WEBVTT 00:00:02.490 --> 00:00:04.830 The following content is provided under a Creative 00:00:04.830 --> 00:00:06.250 Commons license. 00:00:06.250 --> 00:00:08.460 Your support will help MIT OpenCourseWare 00:00:08.460 --> 00:00:12.550 continue to offer high-quality educational resources for free. 00:00:12.550 --> 00:00:15.090 To make a donation or to view additional materials 00:00:15.090 --> 00:00:18.090 from hundreds of MIT courses, visit MIT 00:00:18.090 --> 00:00:23.566 OpenCourseWare at osw.mit.edu, 00:00:23.566 --> 00:00:26.370 YEN-JIE LEE: So welcome, everybody. 00:00:26.370 --> 00:00:27.740 My name is Yen-Jie Lee. 00:00:27.740 --> 00:00:32.210 I am a assistant professor of physics in the physics 00:00:32.210 --> 00:00:36.500 department, and I will be your instructor 00:00:36.500 --> 00:00:39.860 of this semester on 8.03. 00:00:39.860 --> 00:00:44.930 So of course, one first question you have is, 00:00:44.930 --> 00:00:50.240 why do we want to learn about vibrations and waves? 00:00:50.240 --> 00:00:52.460 Why do we learn about this? 00:00:52.460 --> 00:00:55.550 Why do we even care? 00:00:55.550 --> 00:00:58.350 The answer is really, simple. 00:00:58.350 --> 00:01:00.920 If you look at this slide, you can 00:01:00.920 --> 00:01:05.720 see that the reason you can follow this class 00:01:05.720 --> 00:01:11.270 is because I'm producing sound wave by oscillating the air, 00:01:11.270 --> 00:01:14.180 and you can receive those sound waves. 00:01:14.180 --> 00:01:17.120 And you can see me-- 00:01:17.120 --> 00:01:19.790 that's really pretty amazing by itself-- 00:01:19.790 --> 00:01:22.880 because there are a lot of photons 00:01:22.880 --> 00:01:25.580 or electromagnetic waves. 00:01:25.580 --> 00:01:27.700 They are bouncing around in this room, 00:01:27.700 --> 00:01:34.790 and your eye actually receive those electromagnetic waves. 00:01:34.790 --> 00:01:37.430 And that translates into your brain waves. 00:01:37.430 --> 00:01:41.075 You obviously, start to think about what this instructor is 00:01:41.075 --> 00:01:43.400 trying to tell you. 00:01:43.400 --> 00:01:47.030 And of course, all those things we learned from 8.03 00:01:47.030 --> 00:01:50.870 is closely connected to probability density 00:01:50.870 --> 00:01:57.500 waves, which you will learn from 8.04, quantum physics. 00:01:57.500 --> 00:01:59.660 And finally, it's also, of course, 00:01:59.660 --> 00:02:05.030 related to a recent discovery of the gravitational waves. 00:02:05.030 --> 00:02:08.050 When we are sitting here, maybe there 00:02:08.050 --> 00:02:12.800 are already some space-time distortion already passing 00:02:12.800 --> 00:02:16.370 through our body and you don't feel it. 00:02:16.370 --> 00:02:18.550 When I'm moving around like this, 00:02:18.550 --> 00:02:22.370 I am creating also the gravitational waves, 00:02:22.370 --> 00:02:25.430 but it's so small to be detected. 00:02:25.430 --> 00:02:26.960 So that's actually really cool. 00:02:26.960 --> 00:02:32.570 So the take-home message is that we cannot even recognize 00:02:32.570 --> 00:02:37.310 the universe without using waves and the vibrations. 00:02:37.310 --> 00:02:40.810 So that's actually why we care about this subject. 00:02:40.810 --> 00:02:42.575 And the last is actually why this subject 00:02:42.575 --> 00:02:49.170 is so cool even without quantum, without any fancy names. 00:02:49.170 --> 00:02:55.420 So what is actually the relation of 8.03 to other class 00:02:55.420 --> 00:02:57.280 or other field of studies? 00:02:57.280 --> 00:03:00.410 It's closely related to classical mechanics, which 00:03:00.410 --> 00:03:03.050 I will use it immediately, and I hope 00:03:03.050 --> 00:03:05.060 you will still remember what you have 00:03:05.060 --> 00:03:07.480 learned from 8.01 and 8.02. 00:03:07.480 --> 00:03:13.310 Electromagnetic force is actually closely related also, 00:03:13.310 --> 00:03:16.640 and we are going to use a technique we learned 00:03:16.640 --> 00:03:21.270 from this class to understand optics, quantum mechanics, 00:03:21.270 --> 00:03:24.800 and also there are many practical applications, which 00:03:24.800 --> 00:03:29.200 you will learn from this class. 00:03:29.200 --> 00:03:31.390 This is the concrete goal. 00:03:31.390 --> 00:03:36.100 We care about the future of our space time. 00:03:36.100 --> 00:03:40.450 We would like to predict what is going to happen 00:03:40.450 --> 00:03:42.790 when we set up an experiment. 00:03:42.790 --> 00:03:47.290 We would like to design experiments which can improve 00:03:47.290 --> 00:03:50.110 our understanding of nature. 00:03:50.110 --> 00:03:53.850 But without using the most powerful tool 00:03:53.850 --> 00:03:58.030 is very, very difficult to make progress. 00:03:58.030 --> 00:04:03.900 So the most powerful tool we have is mathematics. 00:04:03.900 --> 00:04:07.840 You will see that it really works in this class. 00:04:07.840 --> 00:04:09.610 But the first thing we have to learn 00:04:09.610 --> 00:04:15.610 is how to translate physical situations into mathematics 00:04:15.610 --> 00:04:19.959 so that we can actually include this really wonderful tool 00:04:19.959 --> 00:04:22.960 to help us to solve problems. 00:04:22.960 --> 00:04:25.180 Once we have done that, we will start 00:04:25.180 --> 00:04:29.180 to look at single harmonic oscillator, 00:04:29.180 --> 00:04:32.590 then we try to couple all those oscillators together 00:04:32.590 --> 00:04:36.590 to see how they interact with each other. 00:04:36.590 --> 00:04:41.860 Finally, we go to an infinite number of oscillators. 00:04:41.860 --> 00:04:46.080 Sounds scary, but it's actually not scary after all. 00:04:46.080 --> 00:04:49.810 And we will see waves because waves are actually 00:04:49.810 --> 00:04:52.990 coming from an infinite number of oscillating particles, 00:04:52.990 --> 00:04:55.360 if you think about it. 00:04:55.360 --> 00:04:57.880 Then we would do Fourier decomposition of waves 00:04:57.880 --> 00:04:59.890 to see what we can learn about it. 00:04:59.890 --> 00:05:04.390 We learn how to put together physical systems. 00:05:04.390 --> 00:05:08.800 That brings us to the issue of boundary conditions, 00:05:08.800 --> 00:05:12.520 and we will also enjoy what we have learned 00:05:12.520 --> 00:05:14.620 by looking at the phenomenon related 00:05:14.620 --> 00:05:18.610 to electromagnetic waves and practical application 00:05:18.610 --> 00:05:21.530 and optics. 00:05:21.530 --> 00:05:22.240 Any questions? 00:05:22.240 --> 00:05:26.080 If you have any questions, please stop me any time. 00:05:26.080 --> 00:05:29.040 So if you don't stop me, I'm going to continue talking. 00:05:31.600 --> 00:05:33.950 So that gets started. 00:05:33.950 --> 00:05:37.090 So the first example, the concrete example 00:05:37.090 --> 00:05:43.570 I'm going to talk about is a spring block, a massive block 00:05:43.570 --> 00:05:45.520 system. 