WEBVTT 00:00:02.120 --> 00:00:04.460 The following content is provided under a Creative 00:00:04.460 --> 00:00:05.880 Commons license. 00:00:05.880 --> 00:00:08.090 Your support will help MIT OpenCourseWare 00:00:08.090 --> 00:00:10.970 continue to offer high-quality quality educational resources 00:00:10.970 --> 00:00:12.180 for free. 00:00:12.180 --> 00:00:14.720 To make a donation or to view additional materials 00:00:14.720 --> 00:00:18.680 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:18.680 --> 00:00:19.704 at ocw.mit.edu. 00:00:24.560 --> 00:00:27.020 PROFESSOR: So, I'm back. 00:00:27.020 --> 00:00:30.500 Welcome back, also, to 8.03. 00:00:30.500 --> 00:00:33.640 So today, what we are going to do 00:00:33.640 --> 00:00:36.930 is something really interesting. 00:00:36.930 --> 00:00:40.880 It's to understand how we use symmetry 00:00:40.880 --> 00:00:48.700 to help us with prediction of physical situations. 00:00:48.700 --> 00:00:55.430 So first, I will go through two concrete examples of symmetry, 00:00:55.430 --> 00:00:58.080 and see what we can learn from there. 00:00:58.080 --> 00:01:01.680 And also, today, we are going to go to infinite 00:01:01.680 --> 00:01:03.480 number of coupled oscillators. 00:01:03.480 --> 00:01:04.700 OK? 00:01:04.700 --> 00:01:06.740 I think we are done with finite numbers. 00:01:06.740 --> 00:01:08.330 OK? 00:01:08.330 --> 00:01:08.990 All right. 00:01:08.990 --> 00:01:11.090 So what we have learned last time 00:01:11.090 --> 00:01:14.330 when Bolek was giving lectures, I 00:01:14.330 --> 00:01:17.780 hope we have learned that driving force can 00:01:17.780 --> 00:01:20.420 excite a specific normal mode. 00:01:20.420 --> 00:01:21.060 Right? 00:01:21.060 --> 00:01:24.360 So if you drive the system at the frequency, 00:01:24.360 --> 00:01:27.540 the system like, then the system will respond, 00:01:27.540 --> 00:01:32.300 and will oscillate the driving frequency with large amplitude. 00:01:32.300 --> 00:01:33.860 OK? 00:01:33.860 --> 00:01:38.030 And also, we have learned that the full solution 00:01:38.030 --> 00:01:39.870 of a coupled oscillator is actually 00:01:39.870 --> 00:01:47.860 pretty similar to the situation we got from single oscillators. 00:01:47.860 --> 00:01:50.120 So that where you have a particular solution, 00:01:50.120 --> 00:01:52.190 and a homogeneous solution. 00:01:52.190 --> 00:01:57.650 And the full solution will be a superposition of the two 00:01:57.650 --> 00:02:02.490 component, and all the unknown coefficients 00:02:02.490 --> 00:02:05.590 in the homogeneous part of the solution. 00:02:05.590 --> 00:02:06.470 OK? 00:02:06.470 --> 00:02:09.830 And today, I hope I can help you and convince you 00:02:09.830 --> 00:02:15.590 that symmetry actually can help us to solve the number of modes 00:02:15.590 --> 00:02:20.690 without knowing the detail of M minus one K metrics. 00:02:20.690 --> 00:02:22.790 So that actually sounds really cool, 00:02:22.790 --> 00:02:26.400 and I would like to talk about that in this lecture today. 00:02:26.400 --> 00:02:30.120 So this is actually what we have been doing so far. 00:02:30.120 --> 00:02:33.560 So we tried everything in terms of metrics. 00:02:33.560 --> 00:02:35.930 So we start from the equation of motion, 00:02:35.930 --> 00:02:39.530 and X double dot, you go to minus KX. 00:02:39.530 --> 00:02:44.760 And then we write everything in a complex notation-- 00:02:44.760 --> 00:02:49.240 exponential i omega t plus phi times A-- 00:02:49.240 --> 00:02:50.990 A is actually the vector, right? 00:02:50.990 --> 00:02:53.590 So it's actually A1, A2, A3. 00:02:53.590 --> 00:02:56.810 It's actually the amplitude of the oscillation 00:02:56.810 --> 00:02:59.640 of the first, second, and third and etc. etc. it's 00:02:59.640 --> 00:03:01.850 a component of the system. 00:03:01.850 --> 00:03:03.140 Right? 00:03:03.140 --> 00:03:06.960 Then, we actually found that, in the end of the day, 00:03:06.960 --> 00:03:09.380 we are actually solving this problem 00:03:09.380 --> 00:03:12.390 like eigenvalue problem. 00:03:12.390 --> 00:03:16.750 So basically, we have M minus 1 K metrics describe 00:03:16.750 --> 00:03:21.730 how each component in the system interacts with each other. 00:03:21.730 --> 00:03:22.580 OK? 00:03:22.580 --> 00:03:26.240 Then, what is actually the angle of frequency 00:03:26.240 --> 00:03:27.590 of the the normal modes? 00:03:27.590 --> 00:03:31.290 Essentially, coming from this eigenvalue problem , 00:03:31.290 --> 00:03:35.450 M minus 1 K, A equal to omega square A. 00:03:35.450 --> 00:03:39.200 Then you just go ahead and solve the eigenvalue problem. 00:03:39.200 --> 00:03:41.120 Then you will be able to figure out 00:03:41.120 --> 00:03:44.320 why are there no more mode frequencies, and therefore, 00:03:44.320 --> 00:03:47.060 what are the relative-- the ratio 00:03:47.060 --> 00:03:51.720 of the amplitude in the normal mode, which is actually the A 00:03:51.720 --> 00:03:52.220 vector. 00:03:52.220 --> 00:03:52.720 OK? 00:03:52.720 --> 00:03:53.900 The eigenvector. 00:03:53.900 --> 00:03:57.440 OK, so that's actually what we have been doing. 00:03:57.440 --> 00:03:58.940 OK? 00:03:58.940 --> 00:04:02.270 And today, what I'm going to do is 00:04:02.270 --> 00:04:06.170 to introduce you a very important concept in physics. 00:04:06.170 --> 00:04:10.050 Not only in physics, but also in mathematics, and also art, 00:04:10.050 --> 00:04:10.550 right? 00:04:10.550 --> 00:04:14.120 So you see symmetry in art, for example. 00:04:14.120 --> 00:04:15.210 We can see here-- 00:04:15.210 --> 00:04:17.269 there are several graphs here-- 00:04:17.269 --> 00:04:20.120 and you can see that their apparent symmetry, 00:04:20.120 --> 00:04:25.610 or rotational symmetry, they are refraction symmetry. 00:04:25.610 --> 00:04:30.500 And you can see that when we build the particle 00:04:30.500 --> 00:04:32.240 detector for example lower right plot 00:04:32.240 --> 00:04:37.010 is a CNS detector in the Large Hadron Collider. 00:04:37.010 --> 00:04:41.720 We also try to build this detector symmetric, right? 00:04:41.720 --> 00:04:44.940 Because otherwise, if we get a very complicated shape 00:04:44.940 --> 00:04:47.330 of detector, then the analysis of the data 00:04:47.330 --> 00:04:49.760 will be really complicated. 00:04:49.760 --> 00:04:54.020 So therefore, everybody like symmetry, and everybody 00:04:54.020 --> 00:04:56.690 don't like, really, chaos. 00:04:56.690 --> 00:04:57.330 Right? 00:04:57.330 --> 00:04:57.830 OK? 00:04:57.830 --> 00:04:59.840 So, that's really nice. 00:04:59.840 --> 00:05:04.640 The question is, how do we speak the language 00:05:04.640 --> 00:05:06.950 that the nature speak? 00:05:06.950 --> 00:05:11.460 How do we actually describe symmetry? 00:05:11.460 --> 00:05:13.490 That's actually the question I'm asking, 00:05:13.490 --> 00:05:16.380 and I'm going to show you that, OK, we can actually 00:05:16.380 --> 00:05:20.310 use mathematics to describe symmetry. 00:05:20.310 --> 00:05:25.850 So before we go to infinite number of oscillators, 00:05:25.850 --> 00:05:30.730 let me give you a concrete example of symmetry, 00:05:30.730 --> 00:05:35.020 and then see if we can understand how to use the math 00:05:35.020 --> 00:05:37.060 to describe symmetry. 00:05:37.060 --> 00:05:37.930 OK? 00:05:37.930 --> 00:05:42.220 So there is a two-component system. 00:05:42.220 --> 00:05:46.240 Two pendulums, which we worked together in the last few 00:05:46.240 --> 00:05:50.410 lectures, that they are coupled to each other, 00:05:50.410 --> 00:05:53.350 and there's a parent symmetry of this system. 00:05:53.350 --> 00:05:56.180 Can somebody tell me what is the symmetry, 00:05:56.180 --> 00:05:58.570 you can see from this system? 00:05:58.570 --> 00:05:59.890 Somebody? 00:05:59.890 --> 00:06:02.046 Anybody? 00:06:02.046 --> 00:06:03.010 AUDIENCE: Reflection. 00:06:03.010 --> 00:06:05.160 PROFESSOR: The reflection symmetry. 00:06:05.160 --> 00:06:11.530 So if you reflect this system, as I show you in the slide, 00:06:11.530 --> 00:06:15.670 you can see that if you reflect this picture, 00:06:15.670 --> 00:06:17.630 it looks identical. 00:06:17.630 --> 00:06:18.310 Right? 00:06:18.310 --> 00:06:20.980 So that is actually really, really good news. 00:06:20.980 --> 00:06:24.310 That means if I do this reflection, XY, 00:06:24.310 --> 00:06:26.284 and go to minus X2-- 00:06:26.284 --> 00:06:27.700 you have a minus sign, because you 00:06:27.700 --> 00:06:30.910 can see that after reflection-- the amplitude changes sign. 00:06:30.910 --> 00:06:31.750 Right? 00:06:31.750 --> 00:06:37.120 X2 go to minus X1, the system looks identical, 00:06:37.120 --> 00:06:39.260 and the physics should not change. 00:06:39.260 --> 00:06:39.760 OK? 00:06:39.760 --> 00:06:43.570 So that's actually what we can learn from there. 00:06:43.570 --> 00:06:47.250 So that means if I have-- 00:06:47.250 --> 00:06:53.708 I do this reflection, then I can actually define X tilde-- 00:06:53.708 --> 00:07:01.550 T-- this is equal to minus X2 minus X1. 00:07:01.550 --> 00:07:02.050 OK? 00:07:02.050 --> 00:07:05.091 To become paired with X. OK? 00:07:05.091 --> 00:07:11.830 And this is also going to be the solution of the equation motion 00:07:11.830 --> 00:07:14.990 if the original X is already a solution. 00:07:14.990 --> 00:07:15.590 OK? 00:07:15.590 --> 00:07:19.190 So that's the power of reflection symmetry. 00:07:19.190 --> 00:07:19.690 OK? 00:07:19.690 --> 00:07:25.810 If X is a solution, then I do this reflection, 00:07:25.810 --> 00:07:32.900 and I can figure out that X tilde is also a solution. 00:07:32.900 --> 00:07:34.390 OK? 00:07:34.390 --> 00:07:39.760 So how do I actually describe the symmetry 00:07:39.760 --> 00:07:41.800 in the form of mathematics? 00:07:41.800 --> 00:07:48.550 What we actually do is to define S matrix, symmetry matrix. 00:07:48.550 --> 00:07:53.140 And in this case, when we talk about reflection symmetry, 00:07:53.140 --> 00:07:57.040 it's actually defined as zero minus 1 minus 1, 0. 00:07:57.040 --> 00:07:59.800 This is actually a two by two matrix. 00:07:59.800 --> 00:08:06.430 And if I do this operation, S operate on this X matrix, 00:08:06.430 --> 00:08:10.130 then that is actually is going to give you the X tilde. 00:08:10.130 --> 00:08:10.630 OK? 00:08:10.630 --> 00:08:15.596 So that's the nature of the role of the symmetry matrix. 00:08:15.596 --> 00:08:17.220 OK? 00:08:17.220 --> 00:08:19.620 Any questions? 00:08:19.620 --> 00:08:21.060 OK. 00:08:21.060 --> 00:08:25.560 So now we have defined a symmetry matrix. 00:08:25.560 --> 00:08:30.210 And then you can ask, why do we actually care, 00:08:30.210 --> 00:08:32.970 and why do we actually introduce symmetry matrix? 