00:50:14,870
we are multiplying
by minus K over n.

909
00:50:14,870 --> 00:50:19,760
A j we multiply by 2 K
over n, and A j plus 1,

910
00:50:19,760 --> 00:50:22,440
we are multiplying by
minus K over n, right?

911
00:50:22,440 --> 00:50:25,760
The rest of the terms are all 0.

912
00:50:25,760 --> 00:50:26,260
OK?

913
00:50:26,260 --> 00:50:29,460
It's actually not as
complicated as we thought.

914
00:50:29,460 --> 00:50:30,200
OK?

915
00:50:30,200 --> 00:50:32,620
So, if I write it
down, explicitly,

916
00:50:32,620 --> 00:50:35,630
the left-hand side part,
then what I'm going to get

917
00:50:35,630 --> 00:50:44,291
is minus K over n, capital
K over n, A j minus 1,

918
00:50:44,291 --> 00:50:55,795
plus 2 capital K over n A
j minus capital K over n,

919
00:50:55,795 --> 00:50:59,110
A j plus one.

920
00:50:59,110 --> 00:51:00,820
OK?

921
00:51:00,820 --> 00:51:03,190
So this is actually the j term.

922
00:51:03,190 --> 00:51:08,270
Now I can define
omega 0 square equal--

923
00:51:08,270 --> 00:51:14,380
is defined as capital K over n.

924
00:51:14,380 --> 00:51:17,970
If I do that, then
basically, I can

925
00:51:17,970 --> 00:51:28,790
see that omega square A j will
be equal to omega 0 square.

926
00:51:28,790 --> 00:51:29,290
OK?

927
00:51:29,290 --> 00:51:32,830
I am taking all the K
over n out of the game

928
00:51:32,830 --> 00:51:35,510
and write it down
as omega 0 square.

929
00:51:35,510 --> 00:51:37,660
OK?

930
00:51:37,660 --> 00:51:48,370
Minus A j minus 1 plus
2 A j minus A j plus 1.

931
00:51:48,370 --> 00:51:50,590
OK?

932
00:51:50,590 --> 00:51:55,990
And also we know, from the
previous discussion, S matrix

933
00:51:55,990 --> 00:51:58,570
and the n minus 1 K
matrix should share

934
00:51:58,570 --> 00:52:00,910
the same sets of eigenvectors.

935
00:52:00,910 --> 00:52:05,250
Therefore, I can actually try to
plug in one of the eigenvectors

936
00:52:05,250 --> 00:52:06,621
from S matrix.

937
00:52:06,621 --> 00:52:07,120
Right?

938
00:52:07,120 --> 00:52:09,440
A j equal to beta j.

939
00:52:09,440 --> 00:52:09,940
OK?

940
00:52:09,940 --> 00:52:12,040
I can plug that
in, then basically,

941
00:52:12,040 --> 00:52:17,830
I get omega 0 square minus b--

942
00:52:17,830 --> 00:52:26,130
minus beta, j minus 1 plus
2 beta to the j minus beta

943
00:52:26,130 --> 00:52:29,860
to the j plus 1.

944
00:52:29,860 --> 00:52:34,990
And the left-hand side will be
reading like omega square beta

945
00:52:34,990 --> 00:52:38,000
to the j.

946
00:52:38,000 --> 00:52:39,610
OK?

947
00:52:39,610 --> 00:52:42,950
Questions?

948
00:52:42,950 --> 00:52:44,670
OK.

949
00:52:44,670 --> 00:52:47,240
So now, I can cancel--

950
00:52:47,240 --> 00:52:50,570
I can actually divide everything
by beta to the j, right?

951
00:52:50,570 --> 00:52:52,830
I can get rid of
beta to the j, then

952
00:52:52,830 --> 00:53:00,025
basically, I get omega square
equal to omega 0 square minus 1

953
00:53:00,025 --> 00:53:04,525
over beta plus 2 minus beta.

954
00:53:07,850 --> 00:53:09,600
OK?

955
00:53:09,600 --> 00:53:17,520
And as we discussed before,
beta can have any value.

956
00:53:17,520 --> 00:53:18,190
OK?

957
00:53:18,190 --> 00:53:21,750
And also, you can see
from here that, huh--

958
00:53:21,750 --> 00:53:27,975
once I know the eigenvalue of
S matrix and eigenvector of S

959
00:53:27,975 --> 00:53:31,380
matrix, I also know
what is actually

960
00:53:31,380 --> 00:53:36,604
the corresponding angle of
frequency of the normal mode.

961
00:53:36,604 --> 00:53:37,520
Right?