00:05:45.520 --> 00:05:50.150 So this is actually what I have on that table. 00:05:50.150 --> 00:05:54.740 So basically, I have a highly-idealized spring. 00:05:54.740 --> 00:05:58.480 This is ideal spring with spring constant, k, 00:05:58.480 --> 00:06:01.480 and the natural length L0. 00:06:01.480 --> 00:06:04.150 So that is actually what I have. 00:06:04.150 --> 00:06:13.830 And at t equal to 0, what I am going to do is I am going to-- 00:06:13.830 --> 00:06:16.480 I should remove this mass a little bit, 00:06:16.480 --> 00:06:22.900 and I hold this mass still and release that really carefully. 00:06:22.900 --> 00:06:24.910 So that is actually the experiment, 00:06:24.910 --> 00:06:27.070 which I am going to do. 00:06:27.070 --> 00:06:33.390 And we were wondering what is going to happen afterward. 00:06:33.390 --> 00:06:36.940 Well, the mass as you move, will it stay there 00:06:36.940 --> 00:06:40.790 or it just disappear, I don't know 00:06:40.790 --> 00:06:42.220 before I solved this question. 00:06:44.890 --> 00:06:49.690 Now I have put together a concrete question to you, 00:06:49.690 --> 00:06:52.900 but I don't know how to proceed because you 00:06:52.900 --> 00:06:54.040 say everything works. 00:06:54.040 --> 00:06:55.190 What I am going to do? 00:06:55.190 --> 00:06:58.540 I mean, I don't know. 00:06:58.540 --> 00:07:03.160 So as I mentioned before, there is a pretty powerful tool, 00:07:03.160 --> 00:07:04.780 mathematics. 00:07:04.780 --> 00:07:08.140 So I'm going to use that, even though I don't 00:07:08.140 --> 00:07:11.587 know why mathematics can work. 00:07:11.587 --> 00:07:12.670 Have you thought about it? 00:07:15.340 --> 00:07:19.180 So let's try it and see how we can make progress. 00:07:19.180 --> 00:07:25.540 So the first thing which you can do in order to make progress 00:07:25.540 --> 00:07:29.170 is to define a coordinate system. 00:07:29.170 --> 00:07:32.950 So here I define a coordinate system, which 00:07:32.950 --> 00:07:34.630 is in the horizontal direction. 00:07:34.630 --> 00:07:36.580 It's the x direction. 00:07:36.580 --> 00:07:39.640 And the x equal to 0, the origin, 00:07:39.640 --> 00:07:44.800 is the place which the spring is not stressed, 00:07:44.800 --> 00:07:47.330 is at its natural length. 00:07:47.330 --> 00:07:51.490 That is actually what I define as x equal to 0. 00:07:51.490 --> 00:07:56.020 And once I define this, I can now 00:07:56.020 --> 00:08:01.720 express what is actually the initial position of the mass 00:08:01.720 --> 00:08:03.990 by these coordinates is x0. 00:08:03.990 --> 00:08:08.020 It can be expressed as x initial. 00:08:08.020 --> 00:08:11.830 And also, initially, I said that this mass is not moving. 00:08:11.830 --> 00:08:16.810 Therefore, the velocity at 0 is 0. 00:08:16.810 --> 00:08:20.000 So now I can also formulate my question really concretely 00:08:20.000 --> 00:08:21.460 with some mathematics. 00:08:21.460 --> 00:08:25.330 Basically, you can see that at time equal to t, 00:08:25.330 --> 00:08:28.690 I was wondering where is this mass. 00:08:28.690 --> 00:08:31.060 So actually, the question is, what is actually 00:08:31.060 --> 00:08:34.210 x as a function of t? 00:08:34.210 --> 00:08:38.150 So you can see that once I have the mathematics to help me, 00:08:38.150 --> 00:08:41.490 everything becomes pretty simple. 00:08:41.490 --> 00:08:44.770 So once I have those defined, I would 00:08:44.770 --> 00:08:48.460 like to predict what is going to happen at time equal to t. 00:08:48.460 --> 00:08:54.370 Therefore, I would like to make use physical laws to actually 00:08:54.370 --> 00:08:56.540 help me to solve this problem. 00:08:56.540 --> 00:09:01.200 So apparently what we are going to use is Newton's law. 00:09:01.200 --> 00:09:05.700 And I am going to go through this example 00:09:05.700 --> 00:09:10.100 really slowly so that everybody is on the same page. 00:09:10.100 --> 00:09:12.300 So the first thing which I usually do 00:09:12.300 --> 00:09:18.610 is now I would like to do a force diagram analysis. 00:09:18.610 --> 00:09:19.930 So I have this mass. 00:09:19.930 --> 00:09:25.080 This setup is on Earth, and the question is, 00:09:25.080 --> 00:09:28.240 how many forces are acting on this mass? 00:09:28.240 --> 00:09:32.185 Can anybody answer my question. 00:09:32.185 --> 00:09:33.151 AUDIENCE: Two. 00:09:33.151 --> 00:09:35.566 We got the-- 00:09:35.566 --> 00:09:38.685 So acceleration and the spring force. 00:09:38.685 --> 00:09:41.150 YEN-JIE LEE: OK, so your answer is two. 00:09:41.150 --> 00:09:43.210 Any different? 00:09:43.210 --> 00:09:44.850 Three. 00:09:44.850 --> 00:09:45.450 Very good. 00:09:45.450 --> 00:09:48.120 So we have two and three. 00:09:48.120 --> 00:09:52.750 And the answer actually is three. 00:09:52.750 --> 00:09:54.260 So look at this scene. 00:09:54.260 --> 00:09:57.120 I am drawing in and I have product here. 00:09:57.120 --> 00:10:00.105 So this is actually the most difficult part of the question, 00:10:00.105 --> 00:10:01.220 actually. 00:10:01.220 --> 00:10:05.140 So once you pass this step, everything is straightforward. 00:10:05.140 --> 00:10:06.660 It's just mathematics. 00:10:06.660 --> 00:10:09.150 It's not my problem any more, but the math department, 00:10:09.150 --> 00:10:12.300 they have problem, OK? 00:10:12.300 --> 00:10:13.320 All right. 00:10:13.320 --> 00:10:16.952 So now let's look at this mass. 00:10:16.952 --> 00:10:17.910 There are three forces. 00:10:17.910 --> 00:10:21.915 The first one as you mentioned correctly is F spring. 00:10:21.915 --> 00:10:26.190 It's pulling the mass. 00:10:26.190 --> 00:10:31.020 And since we are working on Earth, 00:10:31.020 --> 00:10:33.240 we have not yet moved the whole class 00:10:33.240 --> 00:10:36.810 to the moon or somewhere else, but there 00:10:36.810 --> 00:10:39.945 would be gravitational force pointing downward. 00:10:42.720 --> 00:10:49.440 But this whole setup is on a table of friction, this table. 00:10:49.440 --> 00:10:51.555 Therefore, there will be no more force. 00:10:54.708 --> 00:10:56.369 So don't forget this one. 00:10:56.369 --> 00:10:57.535 There will be no more force. 00:11:00.080 --> 00:11:02.995 So the answer is that we have three forces. 00:11:05.740 --> 00:11:08.265 The normal force is, actually, a complicated subject, 00:11:08.265 --> 00:11:09.640 which you will need to understand 00:11:09.640 --> 00:11:11.440 that will quantum physics. 00:11:15.160 --> 00:11:18.790 So now I have three force, and now I can actually 00:11:18.790 --> 00:11:24.670 calculate the total force, the total force, 00:11:24.670 --> 00:11:35.510 F. F is equal to Fs plus Fn plus Fg. 00:11:38.830 --> 00:11:43.000 So since we know that the mass is 00:11:43.000 --> 00:11:45.310 moving in the horizontal direction, 00:11:45.310 --> 00:11:49.900 the mass didn't suddenly jump and disappear. 00:11:49.900 --> 00:11:51.760 So it is there. 