00:08:32.970 --> 00:08:33.809 Right? 00:08:33.809 --> 00:08:37.530 Because I can always write down the X tilde in that way. 00:08:37.530 --> 00:08:42.340 That is because I think by the end of this lecture, 00:08:42.340 --> 00:08:47.820 you will find that if S matrix describes 00:08:47.820 --> 00:08:51.600 the symmetry of the system, OK, that 00:08:51.600 --> 00:08:57.930 would mean S matrix will commute with M minus 1 K matrix-- 00:08:57.930 --> 00:09:01.020 which, we don't know commute yet, but I will introduce you-- 00:09:01.020 --> 00:09:06.770 that means M minus 1 K matrix and S can actually swap freely. 00:09:06.770 --> 00:09:08.070 OK? 00:09:08.070 --> 00:09:12.240 If that happens, then S matrix will 00:09:12.240 --> 00:09:17.240 share the same sets of eigenvectors 00:09:17.240 --> 00:09:19.650 as the M minus 1 K matrix. 00:09:19.650 --> 00:09:21.300 What does that mean? 00:09:21.300 --> 00:09:22.920 That means-- OK. 00:09:22.920 --> 00:09:24.840 Before we are doing this solution, 00:09:24.840 --> 00:09:29.060 right, we are solving M minus 1 K matrix eigenvalue problem, 00:09:29.060 --> 00:09:30.630 right? 00:09:30.630 --> 00:09:32.550 And then, we get the eigenvector, 00:09:32.550 --> 00:09:36.940 which is the amplitude ratio of normal modes. 00:09:36.940 --> 00:09:40.950 And that means you have an alternative way 00:09:40.950 --> 00:09:43.170 to get the normal mode. 00:09:43.170 --> 00:09:47.480 You can solve the eigenvalue problem of S matrix, 00:09:47.480 --> 00:09:51.665 then you can get the same set of amplitude ratios 00:09:51.665 --> 00:09:55.550 as M minus 1 K matrix eigenvalue problem. 00:09:55.550 --> 00:09:56.190 OK? 00:09:56.190 --> 00:10:01.170 And then usually, the eigenvalue problem of S matrix 00:10:01.170 --> 00:10:05.710 is far much easier than M minus 1 K matrix. 00:10:05.710 --> 00:10:06.210 OK? 00:10:06.210 --> 00:10:08.740 So that's actually why we're doing this. 00:10:08.740 --> 00:10:09.400 OK? 00:10:09.400 --> 00:10:13.260 So now, I would like to convince you 00:10:13.260 --> 00:10:24.870 that S matrix and M minus 1 K matrix will share eigenvectors. 00:10:31.390 --> 00:10:32.870 OK? 00:10:32.870 --> 00:10:33.950 So. 00:10:33.950 --> 00:10:41.480 Let's go ahead and prove this, or demonstrate this idea. 00:10:41.480 --> 00:10:42.110 OK? 00:10:42.110 --> 00:10:45.230 So the original equation of motion looks like this. 00:10:45.230 --> 00:10:52.580 X double dot equal to M minus 1 K X. Right? 00:10:52.580 --> 00:10:57.530 So now, this is actually the original equation of motion. 00:10:57.530 --> 00:11:03.980 And if this system satisfy the reflection symmetry, 00:11:03.980 --> 00:11:09.620 that means X tilde is also a solution, right? 00:11:09.620 --> 00:11:11.780 Therefore, what does that mean? 00:11:11.780 --> 00:11:16.510 That means X tilde double dot will be also 00:11:16.510 --> 00:11:23.650 equal to M minus 1 K X tilde. 00:11:23.650 --> 00:11:27.010 Because it's also a solution to the equation of motion, right? 00:11:27.010 --> 00:11:28.450 That's pretty natural. 00:11:28.450 --> 00:11:29.530 OK? 00:11:29.530 --> 00:11:31.900 Now. 00:11:31.900 --> 00:11:34.420 I can actually use this expression, 00:11:34.420 --> 00:11:38.890 X tilde is equal to S times X. Right? 00:11:38.890 --> 00:11:41.160 All of those things are matrix, OK? 00:11:41.160 --> 00:11:43.220 Just to be careful. 00:11:43.220 --> 00:11:46.870 That means I can write this like this-- 00:11:46.870 --> 00:12:00.430 S X double dot equal to M minus 1 K S X. OK? 00:12:00.430 --> 00:12:03.460 There's no matrix, and I also replace-- 00:12:03.460 --> 00:12:08.332 I'm just replacing X tilde by S X. OK? 00:12:08.332 --> 00:12:12.830 And also, I call this, actually, 1; I call this actually 2. 00:12:12.830 --> 00:12:13.600 OK? 00:12:13.600 --> 00:12:20.130 I can multiply X from the left-hand side of 1. 00:12:20.130 --> 00:12:20.910 OK? 00:12:20.910 --> 00:12:22.320 And see what will happen. 00:12:22.320 --> 00:12:24.670 So if I do that, then what I am going to get 00:12:24.670 --> 00:12:27.540 is S X double dot-- 00:12:27.540 --> 00:12:28.270 OK? 00:12:28.270 --> 00:12:39.430 That will be equal to S M minus 1 K X. OK? 00:12:39.430 --> 00:12:43.090 If you compare this equation, and the equation number 00:12:43.090 --> 00:12:45.670 three, these two equations, you will see 00:12:45.670 --> 00:12:48.530 that let-hand side is the same. 00:12:48.530 --> 00:12:49.840 Right? 00:12:49.840 --> 00:12:52.570 Right-hand side-- huh! 00:12:52.570 --> 00:12:54.920 Something interesting is happening. 00:12:54.920 --> 00:12:59.020 M minus 1 K S must be equal to S M 00:12:59.020 --> 00:13:02.870 minus 1 K. What does that mean? 00:13:02.870 --> 00:13:04.820 This means that they are the same. 00:13:04.820 --> 00:13:11.980 M minus 1 K S is actually equal to S M minus 1 K. 00:13:11.980 --> 00:13:17.330 So if I say, this distance satisfy a symmetry 00:13:17.330 --> 00:13:21.650 described by S matrix, that means 00:13:21.650 --> 00:13:26.350 X tilde, which is actually the transformed amplitude, 00:13:26.350 --> 00:13:32.230 will be also a solution to the equation of motion. 00:13:32.230 --> 00:13:38.800 And therefore, an inevitable consequence is that M minus 1 00:13:38.800 --> 00:13:43.060 K S will be equal to S M minus 1 K. 00:13:43.060 --> 00:13:46.150 Usually, when you started physics, 00:13:46.150 --> 00:13:48.940 we write this in terms of commutator. 00:13:55.320 --> 00:13:55.920 OK? 00:13:55.920 --> 00:13:59.250 So we call this, these two things actually commute. 00:13:59.250 --> 00:14:00.270 OK? 00:14:00.270 --> 00:14:02.460 So commutator is actually defined 00:14:02.460 --> 00:14:06.990 as A bracket of A and B. This is actually equal-- 00:14:06.990 --> 00:14:13.230 defined as A B minus B A. OK? 00:14:13.230 --> 00:14:16.330 If A and B commute-- 00:14:16.330 --> 00:14:16.830 OK? 00:14:16.830 --> 00:14:20.050 It's this new word, probably, for most of you-- 00:14:20.050 --> 00:14:26.090 if they commute, that means A B in the bracket 00:14:26.090 --> 00:14:28.950 is equal to zero. 00:14:28.950 --> 00:14:30.720 OK? 00:14:30.720 --> 00:14:34.890 So this expression, I can actually 00:14:34.890 --> 00:14:36.440 write it down like this. 00:14:36.440 --> 00:14:43.890 Commutator of S M minus 1 K, that is equal to zero. 00:14:43.890 --> 00:14:47.010 And you will see this really a lot 00:14:47.010 --> 00:14:49.180 when you study quantum physics. 00:14:49.180 --> 00:14:50.010 OK? 00:14:50.010 --> 00:14:51.540 So I hope this actually gives you 00:14:51.540 --> 00:14:54.870 some flavor about commutator. 00:14:54.870 --> 00:14:56.100 OK? 00:14:56.100 --> 00:14:59.100 So now, that's actually pretty nice. 00:14:59.100 --> 00:15:01.950 This means that they commute, OK? 00:15:01.950 --> 00:15:16.120 If I take X of t this is equal to A 1 cosine omega 1 t. 00:15:16.120 --> 00:15:18.810 OK? 00:15:18.810 --> 00:15:24.378 So, this means that A is actually-- 00:15:24.378 --> 00:15:26.670 sorry, X is actually a solution, which 00:15:26.670 --> 00:15:28.570 is a normal mode, a solution. 00:15:28.570 --> 00:15:29.070 Right? 00:15:29.070 --> 00:15:34.680 And A is actually amplitude the vector, the amplitude vector 00:15:34.680 --> 00:15:37.620 of the first normal mode, and omega 1 00:15:37.620 --> 00:15:43.091 is actually the first normal mode frequency. 00:15:43.091 --> 00:15:43.590 OK? 00:15:43.590 --> 00:15:58.110 If this is the case, then I will have X tilde of t 00:15:58.110 --> 00:16:07.490 will be also oppositional to A 1 cosine omega 1 t. 00:16:07.490 --> 00:16:11.300 Because if I actually exchange X1 and X2, 00:16:11.300 --> 00:16:15.004 the oscillation frequency is not going to change. 00:16:15.004 --> 00:16:16.860 Right? 00:16:16.860 --> 00:16:21.860 Therefore, since this system is in the same normal mode 00:16:21.860 --> 00:16:27.100 with angular frequency omega 1, therefore 00:16:27.100 --> 00:16:32.420 the amplitude ratio of the first and second oscillator 00:16:32.420 --> 00:16:34.531 will stay constant. 00:16:34.531 --> 00:16:35.030 Right? 00:16:35.030 --> 00:16:37.271 Because you are in one of the normal modes. 00:16:37.271 --> 00:16:37.770 Right? 00:16:37.770 --> 00:16:42.095 Therefore, I can conclude that X tilde 00:16:42.095 --> 00:16:46.130 is going to be proportional to this expression. 00:16:46.130 --> 00:16:47.890 Because they are in the same normal mode, 00:16:47.890 --> 00:16:50.240 oscillating at the same frequency. 00:16:50.240 --> 00:16:50.780 OK? 00:16:50.780 --> 00:16:52.730 Is that too fast? 00:16:52.730 --> 00:16:54.770 Everybody is following? 00:16:54.770 --> 00:16:55.760 OK. 00:16:55.760 --> 00:16:57.570 So that's nice. 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 00:17:12.710 --> 00:17:14.720 1t, OK? 00:17:14.720 --> 00:17:16.609 So this is actually coming from here, right? 00:17:16.609 --> 00:17:24.270 I am replacing X tilde by S X based on this definition. 00:17:24.270 --> 00:17:26.119 OK? 00:17:26.119 --> 00:17:29.980 Then again, I replace, I write, X explicitly which 00:17:29.980 --> 00:17:32.990 is actually A cosine omega 1 t. 00:17:32.990 --> 00:17:33.890 OK? 00:17:33.890 --> 00:17:35.330 Then you get this expression. 00:17:35.330 --> 00:17:39.090 And from this expression above, you 00:17:39.090 --> 00:17:44.090 see that you conclude that this is proportional to A 1 00:17:44.090 --> 00:17:48.000 cosine omega 1 t. 00:17:48.000 --> 00:17:49.070 That's very nice. 00:17:49.070 --> 00:17:57.240 That means S A cosine omega 1 t is proportional to A 1 cosine 00:17:57.240 --> 00:17:58.620 omega t. 00:17:58.620 --> 00:18:00.865 And you can actually cancel this. 00:18:00.865 --> 00:18:08.060 And you see that S A 1 is proportional to A 1. 00:18:08.060 --> 00:18:17.760 Or I can write it as S A 1 is equal to beta A 1. 00:18:17.760 --> 00:18:19.880 What does that mean? 00:18:19.880 --> 00:18:26.060 This means that A 1 originally-- 00:18:26.060 --> 00:18:27.850 where's A 1 coming from? 00:18:27.850 --> 00:18:32.830 A 1 is the amplitude of all the components 00:18:32.830 --> 00:18:34.340 in the first normal mode. 00:18:34.340 --> 00:18:36.400 Right? 00:18:36.400 --> 00:18:39.770 That's coming from the eigenvalue problem, which 00:18:39.770 --> 00:18:42.700 it actually does in this light. 00:18:42.700 --> 00:18:45.970 Eigenvalue problem M minus 1 K A equal to omega 00:18:45.970 --> 00:18:49.360 square A will give you the solution of normal mode 00:18:49.360 --> 00:18:56.987 and their eigenvectors, which is amplitude ratios of all 00:18:56.987 --> 00:18:58.195 the components in the system. 00:18:58.195 --> 00:18:58.810 Right? 