962
00:53:37,520 --> 00:53:41,260
By using M minus
1 K times A, you

963
00:53:41,260 --> 00:53:45,750
can figure out what is actually
the corresponding omega,

964
00:53:45,750 --> 00:53:48,530
the normal mode frequency.

965
00:53:48,530 --> 00:53:49,840
OK?

966
00:53:49,840 --> 00:53:53,626
So that is actually pretty nice.

967
00:53:53,626 --> 00:53:57,540
But on the other hand,
if you step back and just

968
00:53:57,540 --> 00:54:01,410
think about what we
have been doing, OK?

969
00:54:01,410 --> 00:54:02,100
So very good.

970
00:54:02,100 --> 00:54:05,910
You have a beta, which
is a random value.

971
00:54:05,910 --> 00:54:09,100
You can evaluate
this thing, then

972
00:54:09,100 --> 00:54:14,490
you can get the
corresponding omega.

973
00:54:14,490 --> 00:54:16,510
But then something
doesn't feel right.

974
00:54:16,510 --> 00:54:17,490
Right?

975
00:54:17,490 --> 00:54:23,390
For example, if you
have beta equal to 2,

976
00:54:23,390 --> 00:54:26,390
what is going to happen?

977
00:54:26,390 --> 00:54:29,440
If you have beta equal to
2, what does that mean?

978
00:54:29,440 --> 00:54:35,730
That means A j will be
equal to 2 to the j.

979
00:54:38,971 --> 00:54:39,470
OK?

980
00:54:39,470 --> 00:54:41,420
That's very dangerous.

981
00:54:41,420 --> 00:54:42,540
Hey?

982
00:54:42,540 --> 00:54:46,910
That means-- OK, so I am--

983
00:54:46,910 --> 00:54:50,810
I deploy the whole system,
OK, from the left-hand side

984
00:54:50,810 --> 00:54:52,970
of the universe to
the right-hand side

985
00:54:52,970 --> 00:54:53,870
of the universe.

986
00:54:53,870 --> 00:54:54,530
OK?

987
00:54:54,530 --> 00:54:58,950
So that means, if I go to
the your right-hand side

988
00:54:58,950 --> 00:55:04,940
of the universe, the
amplitude explode.

989
00:55:04,940 --> 00:55:05,440
Right?

990
00:55:05,440 --> 00:55:08,790
It's actually 2 to the
infinite number, right?

991
00:55:08,790 --> 00:55:09,290
OK?

992
00:55:09,290 --> 00:55:12,310
It's not a physical-- doesn't
sound like a physical system

993
00:55:12,310 --> 00:55:13,640
to me.

994
00:55:13,640 --> 00:55:15,040
Right?

995
00:55:15,040 --> 00:55:18,160
If, actually, beta
is greater than 1,

996
00:55:18,160 --> 00:55:21,370
then the right-hand
side A of the universe,

997
00:55:21,370 --> 00:55:25,630
the amplitude
there, will explode.

998
00:55:25,630 --> 00:55:26,430
OK?

999
00:55:26,430 --> 00:55:28,010
Doesn't sound right, right?

1000
00:55:28,010 --> 00:55:28,890
So I don't like that.

1001
00:55:28,890 --> 00:55:29,390
OK?

1002
00:55:29,390 --> 00:55:32,570
Maybe you like it,
but I don't like it.

1003
00:55:32,570 --> 00:55:34,740
For the moment.

1004
00:55:34,740 --> 00:55:39,230
On the other hand, if the beta--

1005
00:55:39,230 --> 00:55:43,220
OK, again, it's not 1,
but smaller than 1--

1006
00:55:43,220 --> 00:55:45,430
what is going to happen?

1007
00:55:45,430 --> 00:55:49,370
If the beta is smaller than
1, what is going to happen

1008
00:55:49,370 --> 00:55:53,050
is that, huh, OK, the right-hand
side of the universe is fine,

1009
00:55:53,050 --> 00:55:56,340
is finite, because the amplitude
has become smaller and smaller.

1010
00:55:56,340 --> 00:55:59,480
But the left-hand side part
of the universe, the amplitude

1011
00:55:59,480 --> 00:56:00,445
still explode.

1012
00:56:03,070 --> 00:56:03,750
Right?

1013
00:56:03,750 --> 00:56:05,920
So what does that mean?

1014
00:56:05,920 --> 00:56:09,990
This means that if beta--

1015
00:56:09,990 --> 00:56:15,290
if the absolute value of
beta is not equal to 1,

1016
00:56:15,290 --> 00:56:20,820
the amplitude, at some
point, goes to infinity.

1017
00:56:20,820 --> 00:56:21,360
OK?