00:11:51.760 --> 00:11:59.020 Therefore, we know that the normal force is actually 00:11:59.020 --> 00:12:07.690 equal to minus Fg, which is actually Ng in the y direction. 00:12:07.690 --> 00:12:13.480 And here I define y is actually pointing up, 00:12:13.480 --> 00:12:16.660 and the x is pointing to the right-hand side. 00:12:16.660 --> 00:12:18.760 Therefore, what is going to happen 00:12:18.760 --> 00:12:22.730 is that the total force is actually just Fs. 00:12:27.740 --> 00:12:31.660 And this is equal to minus k, which 00:12:31.660 --> 00:12:35.290 is the spring constant and x, which 00:12:35.290 --> 00:12:43.230 is the position of the little mass at time equal to t. 00:12:46.350 --> 00:12:55.670 So once we have those forces and the total force, 00:12:55.670 --> 00:12:57.770 actually, we can use Newton's law. 00:13:03.674 --> 00:13:09.240 So F is equal to m times a. 00:13:09.240 --> 00:13:16.760 And this is actually equal to m d squared xt dt squared 00:13:16.760 --> 00:13:20.060 in the x direction, and that is actually 00:13:20.060 --> 00:13:26.880 equal to mx double dot t x. 00:13:26.880 --> 00:13:28.190 So here is my notation. 00:13:28.190 --> 00:13:32.810 I'm going to use each of the dot is actually the differentiation 00:13:32.810 --> 00:13:34.630 with respect to t. 00:13:37.760 --> 00:13:45.600 So this is actually equal to minus kxt in the x direction. 00:13:45.600 --> 00:13:49.730 So you can see that here is actually what you already 00:13:49.730 --> 00:13:51.330 know about Newton's law. 00:13:51.330 --> 00:13:55.250 And that is actually coming from the force analysis. 00:13:55.250 --> 00:13:58.550 So in this example, it's simple enough such 00:13:58.550 --> 00:14:00.830 that you can write it down immediately, 00:14:00.830 --> 00:14:03.110 but in the later examples, things 00:14:03.110 --> 00:14:06.740 will become very complicated and things will be slightly more 00:14:06.740 --> 00:14:08.960 difficult. Therefore, you will really need help 00:14:08.960 --> 00:14:12.080 from the force diagram. 00:14:12.080 --> 00:14:15.320 So now we have everything in the x direction, 00:14:15.320 --> 00:14:18.140 therefore, I can drop the x hat. 00:14:18.140 --> 00:14:25.010 Therefore, finally, my equation of motion is x double dot t. 00:14:25.010 --> 00:14:32.720 And this is equal to minus k over n x of t. 00:14:32.720 --> 00:14:37.280 To make my life easier, I am going 00:14:37.280 --> 00:14:43.460 to define omega equal to square root of k over n. 00:14:43.460 --> 00:14:45.390 You will see why afterward. 00:14:45.390 --> 00:14:48.810 It looks really weird why professor Lee wants to do this, 00:14:48.810 --> 00:14:53.680 but afterward, you will see that omega really have a meaning, 00:14:53.680 --> 00:14:58.000 and that is equal to minus omega squared x. 00:15:00.710 --> 00:15:06.110 So we have solved this problem, actually, as a physicist. 00:15:06.110 --> 00:15:08.420 Now the problem is what is actually 00:15:08.420 --> 00:15:11.180 the solution to this differential 00:15:11.180 --> 00:15:13.440 second-order differential equation. 00:15:13.440 --> 00:15:15.570 And as I mentioned, this is actually 00:15:15.570 --> 00:15:17.530 not the content of 8.03, actually, 00:15:17.530 --> 00:15:20.930 it's a content of 18.03, maybe. 00:15:20.930 --> 00:15:24.650 How many of you actually have taken 18.03? 00:15:24.650 --> 00:15:27.380 Everybody knows the solution, so very good. 00:15:27.380 --> 00:15:30.475 I am safe. 00:15:30.475 --> 00:15:31.475 So what is the solution? 00:15:35.730 --> 00:15:46.940 The solution is x of t equal to a cosine of omega t plus b 00:15:46.940 --> 00:15:49.820 sine omega t. 00:15:49.820 --> 00:15:53.120 So my friends from the math department 00:15:53.120 --> 00:15:58.160 tell me secretly that this is actually the solution. 00:15:58.160 --> 00:16:00.470 And I trust him or her. 00:16:04.340 --> 00:16:07.130 So that's very nice. 00:16:07.130 --> 00:16:09.950 Now I have the solution, and how do I 00:16:09.950 --> 00:16:12.450 know this is the only solution? 00:16:12.450 --> 00:16:14.850 How do I know? 00:16:14.850 --> 00:16:17.330 Actually, there are two unknowns, 00:16:17.330 --> 00:16:20.964 just to remind you what you have learned. 00:16:20.964 --> 00:16:23.810 There are two unknowns. 00:16:23.810 --> 00:16:29.860 And if you plug this thing into this equation, 00:16:29.860 --> 00:16:32.000 you satisfy that equation. 00:16:32.000 --> 00:16:34.880 If you don't trust me, you can do it offline. 00:16:34.880 --> 00:16:37.610 It's always good to check to make 00:16:37.610 --> 00:16:39.290 sure I didn't make a mistake. 00:16:39.290 --> 00:16:40.980 But that's very good news. 00:16:40.980 --> 00:16:46.250 So that means we will have two unknowns, 00:16:46.250 --> 00:16:49.760 and those will satisfy the equation. 00:16:49.760 --> 00:16:54.900 So by uniqueness theorem, this is actually 00:16:54.900 --> 00:17:01.080 the one and the only one solution in my universe, 00:17:01.080 --> 00:17:07.800 also yours, which satisfy the equation because 00:17:07.800 --> 00:17:09.700 of the uniqueness theorem. 00:17:09.700 --> 00:17:12.690 So I hope I have convinced you that we 00:17:12.690 --> 00:17:16.810 have solved this equation. 00:17:16.810 --> 00:17:22.349 So now I take my physicist hat back and now 00:17:22.349 --> 00:17:24.400 it is actually my job again. 00:17:24.400 --> 00:17:27.030 So now we have the solution, and we 00:17:27.030 --> 00:17:29.290 need to determine what is actually 00:17:29.290 --> 00:17:32.440 these two unknown coefficients. 00:17:32.440 --> 00:17:37.770 So what I'm going to use is to use the two initial conditions. 00:17:44.840 --> 00:17:49.180 The first initial condition is x of 0 equal to x initial. 00:17:52.770 --> 00:17:57.540 The second one is that since I released this mass really 00:17:57.540 --> 00:18:03.180 carefully and the initial velocity is 0, therefore, 00:18:03.180 --> 00:18:09.160 I have x dot 0 equal to 0. 00:18:09.160 --> 00:18:11.610 From this, you can solve. 00:18:11.610 --> 00:18:16.000 Plug these two conditions into this equation. 00:18:16.000 --> 00:18:20.370 You can actually figure out that a is equal to x initial. 00:18:26.400 --> 00:18:29.690 And b is equal to 0. 00:18:33.000 --> 00:18:34.110 Any questions so far? 00:18:39.830 --> 00:18:41.460 Very good. 00:18:41.460 --> 00:18:43.530 So now we have the solution. 00:18:43.530 --> 00:18:46.640 So finally, what is actually the solution? 00:18:46.640 --> 00:19:03.660 The solution we get is x of t equal to x initial cosine omega 00:19:03.660 --> 00:19:04.160 t. 00:19:08.120 --> 00:19:12.620 So this is actually the amplitude of the oscillation, 00:19:12.620 --> 00:19:17.240 and this is actually the angular velocity. 00:19:17.240 --> 00:19:20.420 So you may be asking why angular? 00:19:20.420 --> 00:19:22.970 Where is the angular coming from? 00:19:22.970 --> 00:19:25.910 Because this is actually a one-dimensional motion. 