00:18:58.810 --> 00:19:06.350 So that means A 1 is not only M minus 1 K matrix eigenvectors, 00:19:06.350 --> 00:19:11.970 it's also eigenvector of S matrix. 00:19:11.970 --> 00:19:13.350 OK? 00:19:13.350 --> 00:19:15.670 So that is actually very good news. 00:19:15.670 --> 00:19:18.640 And I can also do the same thing for A 2, 00:19:18.640 --> 00:19:21.540 to prove that it also works for A 2-- 00:19:21.540 --> 00:19:23.920 the derivation is identical, so I am not 00:19:23.920 --> 00:19:25.490 going to do that again. 00:19:25.490 --> 00:19:33.220 So that means, actually, starting from here, OK-- 00:19:33.220 --> 00:19:37.960 if X and X tilde are both solutions 00:19:37.960 --> 00:19:40.800 to the equation of motion. 00:19:40.800 --> 00:19:46.390 I will conclude that S matrix and M minus 1 K matrix, 00:19:46.390 --> 00:19:47.800 they commute. 00:19:47.800 --> 00:19:49.690 OK? 00:19:49.690 --> 00:19:55.720 How to tell if a system satisfy a specific symmetry defined 00:19:55.720 --> 00:19:57.550 by my symmetry matrix? 00:19:57.550 --> 00:20:02.940 Is by this way, you can check if M minus 1 K and S commute. 00:20:02.940 --> 00:20:06.700 If they commute, that means the system actually 00:20:06.700 --> 00:20:09.550 satisfy this symmetry. 00:20:09.550 --> 00:20:13.570 And also, the consequence is that from there, 00:20:13.570 --> 00:20:17.090 you will conclude that if you have 00:20:17.090 --> 00:20:20.640 also a set of eigenvectors from M 00:20:20.640 --> 00:20:23.690 minus 1 K matrix eigenvalue problem, then 00:20:23.690 --> 00:20:31.360 that is going to be also the eigenvector of S. OK? 00:20:31.360 --> 00:20:32.725 Any questions? 00:20:36.290 --> 00:20:38.030 OK. 00:20:38.030 --> 00:20:43.400 So M minus 1 K eigenvectors. 00:20:48.610 --> 00:20:52.648 Also S eigenvector. 00:20:55.641 --> 00:20:56.140 OK? 00:20:56.140 --> 00:20:57.670 That's actually what we have learned 00:20:57.670 --> 00:21:00.370 from this small exercise. 00:21:00.370 --> 00:21:03.550 Now, you can say, wait, wait, wait, wait. 00:21:03.550 --> 00:21:05.650 This is actually not what we need, right? 00:21:05.650 --> 00:21:09.910 I would like-- we would like to argue that S matrix-- 00:21:09.910 --> 00:21:13.720 I can solve S matrix eigenvalue problem, 00:21:13.720 --> 00:21:16.760 and I can learn about the solution of M minus 1 K matrix, 00:21:16.760 --> 00:21:17.260 right? 00:21:17.260 --> 00:21:19.770 This logic is actually in the opposite direction, right? 00:21:19.770 --> 00:21:22.420 You said, OK, you solved things already, 00:21:22.420 --> 00:21:25.360 then, actually, it's also S matrix eigenvalue problem. 00:21:25.360 --> 00:21:30.280 So now what I am going to do is to reverse the logic, 00:21:30.280 --> 00:21:31.940 and see if it works. 00:21:31.940 --> 00:21:32.440 OK? 00:21:32.440 --> 00:21:34.790 Again, to see what will happen. 00:21:34.790 --> 00:21:35.830 OK? 00:21:35.830 --> 00:21:41.350 So now, I would like to prove that if I solve S matrix 00:21:41.350 --> 00:21:47.660 eigenvalue problem, I have also solved the eigenvectors for M 00:21:47.660 --> 00:21:48.810 minus 1 K matrix. 00:21:48.810 --> 00:21:51.400 Run the logic in the opposite direction. 00:21:51.400 --> 00:21:52.300 OK? 00:21:52.300 --> 00:21:55.810 So, if I were given two things-- 00:21:55.810 --> 00:22:05.870 one, S A is equal to beta A. Number two, S matrix and M 00:22:05.870 --> 00:22:10.160 minus 1 K matrix commute. 00:22:10.160 --> 00:22:10.660 OK? 00:22:10.660 --> 00:22:13.870 If those are the given conditions, 00:22:13.870 --> 00:22:20.010 then I can actually conclude that S M minus 1 K-- 00:22:20.010 --> 00:22:20.510 OK? 00:22:20.510 --> 00:22:24.280 I can actually contract this expression-- 00:22:24.280 --> 00:22:27.730 I write that S M minus 1 K A, OK? 00:22:27.730 --> 00:22:30.220 Because they commute, right? 00:22:30.220 --> 00:22:35.140 They can actually swap M minus 1 K and S 00:22:35.140 --> 00:22:38.920 safely without actually introducing any more terms. 00:22:38.920 --> 00:22:47.500 This will be equal to M minus 1 K S A. OK? 00:22:47.500 --> 00:22:50.800 And S A, from the first expression, 00:22:50.800 --> 00:22:53.420 S A is equal to beta A. Right? 00:22:53.420 --> 00:22:56.890 Beta is a number, OK? 00:22:56.890 --> 00:23:04.160 Therefore this expression will become beta M minus 1 K A. 00:23:04.160 --> 00:23:06.690 So, beta can penetrate through matrix, 00:23:06.690 --> 00:23:10.070 because beta is just a number, is eigenvalue. 00:23:10.070 --> 00:23:12.110 It's eigenvalue of S matrix. 00:23:12.110 --> 00:23:14.070 OK? 00:23:14.070 --> 00:23:15.200 So what does this mean? 00:23:18.570 --> 00:23:20.320 OK. 00:23:20.320 --> 00:23:22.100 So what does it mean? 00:23:22.100 --> 00:23:30.310 So this means that if you look at this part and that part-- 00:23:30.310 --> 00:23:33.760 you look at the beginning and the end of the expression-- you 00:23:33.760 --> 00:23:41.470 immediately conclude that M minus 1 K A, this expression 00:23:41.470 --> 00:23:46.371 is also an eigenvector of S matrix. 00:23:46.371 --> 00:23:46.870 Right? 00:23:46.870 --> 00:23:52.756 So you have S matrix acting M minus 1 K A. 00:23:52.756 --> 00:23:56.750 And that will give you something proportional 00:23:56.750 --> 00:24:00.370 to M minus 1 K A. You see? 00:24:00.370 --> 00:24:01.810 It's magic, right? 00:24:01.810 --> 00:24:03.810 It's actually not magic, but it's actually just, 00:24:03.810 --> 00:24:06.730 you know, really logical extension. 00:24:06.730 --> 00:24:07.370 Right? 00:24:07.370 --> 00:24:08.700 OK? 00:24:08.700 --> 00:24:09.640 Very cool! 00:24:09.640 --> 00:24:14.140 So that means this is also an eigenvector of S. Right? 00:24:14.140 --> 00:24:16.570 And also, another thing which is interesting 00:24:16.570 --> 00:24:22.203 is that they share the same eigenvalue, beta. 00:24:22.203 --> 00:24:23.030 Right? 00:24:23.030 --> 00:24:24.435 They have the same eigenvalue. 00:24:28.440 --> 00:24:30.960 OK? 00:24:30.960 --> 00:24:34.230 So, if eigenvalues of S-- 00:24:34.230 --> 00:24:36.390 so you can get several eigenvalues, right? 00:24:36.390 --> 00:24:39.300 In this case, two by two matrix, you will get-- 00:24:39.300 --> 00:24:40.050 how many? 00:24:40.050 --> 00:24:41.280 Two, right? 00:24:41.280 --> 00:24:43.080 Two eigenvalues. 00:24:43.080 --> 00:24:45.900 If those two eigenvalues are different, 00:24:45.900 --> 00:24:49.350 then I can conclude that M minus 1 00:24:49.350 --> 00:24:57.580 K A must be proportional to A. Right? 00:24:57.580 --> 00:25:01.750 Because this is actually the same eigenvalue problem, 00:25:01.750 --> 00:25:04.480 and the same eigenvalue, beta. 00:25:04.480 --> 00:25:07.490 Since all the eigenvalues from the solution 00:25:07.490 --> 00:25:12.500 of eigenvalue problem of S A equal to beta A, 00:25:12.500 --> 00:25:14.590 those eigenvalues are all different, 00:25:14.590 --> 00:25:16.450 therefore I can argue that M minus 1 00:25:16.450 --> 00:25:20.770 K A is proportional to A. OK? 00:25:20.770 --> 00:25:31.740 Therefore, M minus 1 K A is equal to omega square A. Omega 00:25:31.740 --> 00:25:34.276 square is actually some constant. 00:25:37.030 --> 00:25:38.370 OK? 00:25:38.370 --> 00:25:41.370 This is actually amazing, because that means given 00:25:41.370 --> 00:25:44.040 the two conditions-- the first one, 00:25:44.040 --> 00:25:48.780 I can figure out the eigenvalue and the eigenvectors of S 00:25:48.780 --> 00:25:54.170 matrix; second, if S matrix and M minus 1 K matrix interaction 00:25:54.170 --> 00:25:57.300 matrix, they commute-- 00:25:57.300 --> 00:26:01.170 then I can actually already figure out 00:26:01.170 --> 00:26:07.340 what are the eigenvectors of M minus 1 K matrix. 00:26:07.340 --> 00:26:09.950 OK? 00:26:09.950 --> 00:26:13.480 And another thing which we've learned from here 00:26:13.480 --> 00:26:16.550 is that, wow, that's good! 00:26:16.550 --> 00:26:19.760 Because the eigenvectors are already solved. 00:26:19.760 --> 00:26:23.690 Therefore, I just have to calculate this. 00:26:23.690 --> 00:26:25.420 It's just a normal operation. 00:26:25.420 --> 00:26:27.320 It's not the eigenvalue problem anymore. 00:26:27.320 --> 00:26:30.530 I just multiply M minus 1 K times A, 00:26:30.530 --> 00:26:34.350 then I can actually get the value omega square. 00:26:34.350 --> 00:26:34.850 You see? 00:26:34.850 --> 00:26:38.260 That's actually much easier than solving the eigenvalue problem 00:26:38.260 --> 00:26:40.505 of M minus 1 K matrix. 00:26:40.505 --> 00:26:41.005 OK? 00:26:43.710 --> 00:26:47.980 That's actually very good news. 00:26:47.980 --> 00:26:53.370 Finally, I think the most important consequence 00:26:53.370 --> 00:27:04.870 is that once we solve this system, which 00:27:04.870 --> 00:27:10.860 satisfy the symmetry described by this S matrix, 00:27:10.860 --> 00:27:16.290 we have solved all the possible systems which 00:27:16.290 --> 00:27:18.390 satisfy the same symmetry. 00:27:18.390 --> 00:27:23.520 For example, in this case, I solve a coupled pendulum 00:27:23.520 --> 00:27:24.670 problem, OK? 00:27:24.670 --> 00:27:26.290 They look symmetric. 00:27:26.290 --> 00:27:27.420 Right? 00:27:27.420 --> 00:27:31.260 And I can, of course, I can draw another one, 00:27:31.260 --> 00:27:32.495 which is like this. 00:27:32.495 --> 00:27:34.140 It's more circular. 00:27:34.140 --> 00:27:36.730 And there are two walls, which is actually-- 00:27:36.730 --> 00:27:40.510 there are three springs connected to the wall. 00:27:40.510 --> 00:27:42.720 This problem is already also solved, right? 00:27:42.720 --> 00:27:46.330 Because it also satisfy the same symmetry. 00:27:46.330 --> 00:27:50.070 And of course-- like, you know, like this, 00:27:50.070 --> 00:27:53.510 go crazy, and even more. 00:27:53.510 --> 00:27:55.180 This is also solved! 00:27:55.180 --> 00:27:55.680 Right? 00:27:55.680 --> 00:27:57.000 Because this is also symmetric. 00:27:57.000 --> 00:27:57.530 Right? 00:27:57.530 --> 00:27:59.420 I can add more. 00:27:59.420 --> 00:28:00.190 Right? 00:28:00.190 --> 00:28:01.240 Like this. 00:28:01.240 --> 00:28:02.750 This is also symmetric. 00:28:02.750 --> 00:28:03.720 Right? 00:28:03.720 --> 00:28:06.150 And this-- let's think. 00:28:06.150 --> 00:28:11.430 The eigenvector of this M minus 1 K matrix eigenvalue problem 00:28:11.430 --> 00:28:15.420 will be identical to what we have already solved here. 00:28:15.420 --> 00:28:16.290 OK? 00:28:16.290 --> 00:28:18.810 So, that is actually really amazing. 00:28:18.810 --> 00:28:22.890 If you speak the right language, and cut into the problem 00:28:22.890 --> 00:28:25.150 in the right angle, you actually find 00:28:25.150 --> 00:28:31.010 that actually, you can solve multiple problems at one time. 00:28:31.010 --> 00:28:32.970 OK? 00:28:32.970 --> 00:28:33.740 Any questions? 