1018
00:56:21,360 --> 00:56:23,920
So that's actually
not very nice.

1019
00:56:23,920 --> 00:56:29,190
That's because A j is actually
proportional to beta to the j.

1020
00:56:29,190 --> 00:56:30,550
OK?

1021
00:56:30,550 --> 00:56:33,520
So in the discussion
we have here,

1022
00:56:33,520 --> 00:56:38,560
we consider beta
equal to 1 case.

1023
00:56:38,560 --> 00:56:39,070
OK?

1024
00:56:39,070 --> 00:56:42,610
Otherwise, it's actually,
things will explode.

1025
00:56:42,610 --> 00:56:43,720
OK?

1026
00:56:43,720 --> 00:56:47,590
So if the absolute value
f beta is equal to 1,

1027
00:56:47,590 --> 00:56:56,450
in general, beta can be
exponential i, small k A.

1028
00:56:56,450 --> 00:56:57,160
Right?

1029
00:56:57,160 --> 00:57:01,400
Then, actually, you can
get absolute valuable of 1.

1030
00:57:01,400 --> 00:57:01,900
OK?

1031
00:57:01,900 --> 00:57:04,813
If beta is equal
to 1, that means

1032
00:57:04,813 --> 00:57:09,620
the amplitude of all the
oscillators are the same.

1033
00:57:09,620 --> 00:57:10,960
OK?

1034
00:57:10,960 --> 00:57:16,030
All right, so now, if we accept
this, we only limit ourself

1035
00:57:16,030 --> 00:57:21,560
to the discussion of beta,
absolute beta, value of beta

1036
00:57:21,560 --> 00:57:26,410
equal to 1, then beta can be
written as exponential i k A.

1037
00:57:26,410 --> 00:57:29,260
Then, if I plug
this back into this,

1038
00:57:29,260 --> 00:57:33,040
basically, what you are
going to get is omega square

1039
00:57:33,040 --> 00:57:45,130
is equal to omega 0 square
2 minus exponential i k A

1040
00:57:45,130 --> 00:57:50,020
plus exponential
minus i k A. Right?

1041
00:57:50,020 --> 00:57:54,640
Because you have minus 1 over
beta, and beta, therefore

1042
00:57:54,640 --> 00:57:58,330
you have exponential i
k A, and the exponential

1043
00:57:58,330 --> 00:57:59,790
minus i k A. OK?

1044
00:57:59,790 --> 00:58:04,070
It's a lot of math
in this lecture,

1045
00:58:04,070 --> 00:58:06,500
but we are getting over to it.

1046
00:58:06,500 --> 00:58:07,960
OK?

1047
00:58:07,960 --> 00:58:08,770
All right.

1048
00:58:08,770 --> 00:58:10,620
So that is actually--

1049
00:58:10,620 --> 00:58:13,150
we actually can identify
this, and this actually

1050
00:58:13,150 --> 00:58:25,960
can be rewritten as 2 omega 0
square 1 minus cosine k A. OK?

1051
00:58:25,960 --> 00:58:30,748
We have arrived a surprisingly
simple expression.

1052
00:58:34,170 --> 00:58:38,780
So let's take a look at
this expression carefully.

1053
00:58:38,780 --> 00:58:45,530
So that means, for each
given k, a small k,

1054
00:58:45,530 --> 00:58:48,050
then I will have a
corresponding angular

1055
00:58:48,050 --> 00:58:50,590
frequency, omega square.

1056
00:58:50,590 --> 00:58:52,490
OK?

1057
00:58:52,490 --> 00:58:57,520
So still, there
are infinite number

1058
00:58:57,520 --> 00:59:01,331
of possible normal modes.

1059
00:59:01,331 --> 00:59:01,830
OK?

1060
00:59:01,830 --> 00:59:02,500
From this.

1061
00:59:07,150 --> 00:59:10,720
So if I take a look
at the amplitude,

1062
00:59:10,720 --> 00:59:14,060
if I select a k value--

1063
00:59:14,060 --> 00:59:22,850
small k value-- if k is given,
I can actually calculate

1064
00:59:22,850 --> 00:59:24,930
the corresponding A j.

1065
00:59:24,930 --> 00:59:29,330
So the A j I can actually
define as a superposition

1066
00:59:29,330 --> 00:59:39,500
of exponential i j k a, and
minus exponential i j k a.

1067
00:59:39,500 --> 00:59:42,500
And that will give you
a sinusoidal shape.

1068
00:59:47,520 --> 00:59:51,250
So if I give you the
k, basically, you'll

1069
00:59:51,250 --> 00:59:58,220
see that if I give you a k, then
you get the corresponding beta.

