1 00:00:02,120 --> 00:00:04,460 The following content is provided under a Creative 2 00:00:04,460 --> 00:00:05,880 Commons license. 3 00:00:05,880 --> 00:00:08,090 Your support will help MIT OpenCourseWare 4 00:00:08,090 --> 00:00:10,970 continue to offer high-quality quality educational resources 5 00:00:10,970 --> 00:00:12,180 for free. 6 00:00:12,180 --> 00:00:14,720 To make a donation or to view additional materials 7 00:00:14,720 --> 00:00:18,680 from hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:18,680 --> 00:00:19,704 at ocw.mit.edu. 9 00:00:24,560 --> 00:00:27,020 PROFESSOR: So, I'm back. 10 00:00:27,020 --> 00:00:30,500 Welcome back, also, to 8.03. 11 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. 13 00:00:36,930 --> 00:00:40,880 It's to understand how we use symmetry 14 00:00:40,880 --> 00:00:48,700 to help us with prediction of physical situations. 15 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. 17 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. 23 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 70 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 73 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? 78 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. 114 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.