00:28:36.710 --> 00:28:37.910 OK. 00:28:37.910 --> 00:28:40.950 So now this is actually very nice, 00:28:40.950 --> 00:28:44.550 and this is actually a very important preparation 00:28:44.550 --> 00:28:48.690 to the next step, actually. 00:28:48.690 --> 00:28:52.500 So now, we have understood coupled oscillator, 00:28:52.500 --> 00:28:55.860 and we have learned a little bit about symmetry. 00:28:55.860 --> 00:28:59.460 Therefore, I would like to go to infinite 00:28:59.460 --> 00:29:01.981 number of coupled oscillator. 00:29:01.981 --> 00:29:02.480 OK? 00:29:02.480 --> 00:29:04.770 So that is actually the next step, which 00:29:04.770 --> 00:29:10.440 we are going to move on in 8.03 00:29:10.440 --> 00:29:13.920 So this is actually one example infinite system. 00:29:13.920 --> 00:29:15.660 OK? 00:29:15.660 --> 00:29:17.501 I cannot write the whole universe. 00:29:17.501 --> 00:29:18.000 Why? 00:29:18.000 --> 00:29:21.480 Because it's infinite, so I couldn't include everything 00:29:21.480 --> 00:29:22.585 in the slide. 00:29:22.585 --> 00:29:24.210 But this is actually an example system. 00:29:24.210 --> 00:29:27.520 Done OK? 00:29:27.520 --> 00:29:30.870 Looks hopeless, right? 00:29:30.870 --> 00:29:34.060 In general, we don't know how to solve 00:29:34.060 --> 00:29:37.030 infinite system, because if you have infinite number of things 00:29:37.030 --> 00:29:41.700 that are connected to each other in random ways, 00:29:41.700 --> 00:29:46.350 then the problem becomes really, really complicated. 00:29:46.350 --> 00:29:46.920 OK? 00:29:46.920 --> 00:29:52.500 In general, I don't know how to solve this problem. 00:29:52.500 --> 00:29:56.930 And if you are a EE major, the first thing, maybe, you 00:29:56.930 --> 00:29:59.760 like to do is, ah, now I have this picture, 00:29:59.760 --> 00:30:01.770 and I can put everything in my computer, 00:30:01.770 --> 00:30:04.710 and see how things evolve as a function of time! 00:30:04.710 --> 00:30:06.810 Right? 00:30:06.810 --> 00:30:08.730 Of course we can rely on the computers, 00:30:08.730 --> 00:30:10.810 and see what we can learn from it. 00:30:10.810 --> 00:30:14.100 And if you made your major of mathematics, 00:30:14.100 --> 00:30:16.344 you will say, no, this is not the problem 00:30:16.344 --> 00:30:17.260 I am going to work on. 00:30:19.900 --> 00:30:20.400 OK? 00:30:20.400 --> 00:30:23.280 I don't care. 00:30:23.280 --> 00:30:26.760 But as a physicist, what we are going to do is that, huh-- 00:30:26.760 --> 00:30:30.366 we look at this infinite system, OK? 00:30:30.366 --> 00:30:31.960 It's kind of interesting, right? 00:30:31.960 --> 00:30:35.040 It's a lot of things, a lot of small balls connected 00:30:35.040 --> 00:30:36.600 to big balls, right? 00:30:36.600 --> 00:30:39.250 Super big ones, and plotting things in log scale. 00:30:39.250 --> 00:30:41.580 So those balls are really, really large 00:30:41.580 --> 00:30:45.210 compared to all the other balls connected to this system. 00:30:45.210 --> 00:30:46.860 Therefore, as a physicist, I'm going 00:30:46.860 --> 00:30:49.440 to ignore all the other balls. 00:30:52.710 --> 00:30:56.760 Oh, if I do that, then it becomes-- 00:30:56.760 --> 00:30:59.800 there is some kind of symmetry you can actually 00:30:59.800 --> 00:31:01.440 see from here, right? 00:31:01.440 --> 00:31:04.960 What is actually the symmetry? you see? 00:31:04.960 --> 00:31:07.960 There are three balls that connected to each other. 00:31:07.960 --> 00:31:10.080 They are equally spaced. 00:31:10.080 --> 00:31:13.500 We have a translation symmetry. 00:31:13.500 --> 00:31:14.670 You see? 00:31:14.670 --> 00:31:16.320 So you can see that, actually, that's 00:31:16.320 --> 00:31:19.490 how we think about a problem. 00:31:19.490 --> 00:31:22.950 Of course, different field have different kind of thinking, 00:31:22.950 --> 00:31:26.220 and different kind of problem they would like to focus on. 00:31:26.220 --> 00:31:28.410 But as a physicist, I would like to know 00:31:28.410 --> 00:31:30.570 how the system will work, and that is actually 00:31:30.570 --> 00:31:32.220 what I'm going to do. 00:31:32.220 --> 00:31:33.880 OK? 00:31:33.880 --> 00:31:36.400 So that's very nice. 00:31:36.400 --> 00:31:41.030 We are going to discuss infinite system. 00:31:41.030 --> 00:31:42.870 So what is actually the infinite system 00:31:42.870 --> 00:31:45.626 I am going to talk about? 00:31:45.626 --> 00:31:52.630 It's actually there is infinite system with space translation 00:31:52.630 --> 00:31:54.580 symmetry. 00:31:54.580 --> 00:32:04.270 So, to save some time, I have already written down the matrix 00:32:04.270 --> 00:32:08.290 involving this system here. 00:32:08.290 --> 00:32:13.790 What I am interested is mass sprint system, OK? 00:32:13.790 --> 00:32:16.810 Infinite number of mass and spring. 00:32:16.810 --> 00:32:22.590 And they actually satisfy space translation symmetry. 00:32:22.590 --> 00:32:25.170 OK? 00:32:25.170 --> 00:32:29.860 They are connected to each other by springs, with natural length 00:32:29.860 --> 00:32:34.780 A and spring constant K. OK? 00:32:34.780 --> 00:32:36.520 And there are infinite number of them, 00:32:36.520 --> 00:32:39.520 actually, lined up from the left-hand side 00:32:39.520 --> 00:32:42.220 of the edge of the universe to the right-hand side 00:32:42.220 --> 00:32:43.520 edge of the universe. 00:32:43.520 --> 00:32:44.020 OK? 00:32:44.020 --> 00:32:45.280 I've prepared this system. 00:32:45.280 --> 00:32:45.780 OK? 00:32:45.780 --> 00:32:47.170 It took me a long time. 00:32:47.170 --> 00:32:49.030 OK? 00:32:49.030 --> 00:32:50.340 All right? 00:32:50.340 --> 00:32:53.420 But it's very difficult to describe this kind of system, 00:32:53.420 --> 00:32:53.920 right? 00:32:53.920 --> 00:32:57.130 So the first thing we have learned from 8.03 00:32:57.130 --> 00:32:59.630 is that in order to describe this system, 00:32:59.630 --> 00:33:03.210 I need to define a coordinate system, right? 00:33:03.210 --> 00:33:07.100 And also have everything properly labeled. 00:33:07.100 --> 00:33:08.830 So I introduce a label-- 00:33:08.830 --> 00:33:13.480 j minus 1 j, j plus one, j plus two-- 00:33:13.480 --> 00:33:17.170 just to name each little mass I'm talking about. 00:33:17.170 --> 00:33:17.740 OK? 00:33:17.740 --> 00:33:19.300 No other purpose. 00:33:19.300 --> 00:33:24.930 Then, once I have the label, I can actually write everything, 00:33:24.930 --> 00:33:29.680 express the displacement of little mass, as X j minus 1, 00:33:29.680 --> 00:33:34.650 X j X j plus one, X j plus two. 00:33:34.650 --> 00:33:39.410 That's just the displacement from the equilibrium position 00:33:39.410 --> 00:33:40.310 of the mass. 00:33:40.310 --> 00:33:42.690 OK? 00:33:42.690 --> 00:33:44.675 And this system will have equation 00:33:44.675 --> 00:33:46.720 of motion looks like this. 00:33:46.720 --> 00:33:53.920 So if now I focus on the little mass, Z. OK? 00:33:53.920 --> 00:33:57.810 Then I can actually write down the equation of motion. 00:33:57.810 --> 00:34:05.560 There are two springs connected to these mass. 00:34:05.560 --> 00:34:06.220 Right? 00:34:06.220 --> 00:34:11.230 Therefore, you are going to have two spring force. 00:34:11.230 --> 00:34:12.040 Right? 00:34:12.040 --> 00:34:15.100 Since this is actually idealize the springs with spring 00:34:15.100 --> 00:34:18.580 constant capital K, therefore, I can write down 00:34:18.580 --> 00:34:21.144 immediately the equation of motion 00:34:21.144 --> 00:34:27.790 is actually equal to M X double dot j is equal to minus 00:34:27.790 --> 00:34:32.784 K X j minus X j minus 1 minus-- 00:34:32.784 --> 00:34:37.300 this is actually the right-hand side spring force-- 00:34:37.300 --> 00:34:41.050 minus K X j minus X j plus 1. 00:34:41.050 --> 00:34:43.139 We have done this exercise before, right, 00:34:43.139 --> 00:34:46.560 with a simpler problem. 00:34:46.560 --> 00:34:47.380 OK? 00:34:47.380 --> 00:34:50.650 As usual, I can collect all the parents 00:34:50.650 --> 00:34:58.000 associated with X j minus 1, X j, and X j plus 1, together. 00:34:58.000 --> 00:35:01.900 Then I get this expression, which actually looks nice. 00:35:01.900 --> 00:35:03.840 OK? 00:35:03.840 --> 00:35:08.550 And I assume that this system is actually undergoing 00:35:08.550 --> 00:35:12.810 some kind of oscillation. 00:35:12.810 --> 00:35:13.440 OK? 00:35:13.440 --> 00:35:17.010 Therefore, I assume that this solution, X j 00:35:17.010 --> 00:35:22.730 will be equal to A j is the amplitude of j's mass. 00:35:22.730 --> 00:35:23.610 OK? 00:35:23.610 --> 00:35:28.380 Cosine omega t plus phi, omega is actually the oscillation 00:35:28.380 --> 00:35:31.020 frequency, and phi is actually the phase, 00:35:31.020 --> 00:35:34.841 and I don't know why this is actually omega and A j yet. 00:35:34.841 --> 00:35:35.340 OK? 00:35:35.340 --> 00:35:37.930 We would like to figure that out. 00:35:37.930 --> 00:35:45.240 And as usual, you can actually write down the M matrix, OK? 00:35:45.240 --> 00:35:50.520 M matrix is actually really simple, in the diagonal terms-- 00:35:50.520 --> 00:35:53.760 diagonal terms are all m, and the off diagonal terms 00:35:53.760 --> 00:35:55.462 are all zero. 00:35:55.462 --> 00:35:57.096 Right? 00:35:57.096 --> 00:35:58.720 And you don't really need to copy them, 00:35:58.720 --> 00:36:01.650 because they're all derived in the lecture notes. 00:36:01.650 --> 00:36:03.411 M minus 1 K matrix-- 00:36:03.411 --> 00:36:03.910 ha! 00:36:03.910 --> 00:36:06.940 I have already arranged my terms here; 00:36:06.940 --> 00:36:08.480 therefore it looks like this. 00:36:08.480 --> 00:36:12.070 It have a strange structure, you have three terms, 00:36:12.070 --> 00:36:17.790 kind of in the diagonal terms, and this actually 00:36:17.790 --> 00:36:23.740 is shifting as a function of number of rows, 00:36:23.740 --> 00:36:29.380 and all the other parts of the matrix actually zero. 00:36:29.380 --> 00:36:29.880 OK? 00:36:29.880 --> 00:36:34.440 It's an infinite times infinite dimension matrix. 00:36:34.440 --> 00:36:37.100 Finally, I would like to also write my 00:36:37.100 --> 00:36:41.080 A matrix is the vector of amplitude, right? 00:36:41.080 --> 00:36:42.880 So you have many, many numbers-- 00:36:42.880 --> 00:36:45.940 A j, A j plus 1, A j plus 2. 00:36:45.940 --> 00:36:47.940 OK? 00:36:47.940 --> 00:36:49.970 And et cetera, et cetera. 00:36:49.970 --> 00:36:50.910 OK? 00:36:50.910 --> 00:36:55.100 Now, very easy, right? 00:36:55.100 --> 00:36:58.220 The question is actually can be solved, right? 00:36:58.220 --> 00:37:02.730 You just have to solve the M minus 1 K matrix, right? 