1070
00:59:58,220 --> 00:59:59,040
Right?

1071
00:59:59,040 --> 01:00:03,950
And you are going to get omega,
the corresponding omega square.

1072
01:00:03,950 --> 01:00:07,750
But one interesting
thing of this expression

1073
01:00:07,750 --> 01:00:13,810
is that if you keep beta,
or keep one over beta,

1074
01:00:13,810 --> 01:00:17,120
you are going to
get the same omega.

1075
01:00:17,120 --> 01:00:25,130
Therefore, I can now use
superposition principle.

1076
01:00:25,130 --> 01:00:30,120
Basically, I can actually add
these two solutions together,

1077
01:00:30,120 --> 01:00:31,980
since they are going
to be oscillating

1078
01:00:31,980 --> 01:00:33,480
at the same frequency.

1079
01:00:33,480 --> 01:00:37,350
Then what I'm going to get is,
huh, interesting thing happen.

1080
01:00:37,350 --> 01:00:41,640
The A j, the amplitude,
as a function of j,

1081
01:00:41,640 --> 01:00:44,660
it's like a sinusoidal function.

1082
01:00:44,660 --> 01:00:45,690
OK?

1083
01:00:45,690 --> 01:00:49,490
So that is actually
what is really predicted

1084
01:00:49,490 --> 01:00:51,390
to an infinity long system.

1085
01:00:51,390 --> 01:00:55,050
For example, if I do this,
you can see that, aha, indeed,

1086
01:00:55,050 --> 01:00:57,700
I can see sinusoidal shape.

1087
01:00:57,700 --> 01:00:58,410
OK?

1088
01:00:58,410 --> 01:01:01,920
And you can see that the
sinusoidal shape is actually

1089
01:01:01,920 --> 01:01:07,510
oscillating up and down,
like a standing wave.

1090
01:01:07,510 --> 01:01:12,730
And that is actually
exactly this expression.

1091
01:01:12,730 --> 01:01:16,840
So that tells you something
really interesting.

1092
01:01:16,840 --> 01:01:22,120
That means the sinusoidal
shape is associated with what?

1093
01:01:22,120 --> 01:01:27,540
Associated with
translation symmetry.

1094
01:01:27,540 --> 01:01:28,040
Right?

1095
01:01:28,040 --> 01:01:33,580
All I have been doing is
to require this translation

1096
01:01:33,580 --> 01:01:38,610
symmetry, and you already
get the amplitude A j.

1097
01:01:38,610 --> 01:01:44,090
And if you choose the
physical beta value,

1098
01:01:44,090 --> 01:01:46,260
then you already
immediately arrive

1099
01:01:46,260 --> 01:01:53,500
at a solution which is
actually like sinusoidal shape.

1100
01:01:53,500 --> 01:01:56,070
Doesn't that sounds
really amazing to you?

1101
01:02:00,620 --> 01:02:01,250
OK.

1102
01:02:01,250 --> 01:02:05,060
So I think it's time to take
a five-minute break, because I

1103
01:02:05,060 --> 01:02:07,730
can see that you are
overwhelmed by the math

1104
01:02:07,730 --> 01:02:11,060
already, and of course, let's
come back in five minutes,

1105
01:02:11,060 --> 01:02:13,730
then we can discuss some
more about what we have

1106
01:02:13,730 --> 01:02:16,490
learned from this mathematics.

1107
01:02:16,490 --> 01:02:19,392
And if you have any
questions, please let me know.

1108
01:02:24,580 --> 01:02:27,580
OK, so welcome back, everybody.

1109
01:02:27,580 --> 01:02:30,420
Of course, you are welcome
to come back here, and play

1110
01:02:30,420 --> 01:02:32,080
with the demonstration.

1111
01:02:32,080 --> 01:02:33,810
OK?

1112
01:02:33,810 --> 01:02:34,380
So very good.

1113
01:02:34,380 --> 01:02:39,150
So during the break, there
are several questions asked,

1114
01:02:39,150 --> 01:02:42,990
which I think, those
are very good questions,

1115
01:02:42,990 --> 01:02:45,990
and that's actually the
purpose of this break.

1116
01:02:45,990 --> 01:02:48,630
So it's a long day
already, right?

1117
01:02:48,630 --> 01:02:52,920
A lot of mathematics, and
I hope everybody survived.

1118
01:02:52,920 --> 01:02:53,450
OK?

1119
01:02:53,450 --> 01:02:55,260
No dead body yet?

1120
01:02:55,260 --> 01:02:59,940
You can see that here, I'm doing
something really crazy, here.

