1 00:00:00,510 --> 00:00:06,060 So now we know that we can solve the equations of motion which have these ones here numerically. 2 00:00:07,140 --> 00:00:11,220 However, as you have seen, the solution is pretty confusing. 3 00:00:11,220 --> 00:00:15,630 So there are a lot of oscillations here and these are not harmonic anymore. 4 00:00:15,630 --> 00:00:20,610 So we could just say, OK, this is not the solution, but we don't really understand what's going on 5 00:00:20,610 --> 00:00:20,910 here. 6 00:00:21,960 --> 00:00:29,280 And to get more of an understanding what these oscillations mean, we must solve the so-called eigenvalue 7 00:00:29,280 --> 00:00:29,710 problem. 8 00:00:30,500 --> 00:00:36,600 And first of all, I will teach you a bit about the mathematics, and I will explain to you why this 9 00:00:36,600 --> 00:00:38,370 is an eigenvalue problem. 10 00:00:39,360 --> 00:00:42,420 So if you find this too mathematical, no problem. 11 00:00:42,420 --> 00:00:43,970 You can just skip ahead. 12 00:00:43,980 --> 00:00:52,140 But let me say you, the the main topic or the main message of this lecture, and this is that we have 13 00:00:52,140 --> 00:00:55,680 this equation of motion, which are three coupled equations. 14 00:00:56,100 --> 00:01:02,670 And you see our one depends on our two as well, and our two depends on our one and our three as well 15 00:01:02,670 --> 00:01:03,270 and so on. 16 00:01:03,690 --> 00:01:05,700 So it's not so easy to solve the problem. 17 00:01:06,690 --> 00:01:15,090 However, it would be much more easy if this matrix here would be a diagonal matrix, or even if it 18 00:01:15,090 --> 00:01:19,380 would just be a scalar, because in this case, we would just have that. 19 00:01:19,380 --> 00:01:27,630 The first coordinate on one would only depend on our one or, as I have written down here, we may accomplish 20 00:01:27,630 --> 00:01:31,110 this by doing some transformation and then we have some new coordinates. 21 00:01:31,110 --> 00:01:36,000 Q and Q one depends only linearly on Q1. 22 00:01:36,000 --> 00:01:39,750 So here the derivative and here not so. 23 00:01:40,620 --> 00:01:46,590 This would, of course, be really nice because then we would have three uncoupled oscillators as we 24 00:01:46,590 --> 00:01:51,000 also had before and one of our examples, and then we could just solve each of them individually. 25 00:01:51,750 --> 00:01:53,610 So this would be really, really simple. 26 00:01:53,610 --> 00:02:01,200 And then we would have a frequency which would be given by the square root of K over m times this value 27 00:02:01,200 --> 00:02:04,080 lambda, which resembles this matrix. 28 00:02:05,160 --> 00:02:12,650 So it turns out that this is the case if the lambdas here are the eigenvalues of this matrix. 29 00:02:12,660 --> 00:02:18,930 So this is why we will continue and solve the eigenvalues of this matrix, and this is why the eigenvalues 30 00:02:18,930 --> 00:02:23,310 are so important and why they can be identified with the frequency. 31 00:02:24,240 --> 00:02:26,400 So now I will talk a bit more about mathematics. 32 00:02:26,400 --> 00:02:31,980 And as I said, if you are not interested in the and this and if you find it too difficult, it's no 33 00:02:31,980 --> 00:02:32,370 problem. 34 00:02:32,370 --> 00:02:35,430 You can just go ahead to the next lecture is not really not important. 35 00:02:35,430 --> 00:02:38,310 It's just that you understand better what's going on here. 36 00:02:39,510 --> 00:02:47,910 So the thing is to transform this equation to that one, we must apply some transformation and this 37 00:02:47,910 --> 00:02:51,150 is typically done by multiplying with the unitary matrix. 38 00:02:52,140 --> 00:02:59,460 So you mean this matrix and unitary means that the inverse times to normal matrix is the same as the 39 00:02:59,460 --> 00:03:02,220 normal matrix times the inverse and is equal to one. 40 00:03:02,940 --> 00:03:09,180 And this is really the key thing here, because of course, you can always multiply one. 41 00:03:09,210 --> 00:03:12,780 For example, you can see matrix times one times this vector. 