00:37:02.730 --> 00:37:03.690 That's easy, right? 00:37:03.690 --> 00:37:07.950 It's an infinite number times infinite number matrix, right? 00:37:07.950 --> 00:37:09.090 Super easy! 00:37:09.090 --> 00:37:09.930 No, actually not. 00:37:09.930 --> 00:37:10.708 Right? 00:37:10.708 --> 00:37:14.790 [LAUGHTER] So we are in trouble. 00:37:14.790 --> 00:37:16.960 I don't know how to solve this problem. 00:37:16.960 --> 00:37:19.250 OK? 00:37:19.250 --> 00:37:22.000 What can we do? 00:37:22.000 --> 00:37:24.698 Anybody have any suggestion to me? 00:37:24.698 --> 00:37:26.547 AUDIENCE: Ask the math department? 00:37:26.547 --> 00:37:27.380 PROFESSOR: Ah, yeah! 00:37:27.380 --> 00:37:29.730 Math department is coming in to help. 00:37:29.730 --> 00:37:31.360 Yes. 00:37:31.360 --> 00:37:33.650 But actually, before asking them, 00:37:33.650 --> 00:37:37.320 we learn some concept, which we just learned, right? 00:37:37.320 --> 00:37:40.070 This-- what kind of property of this system? 00:37:40.070 --> 00:37:41.480 AUDIENCE: Symmetry. 00:37:41.480 --> 00:37:42.650 PROFESSOR: Symmetry! 00:37:42.650 --> 00:37:43.290 Right? 00:37:43.290 --> 00:37:44.590 We have symmetry. 00:37:44.590 --> 00:37:45.350 OK? 00:37:45.350 --> 00:37:50.300 So this M minus 1 K matrix looks horrible. 00:37:50.300 --> 00:37:55.770 But if I write down the symmetry matrix, 00:37:55.770 --> 00:37:58.150 actually, it looks slightly better. 00:37:58.150 --> 00:37:59.570 OK? 00:37:59.570 --> 00:38:03.470 So what is actually the symmetry matrix? 00:38:03.470 --> 00:38:06.090 So one observation we can make from this system 00:38:06.090 --> 00:38:12.380 is that if I shift this system, A, to the left, OK? 00:38:12.380 --> 00:38:15.980 I shift these two mass to the left-hand side, 00:38:15.980 --> 00:38:17.180 I shift all the mass. 00:38:17.180 --> 00:38:19.220 I have to hire many, many students 00:38:19.220 --> 00:38:22.180 to move all the mass from left-hand side of the universe 00:38:22.180 --> 00:38:23.780 to right-hand side of the universe. 00:38:23.780 --> 00:38:24.500 OK? 00:38:24.500 --> 00:38:28.776 And after they have done that, the system looks the same. 00:38:28.776 --> 00:38:29.650 Right? 00:38:29.650 --> 00:38:31.310 That's very good, OK? 00:38:31.310 --> 00:38:34.640 After all the hard work, right? 00:38:34.640 --> 00:38:39.160 So what is actually going to be the symmetry matrix? 00:38:39.160 --> 00:38:39.710 OK. 00:38:39.710 --> 00:38:42.050 Now, I would like to achieve something 00:38:42.050 --> 00:38:46.290 which is A prime equal to S A. And then 00:38:46.290 --> 00:38:53.030 this S actually shift the mass by a distance of A. Right? 00:38:53.030 --> 00:38:56.320 So what would be the functional formula for this S matrix? 00:38:56.320 --> 00:38:58.620 It would look like this. 00:38:58.620 --> 00:39:02.505 It's going to be 0, 1, 0, 0, 0-- 00:39:02.505 --> 00:39:04.940 0, 0, 1, 0, 0, 0-- 00:39:12.130 --> 00:39:14.060 looks like this. 00:39:14.060 --> 00:39:15.300 OK? 00:39:15.300 --> 00:39:20.740 So the next two diagonal term is all one. 00:39:20.740 --> 00:39:24.130 All the rest of the component is zero. 00:39:24.130 --> 00:39:24.940 OK? 00:39:24.940 --> 00:39:29.650 And this looks a lot more friendly compared 00:39:29.650 --> 00:39:31.210 to M minus 1 K matrix, right? 00:39:31.210 --> 00:39:34.540 Still, this is horrible thing to do, 00:39:34.540 --> 00:39:39.630 because this is infinite number times infinite number dimension 00:39:39.630 --> 00:39:40.281 matrix. 00:39:40.281 --> 00:39:40.780 OK? 00:39:44.200 --> 00:39:45.460 So. 00:39:45.460 --> 00:39:51.010 We would like to find the eigenvectors of S matrix. 00:39:51.010 --> 00:39:52.300 OK? 00:39:52.300 --> 00:39:55.540 So this means that if I manage to solve 00:39:55.540 --> 00:39:57.360 the eigenvalue problem, assuming that-- 00:39:57.360 --> 00:40:01.700 OK, I haven't solved it, but assuming that I can solve it, 00:40:01.700 --> 00:40:03.850 then what I'm going to do is going 00:40:03.850 --> 00:40:10.450 to get this S A will be equal to beta A, where A is actually 00:40:10.450 --> 00:40:16.220 a eigenvector of S matrix OK? 00:40:16.220 --> 00:40:20.415 And S A, we just learned from here, 00:40:20.415 --> 00:40:22.018 is actually equal to A prime. 00:40:24.826 --> 00:40:32.950 So beta is the eigenvalue, and A is actually the eigenvector. 00:40:36.160 --> 00:40:38.690 So that means, originally, I have 00:40:38.690 --> 00:40:46.360 A, which is something something A j, A j plus 1, A j plus 2, 00:40:46.360 --> 00:40:47.260 blah blah blah. 00:40:47.260 --> 00:40:48.400 OK? 00:40:48.400 --> 00:40:55.900 And A prime, after I actually multiply A by S matrix, 00:40:55.900 --> 00:41:00.730 I get A prime, which looks like this-- 00:41:00.730 --> 00:41:08.350 A j plus 1, A j plus 2, A j plus 3. 00:41:08.350 --> 00:41:10.320 OK? 00:41:10.320 --> 00:41:12.980 So what I am going to do is-- 00:41:12.980 --> 00:41:15.210 what, actually, this S matrix does 00:41:15.210 --> 00:41:23.350 is to shift the A component one row, right? 00:41:23.350 --> 00:41:24.150 OK? 00:41:24.150 --> 00:41:27.850 So then, we basically get this expression. 00:41:27.850 --> 00:41:31.530 And of course, A 1 is equal to beta, which 00:41:31.530 --> 00:41:33.780 is a constant, times A. Right? 00:41:33.780 --> 00:41:42.220 So if you compare, for example, here, you can get that-- 00:41:42.220 --> 00:41:49.630 A j prime will be equal to beta A j, which is actually 00:41:49.630 --> 00:41:52.091 equal to A j plus 1. 00:41:52.091 --> 00:41:52.590 Right? 00:41:52.590 --> 00:41:56.420 A j prime is actually equal to A j plus 1, right? 00:41:56.420 --> 00:42:01.400 It's just shifting one unique label. 00:42:01.400 --> 00:42:02.470 Right? 00:42:02.470 --> 00:42:04.740 OK. 00:42:04.740 --> 00:42:07.980 So this is actually the expression I'm looking for. 00:42:07.980 --> 00:42:08.970 OK? 00:42:08.970 --> 00:42:12.060 We don't know yet why this is actually beta. 00:42:12.060 --> 00:42:13.680 Beta is a number. 00:42:13.680 --> 00:42:16.590 Assuming that I can solve the eigenvalue problem. 00:42:16.590 --> 00:42:17.900 OK? 00:42:17.900 --> 00:42:25.720 But I do know, if I have A 0, if A 0 is equal to 0, 00:42:25.720 --> 00:42:31.790 from this expression, that means A 1-- 00:42:31.790 --> 00:42:33.810 sorry, A 0 is equal to 1. 00:42:33.810 --> 00:42:36.510 If A 0 is 0, then everything's 0, right? 00:42:36.510 --> 00:42:38.530 And it's not fun, right? 00:42:38.530 --> 00:42:39.030 OK. 00:42:39.030 --> 00:42:41.915 A 0 is equal to 1, then something will happen. 00:42:41.915 --> 00:42:44.020 A 1 will be equal to beta, right? 00:42:44.020 --> 00:42:45.900 From this expression, right? 00:42:45.900 --> 00:42:50.360 Because beta A j is equal to A j plus 1, 00:42:50.360 --> 00:42:54.690 A 2 will be equal to beta square, et cetera, et cetera. 00:42:54.690 --> 00:42:58.380 And then I can say that A j, if I assume 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 00:43:09.070 --> 00:43:11.220 to the j. 00:43:11.220 --> 00:43:12.290 OK? 00:43:12.290 --> 00:43:15.130 Am I going too fast, here? 00:43:15.130 --> 00:43:18.860 Everybody is following? 00:43:18.860 --> 00:43:19.870 No questions? 00:43:19.870 --> 00:43:21.280 No? 00:43:21.280 --> 00:43:22.930 Good. 00:43:22.930 --> 00:43:26.670 Actually, we found that we have already 00:43:26.670 --> 00:43:29.140 solved the eigenvalue problem. 00:43:29.140 --> 00:43:30.000 Right? 00:43:30.000 --> 00:43:33.840 Because I have already the expression for the A j, 00:43:33.840 --> 00:43:37.410 which is actually in the form of beta to the j, right? 00:43:37.410 --> 00:43:43.800 So beta is some kind of number, and the infinite number of beta 00:43:43.800 --> 00:43:49.200 actually can satisfy this eigenvalue problem. 00:43:49.200 --> 00:43:51.790 No matter what kind of beta I choose-- 00:43:51.790 --> 00:43:58.980 it can be 1, it can be 2, 3.14, it can be pi-- 00:43:58.980 --> 00:44:02.640 and what am I going to get is the corresponding A j, 00:44:02.640 --> 00:44:06.670 corresponding A vector, which you have satisfied 00:44:06.670 --> 00:44:08.230 this expression. 00:44:08.230 --> 00:44:08.970 OK? 00:44:08.970 --> 00:44:12.270 So that means some magic happen. 00:44:12.270 --> 00:44:15.690 We have already solved the eigenvalue problem 00:44:15.690 --> 00:44:20.660 without really deriving, you know, a lot of deviation. 00:44:20.660 --> 00:44:21.630 Right? 00:44:21.630 --> 00:44:23.880 Secondly, another thing which we learned 00:44:23.880 --> 00:44:30.030 is that there are infinite number of eigenvalue which 00:44:30.030 --> 00:44:33.660 satisfy this eigenvalue problem. 00:44:33.660 --> 00:44:38.020 The question is, does that make sense, or not? 00:44:38.020 --> 00:44:41.320 Infinite number of eigenvalues can actually 00:44:41.320 --> 00:44:44.600 satisfy this infinity long system. 00:44:44.600 --> 00:44:47.140 It's kind of making sense, right? 00:44:47.140 --> 00:44:50.170 Because we have worked on one oscillator, 00:44:50.170 --> 00:44:53.590 you had one normal mode; two oscillator, 00:44:53.590 --> 00:44:56.050 you have two normal mode; three oscillator, you 00:44:56.050 --> 00:44:57.400 have three normal mode-- 00:44:57.400 --> 00:44:59.410 infinite number of oscillator, you 00:44:59.410 --> 00:45:02.872 should have infinite number of normal modes. 00:45:02.872 --> 00:45:04.120 Right? 00:45:04.120 --> 00:45:08.620 OK, so that is actually a very, very good news, 00:45:08.620 --> 00:45:13.490 because we have already solved the problem, 00:45:13.490 --> 00:45:19.041 and we also know the function of four of eigenvectors. 00:45:19.041 --> 00:45:19.540 OK? 00:45:19.540 --> 00:45:22.150 So let's take a look at those example 00:45:22.150 --> 00:45:26.390 system, which are actually close to infinity long. 00:45:26.390 --> 00:45:31.180 So here, you have a Bell Lab machine, 00:45:31.180 --> 00:45:34.040 which actually can have, actually, 00:45:34.040 --> 00:45:37.450 multiple coupled oscillators. 00:45:37.450 --> 00:45:40.640 Each one of them can oscillate up and down, 00:45:40.640 --> 00:45:44.260 and you can see that, huh, if I actually 00:45:44.260 --> 00:45:50.760 tried to move them up and down, that a complicated kind 00:45:50.760 --> 00:45:54.550 of motion can occur from this system. 00:45:54.550 --> 00:45:58.450 Actually, if I do this, you see that, ah, they are something 00:45:58.450 --> 00:46:00.890 similar to wave is happening. 00:46:00.890 --> 00:46:02.665 And if I do this continuously-- 00:46:05.460 --> 00:46:08.940 oh, some kind of, like, a standing wave 00:46:08.940 --> 00:46:11.100 is produced, right? 00:46:11.100 --> 00:46:14.160 And this system is actually really, really hard 00:46:14.160 --> 00:46:15.820 to describe, right? 