1121
01:02:59,940 --> 01:03:01,490
So, OK.

1122
01:03:01,490 --> 01:03:05,350
Consider-- I think most
of you got this point,

1123
01:03:05,350 --> 01:03:07,820
beta not equal to 1 is not nice.

1124
01:03:07,820 --> 01:03:10,450
Something explode at the
edge of the universe.

1125
01:03:10,450 --> 01:03:11,640
So I don't like that.

1126
01:03:11,640 --> 01:03:14,100
Therefore, I consider
only the case

1127
01:03:14,100 --> 01:03:18,060
which you have absolute
value beta is equal to 1.

1128
01:03:18,060 --> 01:03:21,390
And then we say, OK, it
can be plus 1 and minus 1,

1129
01:03:21,390 --> 01:03:23,580
but that's actually not
the whole story, right?

1130
01:03:23,580 --> 01:03:28,650
You can have, in general,
beta equal to exponential i,

1131
01:03:28,650 --> 01:03:29,400
some number.

1132
01:03:29,400 --> 01:03:30,170
Right?

1133
01:03:30,170 --> 01:03:31,640
Some real number.

1134
01:03:31,640 --> 01:03:32,860
OK?

1135
01:03:32,860 --> 01:03:37,860
And I write, here, a
very fancy expression.

1136
01:03:37,860 --> 01:03:41,232
Beta equal to exponential i k a.

1137
01:03:41,232 --> 01:03:42,960
Why i k a?

1138
01:03:42,960 --> 01:03:44,850
It's a very good
question, right?

1139
01:03:44,850 --> 01:03:46,320
What is a?

1140
01:03:46,320 --> 01:03:49,020
I think most of you
actually already forgot.

1141
01:03:49,020 --> 01:03:50,400
What is a?

1142
01:03:50,400 --> 01:03:55,620
a is actually the natural
length of the spring.

1143
01:03:55,620 --> 01:03:57,450
OK?

1144
01:03:57,450 --> 01:04:00,840
So I was going too fast, because
I would like to get to a break

1145
01:04:00,840 --> 01:04:03,390
to hear your questions.

1146
01:04:03,390 --> 01:04:04,440
So what is a?

1147
01:04:04,440 --> 01:04:06,350
a is the natural length.

1148
01:04:06,350 --> 01:04:07,410
OK?

1149
01:04:07,410 --> 01:04:09,570
And the k-- what is k?

1150
01:04:09,570 --> 01:04:11,490
Later, you will figure that out.

1151
01:04:11,490 --> 01:04:16,720
You'll find that, actually,
k is a wave number.

1152
01:04:16,720 --> 01:04:17,380
OK?

1153
01:04:17,380 --> 01:04:20,510
So that is actually much more
of meaningful now, right?

1154
01:04:20,510 --> 01:04:22,400
After the explanation.

1155
01:04:22,400 --> 01:04:25,010
So you can see that beta
is equal to exponential

1156
01:04:25,010 --> 01:04:30,040
i, some number, and I call it k
a, a fancy name of this number,

1157
01:04:30,040 --> 01:04:32,410
and it has some
physical meaning.

1158
01:04:32,410 --> 01:04:33,370
OK?

1159
01:04:33,370 --> 01:04:34,840
Another thing which
is interesting

1160
01:04:34,840 --> 01:04:42,920
is that if I plug in beta
equal to a, or beta equal to 1

1161
01:04:42,920 --> 01:04:47,540
over a, into the
same expression--

1162
01:04:47,540 --> 01:04:52,370
if I plug in either beta
a or beta equal to 1

1163
01:04:52,370 --> 01:04:54,890
over a to this
expression, I'm going

1164
01:04:54,890 --> 01:04:58,700
to get exactly the same omega.

1165
01:04:58,700 --> 01:05:03,160
So that means, OK,
both of them will

1166
01:05:03,160 --> 01:05:06,050
be-- both value
will be oscillating

1167
01:05:06,050 --> 01:05:08,620
at the same frequency.

1168
01:05:08,620 --> 01:05:09,350
OK?

1169
01:05:09,350 --> 01:05:13,940
So if you choose beta equal
to a, choose beta equal to 1

1170
01:05:13,940 --> 01:05:17,030
minus a, they are oscillating
at the same frequency.

1171
01:05:17,030 --> 01:05:18,280
What does that mean?

1172
01:05:18,280 --> 01:05:24,290
That means linear combination
of eigenvector coming from beta

1173
01:05:24,290 --> 01:05:28,250
equal to a and eigenvector
coming from beta equal to one

1174
01:05:28,250 --> 01:05:31,775
over a, linear combination
of those eigenvectors

1175
01:05:31,775 --> 01:05:41,920
are also eigenvectors of
the M minus 1 K matrix.