00:19:25.910 --> 00:19:28.490 Where is the angular velocity coming from? 00:19:28.490 --> 00:19:34.250 And I will explain that in the later lecture. 00:19:34.250 --> 00:19:38.220 And also this is actually a harmonic oscillation. 00:19:38.220 --> 00:19:40.700 So what we are actually predicting 00:19:40.700 --> 00:19:46.490 is that this mass is going to do this, have a fixed 00:19:46.490 --> 00:19:48.830 amplitude and it's actually going 00:19:48.830 --> 00:19:54.920 to go back and forth with the angular frequency of omega. 00:19:54.920 --> 00:20:00.230 So we can now do an experiment to verify if this is actually 00:20:00.230 --> 00:20:01.940 really the case. 00:20:01.940 --> 00:20:04.400 So there's a small difference. 00:20:04.400 --> 00:20:08.870 There's another spring here, but essentially, the solution 00:20:08.870 --> 00:20:10.290 will be very similar. 00:20:10.290 --> 00:20:15.860 You may get this in a p-set or exam. 00:20:15.860 --> 00:20:20.670 So now I can turn on the air so that I make this surface 00:20:20.670 --> 00:20:23.880 frictionless. 00:20:23.880 --> 00:20:29.120 And you can see that now I actually move this thing 00:20:29.120 --> 00:20:33.390 slightly away from the equilibrium position, 00:20:33.390 --> 00:20:36.410 and I release that carefully. 00:20:36.410 --> 00:20:42.030 So you can see that really it's actually going back and forth 00:20:42.030 --> 00:20:43.080 harmonically. 00:20:46.290 --> 00:20:50.880 I can change the amplitude and see what will happen. 00:20:50.880 --> 00:20:53.540 The amplitude is becoming bigger, 00:20:53.540 --> 00:20:59.120 and you can see that the oscillation amplitude really 00:20:59.120 --> 00:21:03.365 depends on where you put that initially with respect 00:21:03.365 --> 00:21:04.750 to the equilibrium position. 00:21:04.750 --> 00:21:08.330 I can actually make a small amplitude oscillation also. 00:21:08.330 --> 00:21:11.570 Now you can see that now the amplitude is small but still 00:21:11.570 --> 00:21:14.660 oscillating back and forth. 00:21:14.660 --> 00:21:16.401 So that's very encouraging. 00:21:20.730 --> 00:21:25.340 Let's take another example, which I actually rotate 00:21:25.340 --> 00:21:28.520 the whole thing by 90 degrees. 00:21:28.520 --> 00:21:32.150 You are going to get a question about this system 00:21:32.150 --> 00:21:33.510 in your p-set. 00:21:33.510 --> 00:21:38.500 The amazing thing is that the solution is the same. 00:21:38.500 --> 00:21:41.330 What is that? 00:21:41.330 --> 00:21:45.470 And you don't believe me, let me do the experiment. 00:21:45.470 --> 00:21:48.680 I actually shifted the position. 00:21:48.680 --> 00:21:53.220 I changed the position, and I release that really carefully. 00:21:53.220 --> 00:21:56.460 You see that this mass is oscillating up and down. 00:21:56.460 --> 00:22:00.410 The amplitude did not change. 00:22:00.410 --> 00:22:03.890 The frequency did not change as a function of time. 00:22:03.890 --> 00:22:08.960 It really matched with the solution we found here. 00:22:11.690 --> 00:22:14.060 It's truly amazing. 00:22:14.060 --> 00:22:16.040 No? 00:22:16.040 --> 00:22:19.940 The problem is that we are so used to this already. 00:22:19.940 --> 00:22:25.250 You have seen this maybe 100 times before my lecture, 00:22:25.250 --> 00:22:28.040 so therefore, you got so used to this. 00:22:28.040 --> 00:22:30.680 Therefore, when I say, OK, I make a prediction. 00:22:30.680 --> 00:22:34.120 This is what happened, you are just so used to this 00:22:34.120 --> 00:22:36.740 or you don't feel the excitement. 00:22:36.740 --> 00:22:41.510 But for me, after I teach this class so many times, 00:22:41.510 --> 00:22:46.030 I still find this thing really amazing. 00:22:46.030 --> 00:22:48.110 Why is that? 00:22:48.110 --> 00:22:53.120 This means that actually, mathematics 00:22:53.120 --> 00:22:56.520 really works, first of all. 00:22:56.520 --> 00:23:03.560 That means we can use the same tool for the understanding 00:23:03.560 --> 00:23:11.300 of gravitational waves, for the prediction of the Higgs boson, 00:23:11.300 --> 00:23:13.920 for the calculation of the property 00:23:13.920 --> 00:23:17.810 of the quark-gluon plasma in the early universe, 00:23:17.810 --> 00:23:22.280 and also at the same time the motion 00:23:22.280 --> 00:23:25.400 of this spring-mass system. 00:23:25.400 --> 00:23:30.350 We actually use always the same tool, the mathematics, 00:23:30.350 --> 00:23:32.840 to understand this system. 00:23:32.840 --> 00:23:35.360 And nobody will understands why. 00:23:35.360 --> 00:23:37.845 If you understand why, please tell me. 00:23:37.845 --> 00:23:38.720 I would like to know. 00:23:38.720 --> 00:23:39.886 I will be very proud of you. 00:23:43.780 --> 00:23:49.400 Rene Descartes said once, "But in my opinion, 00:23:49.400 --> 00:23:54.190 all things in nature occur mathematically." 00:23:54.190 --> 00:23:55.340 Apparently, he's right. 00:23:58.570 --> 00:24:03.670 Albert Einstein also once said, "The most incomprehensible 00:24:03.670 --> 00:24:08.810 thing about the universe is that it is comprehensible." 00:24:08.810 --> 00:24:13.870 So I would say this is really something 00:24:13.870 --> 00:24:17.470 we need to appreciate the need to think 00:24:17.470 --> 00:24:21.250 about why this is the case. 00:24:21.250 --> 00:24:22.325 Any questions? 00:24:26.790 --> 00:24:29.990 So you may say, oh, come on. 00:24:29.990 --> 00:24:33.990 We just solved the problem of an ideal spring. 00:24:33.990 --> 00:24:35.380 Who cares? 00:24:35.380 --> 00:24:39.120 It's so simple, so easy, and you are making really 00:24:39.120 --> 00:24:41.760 a big thing out of this. 00:24:41.760 --> 00:24:44.490 But actually, what we have been solving 00:24:44.490 --> 00:24:47.110 is really much more than that. 00:24:47.110 --> 00:24:53.850 This equation is much more than just a spring-mass system. 00:24:53.850 --> 00:24:58.800 Actually, if you think about this question carefully, 00:24:58.800 --> 00:25:02.130 there's really no Hooke's law forever. 00:25:02.130 --> 00:25:05.820 Hooke's law will give you a potential proportional 00:25:05.820 --> 00:25:07.800 to x squared. 00:25:07.800 --> 00:25:15.970 And if you are so far away, you pull the spring so really hard, 00:25:15.970 --> 00:25:19.080 you can store the energy of the whole universe. 00:25:19.080 --> 00:25:21.730 Does that make sense? 00:25:21.730 --> 00:25:22.980 No. 00:25:22.980 --> 00:25:25.620 At some point, it should break down. 00:25:25.620 --> 00:25:27.990 So there's really no Hook's law. 00:25:27.990 --> 00:25:31.860 But there's also Hook's law everywhere. 00:25:31.860 --> 00:25:37.230 If you look at this system, it follows the harmonic 00:25:37.230 --> 00:25:38.280 oscillation. 00:25:38.280 --> 00:25:40.950 If you look at this system I perturb this, 00:25:40.950 --> 00:25:42.150 it goes back and forth. 00:25:44.940 --> 00:25:46.560 It's almost like everywhere. 