00:46:15.820 --> 00:46:21.180 If you look at how many things this system can actually do. 00:46:21.180 --> 00:46:22.290 OK? 00:46:22.290 --> 00:46:25.080 Another example is actually-- 00:46:25.080 --> 00:46:27.360 OK, so you can say, come on, this is actually not 00:46:27.360 --> 00:46:29.520 infinitely long system, right? 00:46:29.520 --> 00:46:31.690 You have some final number, right? 00:46:31.690 --> 00:46:36.580 So how about I use this system as a demonstration. 00:46:36.580 --> 00:46:41.010 This is actually a much nicer, or much better, 00:46:41.010 --> 00:46:47.280 approximation, OK, to infinitely long system. 00:46:47.280 --> 00:46:52.310 You can see that, OK, each mass, each-- 00:46:52.310 --> 00:46:56.060 OK, I can say, for example, each small component of the spring, 00:46:56.060 --> 00:47:01.490 essentially, can become seeded as a small m in my graph, 00:47:01.490 --> 00:47:02.050 right? 00:47:02.050 --> 00:47:06.050 And actually, I can, instead of oscillating them 00:47:06.050 --> 00:47:09.940 back and forth, I oscillate them upside down. 00:47:09.940 --> 00:47:10.440 OK? 00:47:10.440 --> 00:47:13.550 And you can see that, huh, they are interesting kind of motion. 00:47:13.550 --> 00:47:15.320 I can have-- 00:47:15.320 --> 00:47:23.040 I can have this, which is like a standing wave; I can do this; 00:47:23.040 --> 00:47:26.775 I can stop this system, and I produce-- 00:47:26.775 --> 00:47:27.590 woo! 00:47:27.590 --> 00:47:29.580 I can produce a wave. 00:47:29.580 --> 00:47:31.490 And then it goes back and forth. 00:47:31.490 --> 00:47:34.310 And I can, whoa, do this crazy, and then you 00:47:34.310 --> 00:47:36.140 see that, how exciting-- 00:47:36.140 --> 00:47:39.680 a much higher frequency normal mode, right? 00:47:39.680 --> 00:47:41.570 And that's really complicated. 00:47:41.570 --> 00:47:47.720 And the question is, how can we actually understand 00:47:47.720 --> 00:47:49.040 this kind of system? 00:47:49.040 --> 00:47:53.000 The thing is that this system is so much, so complicated, 00:47:53.000 --> 00:47:56.330 and have infinite amount of possibilities. 00:47:56.330 --> 00:47:57.920 Right? 00:47:57.920 --> 00:48:02.000 So how are we going to understand this? 00:48:02.000 --> 00:48:06.380 Very good news is that we have solved the normal modes 00:48:06.380 --> 00:48:08.270 of this kind of system, right? 00:48:08.270 --> 00:48:09.940 So the normal mode looks like this-- 00:48:09.940 --> 00:48:13.130 A j equal to beta j. 00:48:13.130 --> 00:48:14.150 OK? 00:48:14.150 --> 00:48:18.710 And the following lecture, the rest of the lecture, 00:48:18.710 --> 00:48:22.430 is to understand what does that mean, 00:48:22.430 --> 00:48:25.670 and also make predictions. 00:48:25.670 --> 00:48:26.170 OK? 00:48:28.700 --> 00:48:32.930 So now we have, actually, the eigenvectors, OK? 00:48:32.930 --> 00:48:34.380 That's really nice. 00:48:34.380 --> 00:48:39.230 So from our previous discussion, if this system actually 00:48:39.230 --> 00:48:43.020 satisfy the symmetry, have the symmetry 00:48:43.020 --> 00:48:47.180 that is acquired by the S matrix, which I have here, that 00:48:47.180 --> 00:48:50.780 means M minus 1 K matrix will share 00:48:50.780 --> 00:48:55.580 the same set of eigenvectors as S matrix. 00:48:55.580 --> 00:49:01.070 So what is actually part of the work is to evaluate this. 00:49:01.070 --> 00:49:07.330 M minus 1 K multiplied by A, and that will give you 00:49:07.330 --> 00:49:11.780 omega squared A. OK? 00:49:11.780 --> 00:49:16.920 So I just need to multiply M minus 1 K matrix by A. 00:49:16.920 --> 00:49:17.650 What is A? 00:49:17.650 --> 00:49:20.050 A is actually here. 00:49:20.050 --> 00:49:22.950 Now what is actually M minus 1 K matrix? 00:49:22.950 --> 00:49:27.170 M minus 1 K matrix is here, have a kind 00:49:27.170 --> 00:49:29.380 of complicated structure. 00:49:29.380 --> 00:49:30.230 OK? 00:49:30.230 --> 00:49:32.660 On the other hand, if I only focused 00:49:32.660 --> 00:49:37.040 on the jth object, the object which is named j, 00:49:37.040 --> 00:49:43.070 have a label j, then actually I can write down, OK, 00:49:43.070 --> 00:49:49.660 the right-hand side is actually just omega square A j, right? 00:49:49.660 --> 00:49:51.650 Because this is actually-- 00:49:51.650 --> 00:49:54.470 if I only focus on the j component, 00:49:54.470 --> 00:49:57.920 OK, left-hand side is actually just M minus 1 K A 00:49:57.920 --> 00:50:02.200 multiplied by A, right? 00:50:02.200 --> 00:50:05.450 OK, so basically, there are only these three 00:50:05.450 --> 00:50:07.750 terms coming into play, right? 00:50:07.750 --> 00:50:11.300 If this is A j minus 1, so anything minus 1, 00:50:11.300 --> 00:50:14.870 we are multiplying by minus K over n. 00:50:14.870 --> 00:50:19.760 A j we multiply by 2 K over n, and A j plus 1, 00:50:19.760 --> 00:50:22.440 we are multiplying by minus K over n, right? 00:50:22.440 --> 00:50:25.760 The rest of the terms are all 0. 00:50:25.760 --> 00:50:26.260 OK? 00:50:26.260 --> 00:50:29.460 It's actually not as complicated as we thought. 00:50:29.460 --> 00:50:30.200 OK? 00:50:30.200 --> 00:50:32.620 So, if I write it down, explicitly, 00:50:32.620 --> 00:50:35.630 the left-hand side part, then what I'm going to get 00:50:35.630 --> 00:50:44.291 is minus K over n, capital K over n, A j minus 1, 00:50:44.291 --> 00:50:55.795 plus 2 capital K over n A j minus capital K over n, 00:50:55.795 --> 00:50:59.110 A j plus one. 00:50:59.110 --> 00:51:00.820 OK? 00:51:00.820 --> 00:51:03.190 So this is actually the j term. 00:51:03.190 --> 00:51:08.270 Now I can define omega 0 square equal-- 00:51:08.270 --> 00:51:14.380 is defined as capital K over n. 00:51:14.380 --> 00:51:17.970 If I do that, then basically, I can 00:51:17.970 --> 00:51:28.790 see that omega square A j will be equal to omega 0 square. 00:51:28.790 --> 00:51:29.290 OK? 00:51:29.290 --> 00:51:32.830 I am taking all the K over n out of the game 00:51:32.830 --> 00:51:35.510 and write it down as omega 0 square. 00:51:35.510 --> 00:51:37.660 OK? 00:51:37.660 --> 00:51:48.370 Minus A j minus 1 plus 2 A j minus A j plus 1. 00:51:48.370 --> 00:51:50.590 OK? 00:51:50.590 --> 00:51:55.990 And also we know, from the previous discussion, S matrix 00:51:55.990 --> 00:51:58.570 and the n minus 1 K matrix should share 00:51:58.570 --> 00:52:00.910 the same sets of eigenvectors. 00:52:00.910 --> 00:52:05.250 Therefore, I can actually try to plug in one of the eigenvectors 00:52:05.250 --> 00:52:06.621 from S matrix. 00:52:06.621 --> 00:52:07.120 Right? 00:52:07.120 --> 00:52:09.440 A j equal to beta j. 00:52:09.440 --> 00:52:09.940 OK? 00:52:09.940 --> 00:52:12.040 I can plug that in, then basically, 00:52:12.040 --> 00:52:17.830 I get omega 0 square minus b-- 00:52:17.830 --> 00:52:26.130 minus beta, j minus 1 plus 2 beta to the j minus beta 00:52:26.130 --> 00:52:29.860 to the j plus 1. 00:52:29.860 --> 00:52:34.990 And the left-hand side will be reading like omega square beta 00:52:34.990 --> 00:52:38.000 to the j. 00:52:38.000 --> 00:52:39.610 OK? 00:52:39.610 --> 00:52:42.950 Questions? 00:52:42.950 --> 00:52:44.670 OK. 00:52:44.670 --> 00:52:47.240 So now, I can cancel-- 00:52:47.240 --> 00:52:50.570 I can actually divide everything by beta to the j, right? 00:52:50.570 --> 00:52:52.830 I can get rid of beta to the j, then 00:52:52.830 --> 00:53:00.025 basically, I get omega square equal to omega 0 square minus 1 00:53:00.025 --> 00:53:04.525 over beta plus 2 minus beta. 00:53:07.850 --> 00:53:09.600 OK? 00:53:09.600 --> 00:53:17.520 And as we discussed before, beta can have any value. 00:53:17.520 --> 00:53:18.190 OK? 00:53:18.190 --> 00:53:21.750 And also, you can see from here that, huh-- 00:53:21.750 --> 00:53:27.975 once I know the eigenvalue of S matrix and eigenvector of S 00:53:27.975 --> 00:53:31.380 matrix, I also know what is actually 00:53:31.380 --> 00:53:36.604 the corresponding angle of frequency of the normal mode. 00:53:36.604 --> 00:53:37.520 Right? 00:53:37.520 --> 00:53:41.260 By using M minus 1 K times A, you 00:53:41.260 --> 00:53:45.750 can figure out what is actually the corresponding omega, 00:53:45.750 --> 00:53:48.530 the normal mode frequency. 00:53:48.530 --> 00:53:49.840 OK? 00:53:49.840 --> 00:53:53.626 So that is actually pretty nice. 00:53:53.626 --> 00:53:57.540 But on the other hand, if you step back and just 00:53:57.540 --> 00:54:01.410 think about what we have been doing, OK? 00:54:01.410 --> 00:54:02.100 So very good. 00:54:02.100 --> 00:54:05.910 You have a beta, which is a random value. 00:54:05.910 --> 00:54:09.100 You can evaluate this thing, then 00:54:09.100 --> 00:54:14.490 you can get the corresponding omega. 00:54:14.490 --> 00:54:16.510 But then something doesn't feel right. 00:54:16.510 --> 00:54:17.490 Right? 00:54:17.490 --> 00:54:23.390 For example, if you have beta equal to 2, 00:54:23.390 --> 00:54:26.390 what is going to happen? 00:54:26.390 --> 00:54:29.440 If you have beta equal to 2, what does that mean? 00:54:29.440 --> 00:54:35.730 That means A j will be equal to 2 to the j. 00:54:38.971 --> 00:54:39.470 OK? 00:54:39.470 --> 00:54:41.420 That's very dangerous. 00:54:41.420 --> 00:54:42.540 Hey? 00:54:42.540 --> 00:54:46.910 That means-- OK, so I am-- 00:54:46.910 --> 00:54:50.810 I deploy the whole system, OK, from the left-hand side 00:54:50.810 --> 00:54:52.970 of the universe to the right-hand side 00:54:52.970 --> 00:54:53.870 of the universe. 00:54:53.870 --> 00:54:54.530 OK? 00:54:54.530 --> 00:54:58.950 So that means, if I go to the your right-hand side 00:54:58.950 --> 00:55:04.940 of the universe, the amplitude explode. 00:55:04.940 --> 00:55:05.440 Right? 00:55:05.440 --> 00:55:08.790 It's actually 2 to the infinite number, right? 00:55:08.790 --> 00:55:09.290 OK? 00:55:09.290 --> 00:55:12.310 It's not a physical-- doesn't sound like a physical system 00:55:12.310 --> 00:55:13.640 to me. 00:55:13.640 --> 00:55:15.040 Right? 00:55:15.040 --> 00:55:18.160 If, actually, beta is greater than 1, 00:55:18.160 --> 00:55:21.370 then the right-hand side A of the universe, 00:55:21.370 --> 00:55:25.630 the amplitude there, will explode. 00:55:25.630 --> 00:55:26.430 OK? 00:55:26.430 --> 00:55:28.010 Doesn't sound right, right? 00:55:28.010 --> 00:55:28.890 So I don't like that. 00:55:28.890 --> 00:55:29.390 OK? 00:55:29.390 --> 00:55:32.570 Maybe you like it, but I don't like it. 00:55:32.570 --> 00:55:34.740 For the moment. 00:55:34.740 --> 00:55:39.230 On the other hand, if the beta-- 00:55:39.230 --> 00:55:43.220 OK, again, it's not 1, but smaller than 1-- 00:55:43.220 --> 00:55:45.430 what is going to happen? 