1176
01:05:41,920 --> 01:05:42,940
OK?

1177
01:05:42,940 --> 01:05:44,860
And that's actually where--

1178
01:05:44,860 --> 01:05:50,280
OK, those are different
eigenvectors for S,

1179
01:05:50,280 --> 01:05:56,612
but the linear combination of
these vectors are all the--

1180
01:05:56,612 --> 01:05:59,790
eigenvector of M minus
1 K matrix and always

1181
01:05:59,790 --> 01:06:03,180
the same eigenvalue
omega square.

1182
01:06:03,180 --> 01:06:03,680
OK?

1183
01:06:03,680 --> 01:06:06,450
So that's another thing
which is important.

1184
01:06:06,450 --> 01:06:09,620
And finally, I
said that there are

1185
01:06:09,620 --> 01:06:12,830
infinite number of choice of k.

1186
01:06:12,830 --> 01:06:13,910
That's valid, right?

1187
01:06:13,910 --> 01:06:16,180
Because you can choose
a little number,

1188
01:06:16,180 --> 01:06:18,710
then you get a
corresponding beta,

1189
01:06:18,710 --> 01:06:20,900
then you get a
corresponding omega.

1190
01:06:20,900 --> 01:06:24,890
So you have infinite
number of normal modes.

1191
01:06:24,890 --> 01:06:29,540
Secondly, if I give you a
k, OK-- if I give you a k,

1192
01:06:29,540 --> 01:06:34,760
or I can give you another
value which is minus k,

1193
01:06:34,760 --> 01:06:40,130
then that means you will
get beta and 1 over beta.

1194
01:06:40,130 --> 01:06:41,480
Right?

1195
01:06:41,480 --> 01:06:44,292
Minus k will give
you 1 over beta.

1196
01:06:44,292 --> 01:06:46,100
Right?

1197
01:06:46,100 --> 01:06:49,960
And as I mentioned before,
beta equal to a and beta

1198
01:06:49,960 --> 01:06:53,540
equal to 1 over a will
give you the same omega.

1199
01:06:53,540 --> 01:06:58,010
Therefore, a linear
combination of the vectors

1200
01:06:58,010 --> 01:07:03,060
are also eigen of
M minus 1 K matrix.

1201
01:07:03,060 --> 01:07:05,780
Though, that's actually
what I am doing here, right?

1202
01:07:05,780 --> 01:07:09,920
So in order to show
you a real amplitude,

1203
01:07:09,920 --> 01:07:14,600
I'm doing a linear combination
of exponential i j k a,

1204
01:07:14,600 --> 01:07:17,450
and exponential minus i j k a.

1205
01:07:17,450 --> 01:07:18,871
It's just a choice.

1206
01:07:18,871 --> 01:07:19,370
OK?

1207
01:07:19,370 --> 01:07:22,250
Of course, you can
say, OK, I choose plus,

1208
01:07:22,250 --> 01:07:24,300
and divide it by 2,
then you get the cosine.

1209
01:07:24,300 --> 01:07:25,100
Right?

1210
01:07:25,100 --> 01:07:28,360
But if I choose this expression,
then what I am going to get

1211
01:07:28,360 --> 01:07:29,780
is that, huh--

1212
01:07:29,780 --> 01:07:33,230
since both of them
are-- both vectors

1213
01:07:33,230 --> 01:07:37,340
are corresponding to the
same eigenvalue omega square,

1214
01:07:37,340 --> 01:07:39,830
therefore, linear
combination of them

1215
01:07:39,830 --> 01:07:44,240
also oscillate at the
angle of frequency omega.

1216
01:07:44,240 --> 01:07:47,750
Therefore, if I calculate
this and make it real,

1217
01:07:47,750 --> 01:07:52,040
then I find that the
amplitude is a function of j.

1218
01:07:52,040 --> 01:07:56,550
Is actually a sinusoidal
function, which is sine j k a.

1219
01:07:56,550 --> 01:07:57,060
OK?

1220
01:07:57,060 --> 01:07:59,920
So what does that mean?

1221
01:07:59,920 --> 01:08:06,290
This means that
if I plug the a--

1222
01:08:06,290 --> 01:08:13,740
if I plug A j as
a function of j,

1223
01:08:13,740 --> 01:08:17,160
this is actually what
I'm going to get.

1224
01:08:17,160 --> 01:08:19,229
It's a sinusoidal shape.

1225
01:08:19,229 --> 01:08:20,660
OK?