00:25:46.560 --> 00:25:48.540 Why is this the case? 00:25:48.540 --> 00:25:51.580 I'm going to answer this question immediately. 00:25:54.280 --> 00:25:58.860 So let's take a look at an example. 00:25:58.860 --> 00:26:01.060 So if I consider a potential, this 00:26:01.060 --> 00:26:03.870 is an artificial potential, which 00:26:03.870 --> 00:26:07.110 you can find in Georgi's book, so v 00:26:07.110 --> 00:26:11.780 is equal to E times L over x plus x over L. 00:26:11.780 --> 00:26:14.640 And if you practice as a function of x, then 00:26:14.640 --> 00:26:16.800 basically you get this funny shape. 00:26:16.800 --> 00:26:20.300 It's not proportional to x squared. 00:26:20.300 --> 00:26:23.130 Therefore, you will see that, OK, 00:26:23.130 --> 00:26:28.710 the resulting motion for the particle in this potential, 00:26:28.710 --> 00:26:31.950 it's not going to be harmonic motion. 00:26:31.950 --> 00:26:35.370 But if I zoom in, zoom in, and zoom in 00:26:35.370 --> 00:26:38.550 and basically, you will see that if I am patient enough, 00:26:38.550 --> 00:26:44.440 I zoom in enough, you'll see that this is a parabola. 00:26:44.440 --> 00:26:50.100 Again, you follow Hooke's law. 00:26:50.100 --> 00:26:52.590 So that is actually really cool. 00:26:52.590 --> 00:26:59.100 So if I consider an arbitrary v of x, 00:26:59.100 --> 00:27:05.740 we can do a Taylor expansion to this potential. 00:27:05.740 --> 00:27:10.920 So basically v of x will be equal to v of 0 00:27:10.920 --> 00:27:16.890 plus v prime 0 divided by 1 factorial times 00:27:16.890 --> 00:27:24.900 x plus v double prime 0 over 2 factorial 00:27:24.900 --> 00:27:30.670 x squared plus v triple prime 0 divided 00:27:30.670 --> 00:27:37.110 by 3 factorial x to the third plus infinite number of terms. 00:27:37.110 --> 00:27:47.010 v 0 is the position of where you have minimum potential. 00:27:47.010 --> 00:27:50.550 So that's actually where the equilibrium position 00:27:50.550 --> 00:27:54.020 is in my coordinate system. 00:27:54.020 --> 00:27:55.770 It's the standard, the coordinate system 00:27:55.770 --> 00:28:02.070 I used for the solving the spring-mass question. 00:28:02.070 --> 00:28:06.900 So if I calculate the force, the force, f of x, 00:28:06.900 --> 00:28:12.690 will be equal to minus d dx v of x. 00:28:12.690 --> 00:28:15.150 And that will be equal to minus v 00:28:15.150 --> 00:28:24.000 prime 0 minus v double prime 0 x minus 1 00:28:24.000 --> 00:28:34.320 over 2 v triple prime 0 x squared plus many other terms. 00:28:34.320 --> 00:28:39.980 Since I have mentioned that v of 0-- 00:28:46.108 --> 00:28:50.160 this will be x. 00:28:50.160 --> 00:28:55.260 v of 0 is actually the position of the minima. 00:28:55.260 --> 00:29:01.002 Therefore, v prime of 0 will be equal to 0. 00:29:03.540 --> 00:29:05.100 Therefore. 00:29:05.100 --> 00:29:07.610 This term is gone. 00:29:11.660 --> 00:29:16.490 So what essentially is left over is the remaining terms here. 00:29:19.910 --> 00:29:26.450 Now, if I assume that x is very small, what is going to happen? 00:29:26.450 --> 00:29:31.090 Anybody know when x is very small, what is going to happen? 00:29:31.090 --> 00:29:32.140 Anybody have the answer? 00:29:32.140 --> 00:29:34.720 AUDIENCE: [INAUDIBLE]. 00:29:34.720 --> 00:29:36.110 YEN-JIE LEE: Exactly. 00:29:36.110 --> 00:29:41.630 So when x is very small, he said that the higher order terms all 00:29:41.630 --> 00:29:43.095 become negligible. 00:29:43.095 --> 00:29:43.910 OK? 00:29:43.910 --> 00:29:45.290 So that is essentially correct. 00:29:45.290 --> 00:29:50.060 So when x is very small, then I only 00:29:50.060 --> 00:29:55.130 need to consider the leading order term. 00:29:55.130 --> 00:29:59.030 But how small is the question. 00:29:59.030 --> 00:30:00.900 How small is small? 00:30:03.760 --> 00:30:07.630 Actually, what you can do is to take the ratio 00:30:07.630 --> 00:30:12.517 between these two terms. 00:30:12.517 --> 00:30:14.350 So if you take the ratio, then basically you 00:30:14.350 --> 00:30:19.300 would get a condition xv triple dot 00:30:19.300 --> 00:30:29.940 0, which will be much smaller than v double prime 0. 00:30:29.940 --> 00:30:32.050 So that is essentially the condition 00:30:32.050 --> 00:30:37.630 which is required to satisfy it so that we actually can ignore 00:30:37.630 --> 00:30:39.340 all the higher-order terms. 00:30:39.340 --> 00:30:42.160 Then the whole question becomes f 00:30:42.160 --> 00:30:48.910 of x equal to minus v double prime 0 x. 00:30:48.910 --> 00:30:52.100 And that essentially, Hooke's law. 00:30:52.100 --> 00:30:54.760 So you can see that first of all, there's 00:30:54.760 --> 00:30:57.190 no Hooke's law in general. 00:30:57.190 --> 00:31:01.240 Secondly, Hook's law essentially applicable 00:31:01.240 --> 00:31:06.490 almost everywhere when you have a well-behaved potential 00:31:06.490 --> 00:31:10.860 and if you only perturb the system really slightly 00:31:10.860 --> 00:31:14.960 with very small amplitude, then it always works. 00:31:14.960 --> 00:31:19.120 So what I would like to say is that after we 00:31:19.120 --> 00:31:23.620 have done this exercise, you will see that, actually, we 00:31:23.620 --> 00:31:28.960 have solved all the possible systems, which 00:31:28.960 --> 00:31:33.630 have a well-behaved potential. 00:31:33.630 --> 00:31:39.830 It has a minima, and if I have the amplitude small enough, 00:31:39.830 --> 00:31:46.400 then the system is going to do simple harmonic oscillation. 00:31:46.400 --> 00:31:47.860 Any questions? 00:31:54.540 --> 00:31:56.065 No question, then we'll continue. 00:32:02.060 --> 00:32:07.040 So let's come back to this equation of motion. 00:32:07.040 --> 00:32:13.360 x double dot plus omega squared x, this is equal to 0. 00:32:16.380 --> 00:32:19.600 There are two important properties 00:32:19.600 --> 00:32:22.930 of this linear equation of motion. 00:32:26.950 --> 00:32:48.130 The first one is that if x1 of t and x2 of t are both solutions, 00:32:48.130 --> 00:32:54.280 then x12, which is the superposition 00:32:54.280 --> 00:33:01.340 of the first and second solution, is also a solution. 00:33:13.930 --> 00:33:17.690 The second thing, which is very interesting about this equation 00:33:17.690 --> 00:33:27.235 of motion, is that there's a time translation invariance. 00:33:31.990 --> 00:33:38.830 So this means that if x of t is a solution, 00:33:38.830 --> 00:33:49.285 then xt prime equal to xt plus a is also a solution. 00:33:52.190 --> 00:33:55.520 So that is really cool, because that 00:33:55.520 --> 00:34:04.370 means if I change t equal to 0, so I shift the 0-th time, 00:34:04.370 --> 00:34:08.090 the whole physics did not change. 