00:55:45.430 --> 00:55:49.370 If the beta is smaller than 1, what is going to happen 00:55:49.370 --> 00:55:53.050 is that, huh, OK, the right-hand side of the universe is fine, 00:55:53.050 --> 00:55:56.340 is finite, because the amplitude has become smaller and smaller. 00:55:56.340 --> 00:55:59.480 But the left-hand side part of the universe, the amplitude 00:55:59.480 --> 00:56:00.445 still explode. 00:56:03.070 --> 00:56:03.750 Right? 00:56:03.750 --> 00:56:05.920 So what does that mean? 00:56:05.920 --> 00:56:09.990 This means that if beta-- 00:56:09.990 --> 00:56:15.290 if the absolute value of beta is not equal to 1, 00:56:15.290 --> 00:56:20.820 the amplitude, at some point, goes to infinity. 00:56:20.820 --> 00:56:21.360 OK? 00:56:21.360 --> 00:56:23.920 So that's actually not very nice. 00:56:23.920 --> 00:56:29.190 That's because A j is actually proportional to beta to the j. 00:56:29.190 --> 00:56:30.550 OK? 00:56:30.550 --> 00:56:33.520 So in the discussion we have here, 00:56:33.520 --> 00:56:38.560 we consider beta equal to 1 case. 00:56:38.560 --> 00:56:39.070 OK? 00:56:39.070 --> 00:56:42.610 Otherwise, it's actually, things will explode. 00:56:42.610 --> 00:56:43.720 OK? 00:56:43.720 --> 00:56:47.590 So if the absolute value f beta is equal to 1, 00:56:47.590 --> 00:56:56.450 in general, beta can be exponential i, small k A. 00:56:56.450 --> 00:56:57.160 Right? 00:56:57.160 --> 00:57:01.400 Then, actually, you can get absolute valuable of 1. 00:57:01.400 --> 00:57:01.900 OK? 00:57:01.900 --> 00:57:04.813 If beta is equal to 1, that means 00:57:04.813 --> 00:57:09.620 the amplitude of all the oscillators are the same. 00:57:09.620 --> 00:57:10.960 OK? 00:57:10.960 --> 00:57:16.030 All right, so now, if we accept this, we only limit ourself 00:57:16.030 --> 00:57:21.560 to the discussion of beta, absolute beta, value of beta 00:57:21.560 --> 00:57:26.410 equal to 1, then beta can be written as exponential i k A. 00:57:26.410 --> 00:57:29.260 Then, if I plug this back into this, 00:57:29.260 --> 00:57:33.040 basically, what you are going to get is omega square 00:57:33.040 --> 00:57:45.130 is equal to omega 0 square 2 minus exponential i k A 00:57:45.130 --> 00:57:50.020 plus exponential minus i k A. Right? 00:57:50.020 --> 00:57:54.640 Because you have minus 1 over beta, and beta, therefore 00:57:54.640 --> 00:57:58.330 you have exponential i k A, and the exponential 00:57:58.330 --> 00:57:59.790 minus i k A. OK? 00:57:59.790 --> 00:58:04.070 It's a lot of math in this lecture, 00:58:04.070 --> 00:58:06.500 but we are getting over to it. 00:58:06.500 --> 00:58:07.960 OK? 00:58:07.960 --> 00:58:08.770 All right. 00:58:08.770 --> 00:58:10.620 So that is actually-- 00:58:10.620 --> 00:58:13.150 we actually can identify this, and this actually 00:58:13.150 --> 00:58:25.960 can be rewritten as 2 omega 0 square 1 minus cosine k A. OK? 00:58:25.960 --> 00:58:30.748 We have arrived a surprisingly simple expression. 00:58:34.170 --> 00:58:38.780 So let's take a look at this expression carefully. 00:58:38.780 --> 00:58:45.530 So that means, for each given k, a small k, 00:58:45.530 --> 00:58:48.050 then I will have a corresponding angular 00:58:48.050 --> 00:58:50.590 frequency, omega square. 00:58:50.590 --> 00:58:52.490 OK? 00:58:52.490 --> 00:58:57.520 So still, there are infinite number 00:58:57.520 --> 00:59:01.331 of possible normal modes. 00:59:01.331 --> 00:59:01.830 OK? 00:59:01.830 --> 00:59:02.500 From this. 00:59:07.150 --> 00:59:10.720 So if I take a look at the amplitude, 00:59:10.720 --> 00:59:14.060 if I select a k value-- 00:59:14.060 --> 00:59:22.850 small k value-- if k is given, I can actually calculate 00:59:22.850 --> 00:59:24.930 the corresponding A j. 00:59:24.930 --> 00:59:29.330 So the A j I can actually define as a superposition 00:59:29.330 --> 00:59:39.500 of exponential i j k a, and minus exponential i j k a. 00:59:39.500 --> 00:59:42.500 And that will give you a sinusoidal shape. 00:59:47.520 --> 00:59:51.250 So if I give you the k, basically, you'll 00:59:51.250 --> 00:59:58.220 see that if I give you a k, then you get the corresponding beta. 00:59:58.220 --> 00:59:59.040 Right? 00:59:59.040 --> 01:00:03.950 And you are going to get omega, the corresponding omega square. 01:00:03.950 --> 01:00:07.750 But one interesting thing of this expression 01:00:07.750 --> 01:00:13.810 is that if you keep beta, or keep one over beta, 01:00:13.810 --> 01:00:17.120 you are going to get the same omega. 01:00:17.120 --> 01:00:25.130 Therefore, I can now use superposition principle. 01:00:25.130 --> 01:00:30.120 Basically, I can actually add these two solutions together, 01:00:30.120 --> 01:00:31.980 since they are going to be oscillating 01:00:31.980 --> 01:00:33.480 at the same frequency. 01:00:33.480 --> 01:00:37.350 Then what I'm going to get is, huh, interesting thing happen. 01:00:37.350 --> 01:00:41.640 The A j, the amplitude, as a function of j, 01:00:41.640 --> 01:00:44.660 it's like a sinusoidal function. 01:00:44.660 --> 01:00:45.690 OK? 01:00:45.690 --> 01:00:49.490 So that is actually what is really predicted 01:00:49.490 --> 01:00:51.390 to an infinity long system. 01:00:51.390 --> 01:00:55.050 For example, if I do this, you can see that, aha, indeed, 01:00:55.050 --> 01:00:57.700 I can see sinusoidal shape. 01:00:57.700 --> 01:00:58.410 OK? 01:00:58.410 --> 01:01:01.920 And you can see that the sinusoidal shape is actually 01:01:01.920 --> 01:01:07.510 oscillating up and down, like a standing wave. 01:01:07.510 --> 01:01:12.730 And that is actually exactly this expression. 01:01:12.730 --> 01:01:16.840 So that tells you something really interesting. 01:01:16.840 --> 01:01:22.120 That means the sinusoidal shape is associated with what? 01:01:22.120 --> 01:01:27.540 Associated with translation symmetry. 01:01:27.540 --> 01:01:28.040 Right? 01:01:28.040 --> 01:01:33.580 All I have been doing is to require this translation 01:01:33.580 --> 01:01:38.610 symmetry, and you already get the amplitude A j. 01:01:38.610 --> 01:01:44.090 And if you choose the physical beta value, 01:01:44.090 --> 01:01:46.260 then you already immediately arrive 01:01:46.260 --> 01:01:53.500 at a solution which is actually like sinusoidal shape. 01:01:53.500 --> 01:01:56.070 Doesn't that sounds really amazing to you? 01:02:00.620 --> 01:02:01.250 OK. 01:02:01.250 --> 01:02:05.060 So I think it's time to take a five-minute break, because I 01:02:05.060 --> 01:02:07.730 can see that you are overwhelmed by the math 01:02:07.730 --> 01:02:11.060 already, and of course, let's come back in five minutes, 01:02:11.060 --> 01:02:13.730 then we can discuss some more about what we have 01:02:13.730 --> 01:02:16.490 learned from this mathematics. 01:02:16.490 --> 01:02:19.392 And if you have any questions, please let me know. 01:02:24.580 --> 01:02:27.580 OK, so welcome back, everybody. 01:02:27.580 --> 01:02:30.420 Of course, you are welcome to come back here, and play 01:02:30.420 --> 01:02:32.080 with the demonstration. 01:02:32.080 --> 01:02:33.810 OK? 01:02:33.810 --> 01:02:34.380 So very good. 01:02:34.380 --> 01:02:39.150 So during the break, there are several questions asked, 01:02:39.150 --> 01:02:42.990 which I think, those are very good questions, 01:02:42.990 --> 01:02:45.990 and that's actually the purpose of this break. 01:02:45.990 --> 01:02:48.630 So it's a long day already, right? 01:02:48.630 --> 01:02:52.920 A lot of mathematics, and I hope everybody survived. 01:02:52.920 --> 01:02:53.450 OK? 01:02:53.450 --> 01:02:55.260 No dead body yet? 01:02:55.260 --> 01:02:59.940 You can see that here, I'm doing something really crazy, here. 01:02:59.940 --> 01:03:01.490 So, OK. 01:03:01.490 --> 01:03:05.350 Consider-- I think most of you got this point, 01:03:05.350 --> 01:03:07.820 beta not equal to 1 is not nice. 01:03:07.820 --> 01:03:10.450 Something explode at the edge of the universe. 01:03:10.450 --> 01:03:11.640 So I don't like that. 01:03:11.640 --> 01:03:14.100 Therefore, I consider only the case 01:03:14.100 --> 01:03:18.060 which you have absolute value beta is equal to 1. 01:03:18.060 --> 01:03:21.390 And then we say, OK, it can be plus 1 and minus 1, 01:03:21.390 --> 01:03:23.580 but that's actually not the whole story, right? 01:03:23.580 --> 01:03:28.650 You can have, in general, beta equal to exponential i, 01:03:28.650 --> 01:03:29.400 some number. 01:03:29.400 --> 01:03:30.170 Right? 01:03:30.170 --> 01:03:31.640 Some real number. 01:03:31.640 --> 01:03:32.860 OK? 01:03:32.860 --> 01:03:37.860 And I write, here, a very fancy expression. 01:03:37.860 --> 01:03:41.232 Beta equal to exponential i k a. 01:03:41.232 --> 01:03:42.960 Why i k a? 01:03:42.960 --> 01:03:44.850 It's a very good question, right? 01:03:44.850 --> 01:03:46.320 What is a? 01:03:46.320 --> 01:03:49.020 I think most of you actually already forgot. 01:03:49.020 --> 01:03:50.400 What is a? 01:03:50.400 --> 01:03:55.620 a is actually the natural length of the spring. 01:03:55.620 --> 01:03:57.450 OK? 01:03:57.450 --> 01:04:00.840 So I was going too fast, because I would like to get to a break 01:04:00.840 --> 01:04:03.390 to hear your questions. 01:04:03.390 --> 01:04:04.440 So what is a? 01:04:04.440 --> 01:04:06.350 a is the natural length. 01:04:06.350 --> 01:04:07.410 OK? 01:04:07.410 --> 01:04:09.570 And the k-- what is k? 01:04:09.570 --> 01:04:11.490 Later, you will figure that out. 01:04:11.490 --> 01:04:16.720 You'll find that, actually, k is a wave number. 01:04:16.720 --> 01:04:17.380 OK? 01:04:17.380 --> 01:04:20.510 So that is actually much more of meaningful now, right? 01:04:20.510 --> 01:04:22.400 After the explanation. 01:04:22.400 --> 01:04:25.010 So you can see that beta is equal to exponential 01:04:25.010 --> 01:04:30.040 i, some number, and I call it k a, a fancy name of this number, 01:04:30.040 --> 01:04:32.410 and it has some physical meaning. 01:04:32.410 --> 01:04:33.370 OK? 01:04:33.370 --> 01:04:34.840 Another thing which is interesting 01:04:34.840 --> 01:04:42.920 is that if I plug in beta equal to a, or beta equal to 1 01:04:42.920 --> 01:04:47.540 over a, into the same expression-- 01:04:47.540 --> 01:04:52.370 if I plug in either beta a or beta equal to 1 01:04:52.370 --> 01:04:54.890 over a to this expression, I'm going 01:04:54.890 --> 01:04:58.700 to get exactly the same omega. 01:04:58.700 --> 01:05:03.160 So that means, OK, both of them will 01:05:03.160 --> 01:05:06.050 be-- both value will be oscillating 01:05:06.050 --> 01:05:08.620 at the same frequency. 01:05:08.620 --> 01:05:09.350 OK? 01:05:09.350 --> 01:05:13.940 So if you choose beta equal to a, choose beta equal to 1 01:05:13.940 --> 01:05:17.030 minus a, they are oscillating at the same frequency. 