1226
01:08:20,660 --> 01:08:28,410
And we know that x j is actually
equal to A j cosine omega

1227
01:08:28,410 --> 01:08:30,649
t plus phi.

1228
01:08:30,649 --> 01:08:32,260
Right?

1229
01:08:32,260 --> 01:08:36,050
Omega, I can actually
evaluate that, right?

1230
01:08:36,050 --> 01:08:37,260
From here, right?

1231
01:08:37,260 --> 01:08:39,100
Just a reminder.

1232
01:08:39,100 --> 01:08:42,750
And what we are going to
get is, when this system is

1233
01:08:42,750 --> 01:08:45,805
thinking of normal mode, OK--

1234
01:08:45,805 --> 01:08:52,350
actually, this system is still
a discrete system, so i--

1235
01:08:52,350 --> 01:08:57,573
actually, would like to point
out that as a function of j,

1236
01:08:57,573 --> 01:09:03,271
only discrete
location have mass.

1237
01:09:03,271 --> 01:09:03,770
Right?

1238
01:09:03,770 --> 01:09:06,770
So you see that those
are individual mass.

1239
01:09:06,770 --> 01:09:10,420
They are oscillating
up and down.

1240
01:09:10,420 --> 01:09:11,479
OK?

1241
01:09:11,479 --> 01:09:13,850
And you can see
that, OK, since they

1242
01:09:13,850 --> 01:09:18,450
are oscillating up
and down, therefore,

1243
01:09:18,450 --> 01:09:21,603
the oscillation, essentially,
going up and down.

1244
01:09:21,603 --> 01:09:27,130
Therefore, what is the
actually the normal mode

1245
01:09:27,130 --> 01:09:28,819
of this infinity long system?

1246
01:09:28,819 --> 01:09:33,700
The normal mode are
actually standing waves.

1247
01:09:33,700 --> 01:09:39,370
But they actually only appear
in the discrete value of j.

1248
01:09:39,370 --> 01:09:42,200
And it has a functional
form of something

1249
01:09:42,200 --> 01:09:45,090
like a sinusoidal
shape, or cosine.

1250
01:09:45,090 --> 01:09:45,590
OK?

1251
01:09:45,590 --> 01:09:48,300
So that's actually what we
learn, and actually, you

1252
01:09:48,300 --> 01:09:50,220
can see that from here.

1253
01:09:50,220 --> 01:09:56,461
So if I oscillate this at
some selected amplitude--

1254
01:09:59,407 --> 01:10:00,389
OK?

1255
01:10:00,389 --> 01:10:04,317
Not quite get it.

1256
01:10:04,317 --> 01:10:04,820
Yeah.

1257
01:10:04,820 --> 01:10:08,700
So you see that, OK, it's
roughly like a standing wave.

1258
01:10:08,700 --> 01:10:11,420
It's a fixed frequency.

1259
01:10:11,420 --> 01:10:12,820
OK?

1260
01:10:12,820 --> 01:10:18,880
I would like to discuss with you
a really interesting selection.

1261
01:10:18,880 --> 01:10:22,690
So if I now take a look at--

1262
01:10:22,690 --> 01:10:24,730
so we have went through
a lot of math, right?

1263
01:10:24,730 --> 01:10:27,940
So now is the time to enjoy
what we have learned, right?

1264
01:10:27,940 --> 01:10:34,840
So if I now take a extreme
value, cosine k a, OK,

1265
01:10:34,840 --> 01:10:37,110
equal to minus 1.

1266
01:10:37,110 --> 01:10:38,520
OK?

1267
01:10:38,520 --> 01:10:44,550
Then I am reaching the
maximal oscillation frequency.

1268
01:10:44,550 --> 01:10:45,380
Right?

1269
01:10:45,380 --> 01:10:53,790
So if I choose cosine k,
small k, a equal to minus 1,

1270
01:10:53,790 --> 01:10:57,150
OK-- what is going to
happen is like this.

1271
01:10:57,150 --> 01:11:00,530
It is as a function
of j, by product

1272
01:11:00,530 --> 01:11:04,180
A j is a function of j,
what you are going to get

1273
01:11:04,180 --> 01:11:05,180
is starting like this.

1274
01:11:05,180 --> 01:11:10,550
Those are actually the
amplitude of individual mass.

1275
01:11:10,550 --> 01:11:16,400
So you can see that if cosine
k a is equal to minus 1,

1276
01:11:16,400 --> 01:11:20,240
omega square, based
on that expression--

1277
01:11:20,240 --> 01:11:23,880
1 minus, minus 1, you get 2--

1278
01:11:23,880 --> 01:11:29,200
therefore, you get omega square
equal to 4 omega 0 square.