00:34:10.989 --> 00:34:13.650 So this is actually because of the chain law. 00:34:13.650 --> 00:34:19.929 So if you have chain law dx t plus a dt, that 00:34:19.929 --> 00:34:35.690 is equal to d t plus a dt, dx t prime dt prime 00:34:35.690 --> 00:34:39.949 evaluated at t prime equal to t plus a. 00:34:39.949 --> 00:34:52.969 And that is equal to dx t prime dt, t prime equal to t plus a. 00:34:52.969 --> 00:35:00.020 So that means if I have changed the t equal to 0 00:35:00.020 --> 00:35:04.670 to other place, the whole equation of motion 00:35:04.670 --> 00:35:06.860 is still the same. 00:35:06.860 --> 00:35:10.880 On the other hand, if the k, or say the potential, 00:35:10.880 --> 00:35:14.640 is time dependent, then that may break this symmetry. 00:35:17.970 --> 00:35:18.932 Any questions? 00:35:22.150 --> 00:35:27.670 So before we take a five minute break, 00:35:27.670 --> 00:35:33.640 I would like to discuss further about this point, this linear 00:35:33.640 --> 00:35:34.990 and nonlinear event. 00:35:34.990 --> 00:35:38.230 So you can see that the force is actually 00:35:38.230 --> 00:35:42.760 linearly dependent on x. 00:35:42.760 --> 00:35:49.420 But what will happen if I increase x more? 00:35:49.420 --> 00:35:50.350 Something will happen. 00:35:50.350 --> 00:35:54.850 That means the higher-ordered term should also be 00:35:54.850 --> 00:35:57.370 taken into account carefully. 00:35:57.370 --> 00:36:01.210 So that means the solution of this kind, x 00:36:01.210 --> 00:36:07.070 initial cosine omega t, will not work perfectly. 00:36:07.070 --> 00:36:12.690 In 8.03, we only consider the linear term most of the time. 00:36:12.690 --> 00:36:15.700 But actually, I would like to make sure 00:36:15.700 --> 00:36:17.450 that everybody can at this point, 00:36:17.450 --> 00:36:21.220 the higher-order contribution is actually 00:36:21.220 --> 00:36:23.290 visible in our daily life. 00:36:23.290 --> 00:36:28.710 So let me actually give you a concrete example. 00:36:28.710 --> 00:36:33.520 So here I have two pendulums. 00:36:33.520 --> 00:36:36.550 So I can now perturb this pendulum slightly. 00:36:36.550 --> 00:36:42.970 And you you'll see that it goes back and forth and following 00:36:42.970 --> 00:36:45.610 simple harmonic emotion. 00:36:45.610 --> 00:36:48.670 So if I have both things slightly 00:36:48.670 --> 00:36:52.810 oscillating with small amplitude, what 00:36:52.810 --> 00:36:58.140 is going to happen is that both pendulums reach maxima 00:36:58.140 --> 00:37:00.800 amplitude at the same time. 00:37:00.800 --> 00:37:05.360 You can see that very clearly. 00:37:05.360 --> 00:37:08.916 I don't need to do this carefully. 00:37:08.916 --> 00:37:14.150 You see that they always reach maxima at the same time 00:37:14.150 --> 00:37:16.040 when the amplitude is small. 00:37:16.040 --> 00:37:17.250 Why? 00:37:17.250 --> 00:37:24.000 That is because the higher-order terms are not important. 00:37:24.000 --> 00:37:26.350 So now let's do a experiment. 00:37:26.350 --> 00:37:29.010 And now I go crazy. 00:37:29.010 --> 00:37:33.840 I make the amplitude very large so that I 00:37:33.840 --> 00:37:37.800 break that approximation. 00:37:37.800 --> 00:37:39.300 So let's see what will happen. 00:37:39.300 --> 00:37:41.805 So now I do this then. 00:37:41.805 --> 00:37:47.170 I release at the same time and see what will happen. 00:37:47.170 --> 00:37:49.410 You see that originally they are in phase. 00:37:49.410 --> 00:37:53.830 They are reaching maxima at the same time. 00:37:53.830 --> 00:37:57.110 But if we are patient enough, you see that now? 00:37:57.110 --> 00:38:01.020 They are is oscillating, actually, 00:38:01.020 --> 00:38:02.920 at different frequencies. 00:38:02.920 --> 00:38:06.150 Originally, the solution, the omega, 00:38:06.150 --> 00:38:09.810 is really independent of the amplitude. 00:38:09.810 --> 00:38:11.970 So they should, actually, be isolating 00:38:11.970 --> 00:38:13.810 at the same frequency. 00:38:13.810 --> 00:38:16.260 But clearly you can see here, when 00:38:16.260 --> 00:38:19.380 you increase the amplitude, then you 00:38:19.380 --> 00:38:24.600 need to consider also the nonlinear effects. 00:38:24.600 --> 00:38:30.067 So any questions before we take a five-minute break. 00:38:30.067 --> 00:38:32.150 So if not, then we would take a five-minute break, 00:38:32.150 --> 00:38:34.850 and we come back at 25. 00:38:40.590 --> 00:38:43.290 So welcome back, everybody. 00:38:43.290 --> 00:38:45.630 So we will continue the discussion 00:38:45.630 --> 00:38:51.170 of this equation of motion, x double dot plus omega square x 00:38:51.170 --> 00:38:54.120 equal to 0. 00:38:54.120 --> 00:38:58.750 So there are three possible way to like 00:38:58.750 --> 00:39:01.935 the solution to this equation. 00:39:01.935 --> 00:39:04.860 So the first one as I mentioned before, 00:39:04.860 --> 00:39:15.960 x of t equal to a cosine omega t plus b sine omega t. 00:39:15.960 --> 00:39:17.800 So this is actually the functional form 00:39:17.800 --> 00:39:21.010 we have been using before. 00:39:21.010 --> 00:39:26.520 And we can actually also rewrite it in a different way. 00:39:26.520 --> 00:39:39.600 So x or t equal to capital A cosine omega t plus phi. 00:39:39.600 --> 00:39:41.640 You may say, wait a second. 00:39:41.640 --> 00:39:44.190 You just promised me that this is the first one, 00:39:44.190 --> 00:39:47.190 the one is the one and only one solution 00:39:47.190 --> 00:39:51.750 in the universe, which actually satisfy the equation of motion. 00:39:51.750 --> 00:39:53.960 Now you write another one. 00:39:53.960 --> 00:39:56.210 What is going on? 00:39:56.210 --> 00:39:57.680 Why? 00:39:57.680 --> 00:39:59.060 But actually, they are the same. 00:40:01.770 --> 00:40:09.740 This is actually A cosine phi cosine omega t 00:40:09.740 --> 00:40:16.250 minus A sine phi sine omega t. 00:40:19.250 --> 00:40:23.390 So the good thing is that A and phi 00:40:23.390 --> 00:40:27.770 are arbitrary constant so that it should be you 00:40:27.770 --> 00:40:31.040 can use two initial conditions to determine 00:40:31.040 --> 00:40:32.570 the arbitrary constant. 00:40:32.570 --> 00:40:39.050 So you can see that one and two are completely equivalent. 00:40:39.050 --> 00:40:44.390 So I hope that solves some of the questions 00:40:44.390 --> 00:40:48.320 because you really find it confusing 00:40:48.320 --> 00:40:53.960 why we have different presentations of the solution. 00:40:53.960 --> 00:40:59.600 So there's a third one, which is actually much more fancier. 00:40:59.600 --> 00:41:04.270 The third one is that I have x of t. 00:41:04.270 --> 00:41:10.575 This is actually a real part of A-- 00:41:10.575 --> 00:41:16.650 again, the amplitude-- exponential i omega t 00:41:16.650 --> 00:41:24.030 plus phi, where i is equal to the square root of minus 1. 00:41:27.800 --> 00:41:28.460 Wait a second. 00:41:28.460 --> 00:41:32.530 We will say, well, professor, why are you writing 00:41:32.530 --> 00:41:34.975 such a horrible solution? 00:41:38.820 --> 00:41:40.150 Right? 00:41:40.150 --> 00:41:41.150 Really strange. 