01:05:17.030 --> 01:05:18.280 What does that mean? 01:05:18.280 --> 01:05:24.290 That means linear combination of eigenvector coming from beta 01:05:24.290 --> 01:05:28.250 equal to a and eigenvector coming from beta equal to one 01:05:28.250 --> 01:05:31.775 over a, linear combination of those eigenvectors 01:05:31.775 --> 01:05:41.920 are also eigenvectors of the M minus 1 K matrix. 01:05:41.920 --> 01:05:42.940 OK? 01:05:42.940 --> 01:05:44.860 And that's actually where-- 01:05:44.860 --> 01:05:50.280 OK, those are different eigenvectors for S, 01:05:50.280 --> 01:05:56.612 but the linear combination of these vectors are all the-- 01:05:56.612 --> 01:05:59.790 eigenvector of M minus 1 K matrix and always 01:05:59.790 --> 01:06:03.180 the same eigenvalue omega square. 01:06:03.180 --> 01:06:03.680 OK? 01:06:03.680 --> 01:06:06.450 So that's another thing which is important. 01:06:06.450 --> 01:06:09.620 And finally, I said that there are 01:06:09.620 --> 01:06:12.830 infinite number of choice of k. 01:06:12.830 --> 01:06:13.910 That's valid, right? 01:06:13.910 --> 01:06:16.180 Because you can choose a little number, 01:06:16.180 --> 01:06:18.710 then you get a corresponding beta, 01:06:18.710 --> 01:06:20.900 then you get a corresponding omega. 01:06:20.900 --> 01:06:24.890 So you have infinite number of normal modes. 01:06:24.890 --> 01:06:29.540 Secondly, if I give you a k, OK-- if I give you a k, 01:06:29.540 --> 01:06:34.760 or I can give you another value which is minus k, 01:06:34.760 --> 01:06:40.130 then that means you will get beta and 1 over beta. 01:06:40.130 --> 01:06:41.480 Right? 01:06:41.480 --> 01:06:44.292 Minus k will give you 1 over beta. 01:06:44.292 --> 01:06:46.100 Right? 01:06:46.100 --> 01:06:49.960 And as I mentioned before, beta equal to a and beta 01:06:49.960 --> 01:06:53.540 equal to 1 over a will give you the same omega. 01:06:53.540 --> 01:06:58.010 Therefore, a linear combination of the vectors 01:06:58.010 --> 01:07:03.060 are also eigen of M minus 1 K matrix. 01:07:03.060 --> 01:07:05.780 Though, that's actually what I am doing here, right? 01:07:05.780 --> 01:07:09.920 So in order to show you a real amplitude, 01:07:09.920 --> 01:07:14.600 I'm doing a linear combination of exponential i j k a, 01:07:14.600 --> 01:07:17.450 and exponential minus i j k a. 01:07:17.450 --> 01:07:18.871 It's just a choice. 01:07:18.871 --> 01:07:19.370 OK? 01:07:19.370 --> 01:07:22.250 Of course, you can say, OK, I choose plus, 01:07:22.250 --> 01:07:24.300 and divide it by 2, then you get the cosine. 01:07:24.300 --> 01:07:25.100 Right? 01:07:25.100 --> 01:07:28.360 But if I choose this expression, then what I am going to get 01:07:28.360 --> 01:07:29.780 is that, huh-- 01:07:29.780 --> 01:07:33.230 since both of them are-- both vectors 01:07:33.230 --> 01:07:37.340 are corresponding to the same eigenvalue omega square, 01:07:37.340 --> 01:07:39.830 therefore, linear combination of them 01:07:39.830 --> 01:07:44.240 also oscillate at the angle of frequency omega. 01:07:44.240 --> 01:07:47.750 Therefore, if I calculate this and make it real, 01:07:47.750 --> 01:07:52.040 then I find that the amplitude is a function of j. 01:07:52.040 --> 01:07:56.550 Is actually a sinusoidal function, which is sine j k a. 01:07:56.550 --> 01:07:57.060 OK? 01:07:57.060 --> 01:07:59.920 So what does that mean? 01:07:59.920 --> 01:08:06.290 This means that if I plug the a-- 01:08:06.290 --> 01:08:13.740 if I plug A j as a function of j, 01:08:13.740 --> 01:08:17.160 this is actually what I'm going to get. 01:08:17.160 --> 01:08:19.229 It's a sinusoidal shape. 01:08:19.229 --> 01:08:20.660 OK? 01:08:20.660 --> 01:08:28.410 And we know that x j is actually equal to A j cosine omega 01:08:28.410 --> 01:08:30.649 t plus phi. 01:08:30.649 --> 01:08:32.260 Right? 01:08:32.260 --> 01:08:36.050 Omega, I can actually evaluate that, right? 01:08:36.050 --> 01:08:37.260 From here, right? 01:08:37.260 --> 01:08:39.100 Just a reminder. 01:08:39.100 --> 01:08:42.750 And what we are going to get is, when this system is 01:08:42.750 --> 01:08:45.805 thinking of normal mode, OK-- 01:08:45.805 --> 01:08:52.350 actually, this system is still a discrete system, so i-- 01:08:52.350 --> 01:08:57.573 actually, would like to point out that as a function of j, 01:08:57.573 --> 01:09:03.271 only discrete location have mass. 01:09:03.271 --> 01:09:03.770 Right? 01:09:03.770 --> 01:09:06.770 So you see that those are individual mass. 01:09:06.770 --> 01:09:10.420 They are oscillating up and down. 01:09:10.420 --> 01:09:11.479 OK? 01:09:11.479 --> 01:09:13.850 And you can see that, OK, since they 01:09:13.850 --> 01:09:18.450 are oscillating up and down, therefore, 01:09:18.450 --> 01:09:21.603 the oscillation, essentially, going up and down. 01:09:21.603 --> 01:09:27.130 Therefore, what is the actually the normal mode 01:09:27.130 --> 01:09:28.819 of this infinity long system? 01:09:28.819 --> 01:09:33.700 The normal mode are actually standing waves. 01:09:33.700 --> 01:09:39.370 But they actually only appear in the discrete value of j. 01:09:39.370 --> 01:09:42.200 And it has a functional form of something 01:09:42.200 --> 01:09:45.090 like a sinusoidal shape, or cosine. 01:09:45.090 --> 01:09:45.590 OK? 01:09:45.590 --> 01:09:48.300 So that's actually what we learn, and actually, you 01:09:48.300 --> 01:09:50.220 can see that from here. 01:09:50.220 --> 01:09:56.461 So if I oscillate this at some selected amplitude-- 01:09:59.407 --> 01:10:00.389 OK? 01:10:00.389 --> 01:10:04.317 Not quite get it. 01:10:04.317 --> 01:10:04.820 Yeah. 01:10:04.820 --> 01:10:08.700 So you see that, OK, it's roughly like a standing wave. 01:10:08.700 --> 01:10:11.420 It's a fixed frequency. 01:10:11.420 --> 01:10:12.820 OK? 01:10:12.820 --> 01:10:18.880 I would like to discuss with you a really interesting selection. 01:10:18.880 --> 01:10:22.690 So if I now take a look at-- 01:10:22.690 --> 01:10:24.730 so we have went through a lot of math, right? 01:10:24.730 --> 01:10:27.940 So now is the time to enjoy what we have learned, right? 01:10:27.940 --> 01:10:34.840 So if I now take a extreme value, cosine k a, OK, 01:10:34.840 --> 01:10:37.110 equal to minus 1. 01:10:37.110 --> 01:10:38.520 OK? 01:10:38.520 --> 01:10:44.550 Then I am reaching the maximal oscillation frequency. 01:10:44.550 --> 01:10:45.380 Right? 01:10:45.380 --> 01:10:53.790 So if I choose cosine k, small k, a equal to minus 1, 01:10:53.790 --> 01:10:57.150 OK-- what is going to happen is like this. 01:10:57.150 --> 01:11:00.530 It is as a function of j, by product 01:11:00.530 --> 01:11:04.180 A j is a function of j, what you are going to get 01:11:04.180 --> 01:11:05.180 is starting like this. 01:11:05.180 --> 01:11:10.550 Those are actually the amplitude of individual mass. 01:11:10.550 --> 01:11:16.400 So you can see that if cosine k a is equal to minus 1, 01:11:16.400 --> 01:11:20.240 omega square, based on that expression-- 01:11:20.240 --> 01:11:23.880 1 minus, minus 1, you get 2-- 01:11:23.880 --> 01:11:29.200 therefore, you get omega square equal to 4 omega 0 square. 01:11:29.200 --> 01:11:30.110 OK? 01:11:30.110 --> 01:11:34.500 And if you plug the A j as a function of j, 01:11:34.500 --> 01:11:37.130 that is actually what you are going to get. 01:11:37.130 --> 01:11:40.970 You actually have maximal stretch to the system. 01:11:40.970 --> 01:11:41.870 Right? 01:11:41.870 --> 01:11:45.530 You can see that it's actually positive, negative, positive, 01:11:45.530 --> 01:11:47.390 negative, positive, negative. 01:11:47.390 --> 01:11:51.950 That would reach the maximum speed of the oscillation. 01:11:51.950 --> 01:11:55.340 And of course, we cannot demo-- 01:11:55.340 --> 01:11:59.860 we cannot demo maximum, infinite number of oscillator, 01:11:59.860 --> 01:12:04.760 but of course, I can demo a system with 10 oscillators. 01:12:04.760 --> 01:12:09.100 So you can see that now, I maximize the amplitude 01:12:09.100 --> 01:12:13.550 of the highest frequency normal mode. 01:12:13.550 --> 01:12:17.610 And then I let go, and you see that this is actually 01:12:17.610 --> 01:12:21.460 exactly what is going to happen when I have cosine 01:12:21.460 --> 01:12:23.780 k a equal to minus 1. 01:12:23.780 --> 01:12:28.670 Then the wavelengths-- it's very small-- 01:12:28.670 --> 01:12:32.450 and you actually reach the maximum speed. 01:12:32.450 --> 01:12:34.370 Maximum speed is actually to become 01:12:34.370 --> 01:12:40.470 paired with, for example, lower frequency modes like this one. 01:12:40.470 --> 01:12:45.500 This is actually oscillating at a much lower frequency. 01:12:45.500 --> 01:12:48.530 And you can ask, OK, does that make sense? 01:12:48.530 --> 01:12:54.200 If I have this really, really zig-zag 01:12:54.200 --> 01:12:59.750 shape, why this system should be oscillating at the highest 01:12:59.750 --> 01:13:02.620 possible frequency? 01:13:02.620 --> 01:13:04.280 Why is that? 01:13:04.280 --> 01:13:05.870 It also makes sense, right? 01:13:05.870 --> 01:13:08.475 If you have that set up, then you 01:13:08.475 --> 01:13:14.480 are stretching this system to the maxima possible amount. 01:13:14.480 --> 01:13:15.280 Right? 01:13:15.280 --> 01:13:20.000 So, actually, now the springs looks like this. 01:13:20.000 --> 01:13:22.430 You are stretching this really hard, 01:13:22.430 --> 01:13:27.720 and therefore, the restorative force is going to be large. 01:13:27.720 --> 01:13:31.373 Therefore, you get high frequency. 01:13:31.373 --> 01:13:33.220 OK? 01:13:33.220 --> 01:13:38.650 OK, so I hope you actually enjoy the lecture today. 01:13:38.650 --> 01:13:41.600 It's a lot of mathematics, but what we have learned 01:13:41.600 --> 01:13:44.170 is really a lot. 01:13:44.170 --> 01:13:49.520 We learned how to actually describe system, 01:13:49.520 --> 01:13:54.675 how to actually solve a system without actually touching 01:13:54.675 --> 01:13:57.360 the M minus 1 K matrix; we can actually 01:13:57.360 --> 01:14:00.450 already get the eigenvectors. 01:14:00.450 --> 01:14:03.600 And using the M minus 1 K matrix, 01:14:03.600 --> 01:14:09.850 we can actually evaluate omega as a function of the input 01:14:09.850 --> 01:14:13.680 parameter from S eigenvalue. 01:14:13.680 --> 01:14:18.030 And the next lectures, we are going to discuss more examples, 01:14:18.030 --> 01:14:20.880 and make the whole system continuous. 01:14:20.880 --> 01:14:23.610 Thank you very much, and if you have any more questions, 01:14:23.610 --> 01:14:24.460 I will be here. 01:14:24.460 --> 01:14:27.850 I'm very happy to answer your questions.