42 00:03:13,230 --> 00:03:14,670 This is still the same thing. 43 00:03:15,510 --> 00:03:23,370 So what we do is we write this equation vector our second derivative sequence of minus K over m times 44 00:03:23,370 --> 00:03:26,760 two Matrix A, which is this one times two vector. 45 00:03:28,110 --> 00:03:34,140 And now we can multiply here times one because this doesn't change anything and we say one is equal 46 00:03:34,140 --> 00:03:37,710 to you, times you inverse. 47 00:03:38,580 --> 00:03:40,950 So these two equations are still the same. 48 00:03:42,000 --> 00:03:44,340 And now we can just change the brackets here. 49 00:03:44,940 --> 00:03:52,530 So this means we multiply u minus one times r we multiply u minus one times, eight times u and we have 50 00:03:52,530 --> 00:03:53,820 ten here missing u. 51 00:03:54,150 --> 00:03:57,360 And so we can see multiply from the left with you. 52 00:03:57,360 --> 00:03:58,650 Two The power of minus one. 53 00:03:59,070 --> 00:04:02,040 So then we have here u minus one times u, which is one. 54 00:04:02,310 --> 00:04:06,960 So we can leave it out and we must, of course, multiply it also to the left hand side. 55 00:04:06,960 --> 00:04:08,910 So we have u minus one. 56 00:04:09,300 --> 00:04:12,210 And then this vector R. And the second derivative. 57 00:04:13,410 --> 00:04:24,360 And so now I I introduce that we must find an I introduce new variables Q, which are u minus one times 58 00:04:24,360 --> 00:04:24,720 R. 59 00:04:25,230 --> 00:04:30,480 And then the same applies, of course, for the second derivative because this is not time dependent 60 00:04:30,480 --> 00:04:30,720 here. 61 00:04:30,720 --> 00:04:37,260 So we can just write that u minus one second or the derivative of R is second order the relative of 62 00:04:37,260 --> 00:04:37,680 Q. 63 00:04:38,940 --> 00:04:45,120 And then if we want to say that this is equal to lambda, then this one here you two the power of minus 64 00:04:45,120 --> 00:04:47,820 one times eight times you must be equal to lambda. 65 00:04:48,510 --> 00:04:51,210 And this is how we indirectly define you. 66 00:04:52,410 --> 00:04:59,370 So the thing is, you can show mathematically that it is possible to find such matrices you. 67 00:05:00,370 --> 00:05:04,150 Transformation here is correct on these, this equation is correct. 68 00:05:04,660 --> 00:05:10,180 And if you use this, then you can see that you can transform this matrix equation to this equation. 69 00:05:11,650 --> 00:05:18,790 So it means that lambda is defined like this or not, lambda as the out is more, you could say you 70 00:05:18,790 --> 00:05:20,500 is defined by eight and lambda. 71 00:05:20,950 --> 00:05:28,690 And so this is the equation that we have used for the definition, and now you can multiply with you 72 00:05:28,690 --> 00:05:29,500 from the left. 73 00:05:29,530 --> 00:05:31,900 So this one disappears and goes here. 74 00:05:32,590 --> 00:05:39,490 And you can also say, if you multiply here, another vector, that you have to solve this equation 75 00:05:39,490 --> 00:05:39,820 here. 76 00:05:40,450 --> 00:05:48,040 And so it turns out that we must solve the eigenvalues of the matrix a and that's the eigenvalues of 77 00:05:48,040 --> 00:05:48,580 the Matrix. 78 00:05:48,580 --> 00:05:54,130 Eight are lambda because this year's and IGen eigenvalue equation. 79 00:05:55,450 --> 00:06:02,980 So we have transformed, in short this equation to that one, and we figured out that lambda already 80 00:06:02,980 --> 00:06:04,870 eigenvalues of this matrix. 81 00:06:05,620 --> 00:06:12,580 And furthermore, we have figured out from our physical understanding that the eigenvalues lambda can 82 00:06:12,580 --> 00:06:19,450 be identified with the frequency not directly, but by the square root K over m times lambda relation. 83 00:06:20,260 --> 00:06:26,710 So in the following, we will analyze this matrix and determine the eigenvalues and then calculate the 84 00:06:26,710 --> 00:06:29,830 EIGEN frequencies for this matrix. 85 00:06:29,830 --> 00:06:35,770 And then we will compare these IGen frequencies with the frequencies that we see in our numerical solution.