00:41:41.150 --> 00:41:42.400 But that will explain you why. 00:41:45.030 --> 00:41:48.120 So three is actually a mathematical trick. 00:41:48.120 --> 00:41:53.220 I'm not going to prove anything here because I'm a physicist, 00:41:53.220 --> 00:41:58.600 but I would like to share with you what I think is going on. 00:41:58.600 --> 00:42:01.410 I think three is really just a mathematical trick 00:42:01.410 --> 00:42:05.310 from the math department. 00:42:05.310 --> 00:42:10.780 In principle, I can drive it an even more horrible way. 00:42:10.780 --> 00:42:24.486 x of t equal to a real part of A cosine omega t plus phi plus i 00:42:24.486 --> 00:42:29.501 f of t. 00:42:29.501 --> 00:42:36.910 And f of t is a real function. 00:42:36.910 --> 00:42:38.730 In principle, I can do that. 00:42:38.730 --> 00:42:41.870 It's even more horrible. 00:42:41.870 --> 00:42:43.500 Why is that? 00:42:43.500 --> 00:42:46.470 Because I now have this function. 00:42:46.470 --> 00:42:48.870 I take the real part, and I actually 00:42:48.870 --> 00:42:56.200 take the two out of this operation. 00:42:56.200 --> 00:42:59.900 So f of t is actually the real function. 00:42:59.900 --> 00:43:02.170 It can be something arbitrary. 00:43:02.170 --> 00:43:11.690 And i can now plot the locus of this function, the solution 00:43:11.690 --> 00:43:15.280 on the complex print. 00:43:15.280 --> 00:43:19.800 Now I'm plotting this solution on this complex print. 00:43:19.800 --> 00:43:23.780 What is going to happen is that you're going to have-- 00:43:31.250 --> 00:43:33.310 That is what you are going to get. 00:43:33.310 --> 00:43:36.520 If I am lucky, if this f of t is confined 00:43:36.520 --> 00:43:39.250 in some specific region, if I not lucky, 00:43:39.250 --> 00:43:41.160 then it goes out of the print there. 00:43:41.160 --> 00:43:42.490 I couldn't see it. 00:43:42.490 --> 00:43:44.490 Maybe it go to the moon or something. 00:43:47.150 --> 00:43:50.090 But if you are smart enough, and I'm 00:43:50.090 --> 00:44:00.080 sure you are, if I choose f of t equal to A sine omega 00:44:00.080 --> 00:44:06.410 t plus phi, can anybody tell me what is going to happen? 00:44:09.314 --> 00:44:10.766 AUDIENCE: [INAUDIBLE]. 00:44:15.122 --> 00:44:16.950 YEN-JIE LEE: Would you count a circle? 00:44:19.920 --> 00:44:22.770 Very good. 00:44:22.770 --> 00:44:27.700 If I plot the locus again of this function, 00:44:27.700 --> 00:44:33.970 the real axis, imaginary axis, then you should get a circle. 00:44:33.970 --> 00:44:37.030 Some miracle happened. 00:44:37.030 --> 00:44:41.530 If you choose the f of t correctly, 00:44:41.530 --> 00:44:50.400 wisely, then you can actually turn all this mess into order. 00:44:50.400 --> 00:44:51.649 Any questions? 00:44:56.440 --> 00:44:59.590 So I can now follow up about this. 00:45:09.380 --> 00:45:20.110 So now I have x of t is equal to the real part of A cosine omega 00:45:20.110 --> 00:45:28.260 t plus phi plus iA sine omega t plus phi. 00:45:31.390 --> 00:45:36.120 And just a reminder, exponential i theta 00:45:36.120 --> 00:45:43.830 is equal to cosine theta plus i sine theta. 00:45:43.830 --> 00:45:45.920 Therefore, I arrive this. 00:45:45.920 --> 00:45:54.020 This is a real part of A exponential i omega t plus phi. 00:45:58.840 --> 00:46:01.090 So if I do this really carefully, 00:46:01.090 --> 00:46:08.680 I look at this the position of the point at a specific time. 00:46:08.680 --> 00:46:13.500 So now time is equal to t. 00:46:13.500 --> 00:46:16.915 And this is the real axis, and this is the imaginary axis. 00:46:16.915 --> 00:46:21.100 So I have this circle here. 00:46:21.100 --> 00:46:24.520 So at time equal to t, what you are getting 00:46:24.520 --> 00:46:27.400 is that x is actually-- 00:46:27.400 --> 00:46:31.430 before taking the real part, A, exponential i omega t plus phi, 00:46:31.430 --> 00:46:33.730 it's actually here. 00:46:33.730 --> 00:46:39.380 And this vector actually shows the amplitude. 00:46:39.380 --> 00:46:46.540 Amplitude is A. And the angle between this vector pointing 00:46:46.540 --> 00:46:54.460 to the position of this function is omega t plus phi. 00:46:54.460 --> 00:47:00.100 So this is actually the angle between this vector 00:47:00.100 --> 00:47:03.500 and the real axis. 00:47:03.500 --> 00:47:06.060 So that's pretty cool. 00:47:06.060 --> 00:47:06.770 Why? 00:47:06.770 --> 00:47:12.410 Because now I understand why I call this omega angular 00:47:12.410 --> 00:47:15.830 velocity or angular frequency. 00:47:15.830 --> 00:47:22.140 Because the solution to the equation of motion, 00:47:22.140 --> 00:47:26.325 which we have actually derived before, 00:47:26.325 --> 00:47:35.610 is actually the real part of rotation in a complex print. 00:47:35.610 --> 00:47:39.540 If you think about it, that means now 00:47:39.540 --> 00:47:45.910 I see this particle going up and down. 00:47:45.910 --> 00:47:49.160 I see this particle going up and down. 00:47:49.160 --> 00:47:51.950 You can think about that, this is Earth. 00:47:51.950 --> 00:47:55.370 If there is an extra dimension mention, 00:47:55.370 --> 00:47:58.620 which you couldn't see. 00:47:58.620 --> 00:48:02.620 Actually, this particle in the dimension 00:48:02.620 --> 00:48:06.200 where we can see into the extra dimension, which is hidden 00:48:06.200 --> 00:48:09.180 is actually rotating. 00:48:09.180 --> 00:48:12.470 And while we see that reality, it's 00:48:12.470 --> 00:48:16.040 a projection to the real axis. 00:48:16.040 --> 00:48:16.650 You see? 00:48:16.650 --> 00:48:25.210 So in reality, this particle is actually rotating, 00:48:25.210 --> 00:48:29.450 if you add the image and the extra dimension. 00:48:29.450 --> 00:48:33.620 So that is actually pretty cool, but the purity artificial. 00:48:33.620 --> 00:48:37.580 So you can see that I can choose f of t 00:48:37.580 --> 00:48:42.530 to be a different function, and then this whole picture 00:48:42.530 --> 00:48:43.970 is different. 00:48:43.970 --> 00:48:46.490 But I also would create a lot of trouble 00:48:46.490 --> 00:48:49.250 because then the mathematics become complicated. 00:48:49.250 --> 00:48:50.880 I didn't gain anything. 00:48:50.880 --> 00:48:58.240 But by choosing this functional form, 00:48:58.240 --> 00:49:02.280 you actually write a very beautiful picture. 00:49:02.280 --> 00:49:05.920 Another thing, which is very cool about this 00:49:05.920 --> 00:49:12.480 is that if I write this thing in the exponential functional 00:49:12.480 --> 00:49:18.340 form, since we are dealing with differential equations, 00:49:18.340 --> 00:49:22.170 there is a very good property about exponential function. 00:49:22.170 --> 00:49:26.856 That is it is essentially a phoenix function. 00:49:26.856 --> 00:49:30.480 Do you know what is a phoenix? 00:49:30.480 --> 00:49:36.060 Phoenix is actually some kind of animal, a long-beaked bird, 00:49:36.060 --> 00:49:41.100 which is cyclically called the regenerated or reborn. 