1279
01:11:29,200 --> 01:11:30,110
OK?

1280
01:11:30,110 --> 01:11:34,500
And if you plug the A
j as a function of j,

1281
01:11:34,500 --> 01:11:37,130
that is actually what
you are going to get.

1282
01:11:37,130 --> 01:11:40,970
You actually have maximal
stretch to the system.

1283
01:11:40,970 --> 01:11:41,870
Right?

1284
01:11:41,870 --> 01:11:45,530
You can see that it's actually
positive, negative, positive,

1285
01:11:45,530 --> 01:11:47,390
negative, positive, negative.

1286
01:11:47,390 --> 01:11:51,950
That would reach the maximum
speed of the oscillation.

1287
01:11:51,950 --> 01:11:55,340
And of course, we cannot demo--

1288
01:11:55,340 --> 01:11:59,860
we cannot demo maximum,
infinite number of oscillator,

1289
01:11:59,860 --> 01:12:04,760
but of course, I can demo a
system with 10 oscillators.

1290
01:12:04,760 --> 01:12:09,100
So you can see that now,
I maximize the amplitude

1291
01:12:09,100 --> 01:12:13,550
of the highest
frequency normal mode.

1292
01:12:13,550 --> 01:12:17,610
And then I let go, and you
see that this is actually

1293
01:12:17,610 --> 01:12:21,460
exactly what is going to
happen when I have cosine

1294
01:12:21,460 --> 01:12:23,780
k a equal to minus 1.

1295
01:12:23,780 --> 01:12:28,670
Then the wavelengths--
it's very small--

1296
01:12:28,670 --> 01:12:32,450
and you actually reach
the maximum speed.

1297
01:12:32,450 --> 01:12:34,370
Maximum speed is
actually to become

1298
01:12:34,370 --> 01:12:40,470
paired with, for example, lower
frequency modes like this one.

1299
01:12:40,470 --> 01:12:45,500
This is actually oscillating
at a much lower frequency.

1300
01:12:45,500 --> 01:12:48,530
And you can ask, OK,
does that make sense?

1301
01:12:48,530 --> 01:12:54,200
If I have this
really, really zig-zag

1302
01:12:54,200 --> 01:12:59,750
shape, why this system should
be oscillating at the highest

1303
01:12:59,750 --> 01:13:02,620
possible frequency?

1304
01:13:02,620 --> 01:13:04,280
Why is that?

1305
01:13:04,280 --> 01:13:05,870
It also makes sense, right?

1306
01:13:05,870 --> 01:13:08,475
If you have that
set up, then you

1307
01:13:08,475 --> 01:13:14,480
are stretching this system to
the maxima possible amount.

1308
01:13:14,480 --> 01:13:15,280
Right?

1309
01:13:15,280 --> 01:13:20,000
So, actually, now the
springs looks like this.

1310
01:13:20,000 --> 01:13:22,430
You are stretching
this really hard,

1311
01:13:22,430 --> 01:13:27,720
and therefore, the restorative
force is going to be large.

1312
01:13:27,720 --> 01:13:31,373
Therefore, you get
high frequency.

1313
01:13:31,373 --> 01:13:33,220
OK?

1314
01:13:33,220 --> 01:13:38,650
OK, so I hope you actually
enjoy the lecture today.

1315
01:13:38,650 --> 01:13:41,600
It's a lot of mathematics,
but what we have learned

1316
01:13:41,600 --> 01:13:44,170
is really a lot.

1317
01:13:44,170 --> 01:13:49,520
We learned how to
actually describe system,

1318
01:13:49,520 --> 01:13:54,675
how to actually solve a system
without actually touching

1319
01:13:54,675 --> 01:13:57,360
the M minus 1 K
matrix; we can actually

1320
01:13:57,360 --> 01:14:00,450
already get the eigenvectors.

1321
01:14:00,450 --> 01:14:03,600
And using the M
minus 1 K matrix,

1322
01:14:03,600 --> 01:14:09,850
we can actually evaluate omega
as a function of the input

1323
01:14:09,850 --> 01:14:13,680
parameter from S eigenvalue.

1324
01:14:13,680 --> 01:14:18,030
And the next lectures, we are
going to discuss more examples,

1325
01:14:18,030 --> 01:14:20,880
and make the whole
system continuous.

1326
01:14:20,880 --> 01:14:23,610
Thank you very much, and if
you have any more questions,

1327
01:14:23,610 --> 01:14:24,460
I will be here.

1328
01:14:24,460 --> 01:14:27,850
I'm very happy to
answer your questions.