00:49:41.100 --> 00:49:44.100 So basically, when this phoenix die, 00:49:44.100 --> 00:49:48.820 you will lay the eggs in the fire and you were reborn. 00:49:48.820 --> 00:49:50.670 This is actually the same as this function. 00:49:54.240 --> 00:49:59.650 I can do differentiation, still an exponential function, 00:49:59.650 --> 00:50:02.460 and differentiate, differentiate, differentiate. 00:50:02.460 --> 00:50:05.460 Still exponential function. 00:50:05.460 --> 00:50:07.620 So that is very nice because when 00:50:07.620 --> 00:50:13.350 we deal with differential equation, 00:50:13.350 --> 00:50:16.590 then you can actually remove all those dots 00:50:16.590 --> 00:50:20.940 and make them become just exponential function. 00:50:20.940 --> 00:50:25.770 So essentially, a very nice property. 00:50:25.770 --> 00:50:32.910 So the first property, which is very nice is that it cannot be 00:50:32.910 --> 00:50:39.480 killed by differentiation. 00:50:39.480 --> 00:50:46.980 You will see how useful this is in the following lectures. 00:50:49.620 --> 00:50:52.230 The second thing, which is really nice 00:50:52.230 --> 00:50:55.170 is that it has a very nice property. 00:50:55.170 --> 00:51:02.090 So basically the exponential i theta 1 times exponential i 00:51:02.090 --> 00:51:07.410 theta 2, and that will give you exponential i theta 00:51:07.410 --> 00:51:08.980 1 plus theta 2. 00:51:11.610 --> 00:51:13.420 So what does that mean? 00:51:13.420 --> 00:51:22.110 That means if I have a solution in this form, A exponential i 00:51:22.110 --> 00:51:25.400 omega t plus phi. 00:51:29.040 --> 00:51:40.200 And I do a times translation, t become t plus A. Then 00:51:40.200 --> 00:51:47.650 this become A exponential i omega t plus A plus phi. 00:51:51.350 --> 00:51:59.430 So this means that times translation in this rotation 00:51:59.430 --> 00:52:04.880 is just a rotation in complex print. 00:52:04.880 --> 00:52:05.730 You see? 00:52:05.730 --> 00:52:08.900 So now t becomes t plus A. Then you are actually 00:52:08.900 --> 00:52:16.190 just changing the angle between this vector and the x-axis. 00:52:16.190 --> 00:52:18.140 So as time goes on, what is going 00:52:18.140 --> 00:52:21.300 to happen is that this thing will go around and around 00:52:21.300 --> 00:52:28.100 and around and the physics is always the set, no matter when 00:52:28.100 --> 00:52:32.340 you start counting, and the translation is just 00:52:32.340 --> 00:52:35.790 the rotation in this print. 00:52:35.790 --> 00:52:37.165 Any questions? 00:52:44.940 --> 00:52:50.160 So I think this is actually a basic slide just to remind you 00:52:50.160 --> 00:52:52.710 about Euler's formula. 00:52:52.710 --> 00:52:55.460 So basically, the explanation i phi 00:52:55.460 --> 00:52:58.920 is equal to cosine phi plus i sine phi. 00:52:58.920 --> 00:53:03.300 And I think it will be useful if you are not familiar with this. 00:53:03.300 --> 00:53:06.180 It is useful to actually review a little bit 00:53:06.180 --> 00:53:10.230 about exponential function, which will 00:53:10.230 --> 00:53:12.050 be very useful for this class. 00:53:17.350 --> 00:53:20.260 So I'm running a bit faster today. 00:53:20.260 --> 00:53:25.930 So let's take a look at what we have learned today. 00:53:25.930 --> 00:53:31.060 We have analyzed the physics of a harmonic oscillator. 00:53:31.060 --> 00:53:37.830 So basically, we start by asking really just a verbal question, 00:53:37.830 --> 00:53:40.650 what is going to happen to this mass 00:53:40.650 --> 00:53:44.310 on the table attached to a spring. 00:53:44.310 --> 00:53:49.620 And what we have learned is that we actually use mathematics. 00:53:49.620 --> 00:53:57.750 Basically, we translate all what we have learned about this mass 00:53:57.750 --> 00:54:03.480 into mathematics by first define a coordinate system. 00:54:03.480 --> 00:54:09.130 Then I'd write everything using that coordinate system. 00:54:09.130 --> 00:54:13.125 Then I use Newton's law to help us to solve this question. 00:54:15.870 --> 00:54:19.710 And we have analyzed the physics of this harmonic oscillator. 00:54:19.710 --> 00:54:23.700 And Hooke's law, we found that he actually, not only 00:54:23.700 --> 00:54:30.380 works for this spring-mass system, 00:54:30.380 --> 00:54:37.260 it also works for all kinds of different small oscillations 00:54:37.260 --> 00:54:40.530 about a point of equilibrium. 00:54:40.530 --> 00:54:44.700 So basically, it's actually a universal solution 00:54:44.700 --> 00:54:47.070 what we have been doing. 00:54:47.070 --> 00:54:51.990 And we have found out a complex exponential function 00:54:51.990 --> 00:54:55.450 is actually a beautiful way to present 00:54:55.450 --> 00:54:59.910 the solution to the equation of motion we have been studying. 00:54:59.910 --> 00:55:02.240 So everything is nice and good. 00:55:02.240 --> 00:55:06.960 However, life is hard because there 00:55:06.960 --> 00:55:13.010 are many things which actually, we ignored in this example. 00:55:13.010 --> 00:55:16.620 One apparent thing, which we actually ignore, 00:55:16.620 --> 00:55:18.840 is the direct force. 00:55:18.840 --> 00:55:24.420 So you can see that before I was actually making this pendulum 00:55:24.420 --> 00:55:27.480 oscillate back and forth. 00:55:27.480 --> 00:55:29.430 What is happening now? 00:55:29.430 --> 00:55:31.930 There are not oscillating anymore. 00:55:31.930 --> 00:55:33.660 Why? 00:55:33.660 --> 00:55:37.020 Well, they stopped being. 00:55:37.020 --> 00:55:40.440 Apparently, something is missing. 00:55:40.440 --> 00:55:46.930 When I actually moved this system, 00:55:46.930 --> 00:55:51.490 if I turn off the air so that there's friction, 00:55:51.490 --> 00:55:53.590 then it doesn't really move. 00:55:53.590 --> 00:55:58.820 If I increase a bit, the air so that the slide have 00:55:58.820 --> 00:56:02.330 some slight freedom, then actually, you 00:56:02.330 --> 00:56:05.990 can see that you move a bit then you stop. 00:56:05.990 --> 00:56:12.625 If I increase this some more, you 00:56:12.625 --> 00:56:18.660 can see that the amplitude becomes smaller and smaller. 00:56:18.660 --> 00:56:22.400 So in the following lecture, what we are going to do 00:56:22.400 --> 00:56:27.110 is to study how to actually include a direct force into it 00:56:27.110 --> 00:56:31.210 again and of course, using the same machinery which we have 00:56:31.210 --> 00:56:34.160 learned from here and see if we can actually 00:56:34.160 --> 00:56:35.890 solve this problem. 00:56:35.890 --> 00:56:36.950 Thank you very much. 00:56:36.950 --> 00:56:39.360 We actually end up earlier today. 00:56:39.360 --> 00:56:40.520 Sorry for that. 00:56:40.520 --> 00:56:43.280 And maybe I will make the lecture longer next time. 00:56:45.890 --> 00:56:48.340 And if you have any questions about what 00:56:48.340 --> 00:56:55.480 we have covered today, I'm here available to help you.