1 00:00:00,060 --> 00:00:05,400 The throughout the section, we have already solved several ordinary differential equations, but to 2 00:00:05,400 --> 00:00:08,700 be honest, all of them, we could have also solved analytically. 3 00:00:09,630 --> 00:00:16,770 So so far, our new routine didn't serve any purpose, but now we will finally come to an example which 4 00:00:16,770 --> 00:00:18,150 we cannot solve analytically. 5 00:00:18,690 --> 00:00:24,870 So here we can really use our new routine, which we have to find and called Euler to to solve second 6 00:00:24,870 --> 00:00:27,120 order ordinary differential equations. 7 00:00:27,810 --> 00:00:32,880 And the example will be the harmonic oscillator or more precisely, a pendulum. 8 00:00:34,020 --> 00:00:35,730 So you probably know this example. 9 00:00:36,150 --> 00:00:43,260 There's a pendulum swinging and it's described by the angle, so you could also write that down in terms 10 00:00:43,260 --> 00:00:44,070 of the coordinates. 11 00:00:44,370 --> 00:00:49,500 But I will write it down here in terms of the angle and the differential equation looks like this. 12 00:00:49,800 --> 00:00:57,020 So you have the first term, which is basically characterizing the force acting on the angle or, you 13 00:00:57,030 --> 00:01:03,750 know, the the force acting on the pendulum, which is characterized by a second or the derivative of 14 00:01:03,750 --> 00:01:04,349 the angle. 15 00:01:04,860 --> 00:01:08,730 Then we have here a damping term, which is friction basically. 16 00:01:08,740 --> 00:01:09,930 So loss of energy. 17 00:01:10,530 --> 00:01:14,400 And then we have a third term which describes the gravity force. 18 00:01:15,060 --> 00:01:23,130 So the force will always point straight down and it's not perpendicular to the pendulum, so not along 19 00:01:23,130 --> 00:01:28,680 the radial direction, and therefore it will not be linear and theto as you sometimes see this. 20 00:01:29,490 --> 00:01:36,120 But this is in general not correct because in general, the correct differential equation is characterized 21 00:01:36,120 --> 00:01:43,890 by a sign of theta, which is just due to the fact that the force is not always radial, but it is pointing, 22 00:01:43,890 --> 00:01:45,360 always straight down. 23 00:01:46,830 --> 00:01:48,300 So you have these three terms. 24 00:01:48,720 --> 00:01:56,130 B is the damping parameter, and C is basically determined by the pendulum length it's given by the 25 00:01:56,130 --> 00:01:59,790 gravity constant, divided by the length of the pendulum. 26 00:02:01,320 --> 00:02:07,260 So before we really solve this equation, which cannot be solved analytically, that has come to the 27 00:02:07,260 --> 00:02:09,150 case that you probably know already. 28 00:02:09,300 --> 00:02:13,170 This is called the small angle of a small angle approximation. 29 00:02:14,130 --> 00:02:20,560 So if theta is really, really small, then you can approximate the sign of theta to be equal to the 30 00:02:20,560 --> 00:02:21,060 theater. 31 00:02:22,110 --> 00:02:26,460 And if you then consider a case without damping, then you have the differential equation. 32 00:02:26,880 --> 00:02:32,520 Theta second derivative is a is equal to minus C times sign of theta. 33 00:02:33,030 --> 00:02:35,220 So this becomes this one here. 34 00:02:35,610 --> 00:02:39,000 And so this is an equation that is linear in the terms of theta. 35 00:02:40,020 --> 00:02:46,020 And also, it's very easy to solve because a function that solves this is, of course, the sign or 36 00:02:46,020 --> 00:02:47,160 the cosine function. 37 00:02:47,730 --> 00:02:55,110 So a possible solution would be this one here where you have a course in function square root of g over 38 00:02:55,110 --> 00:02:56,070 all times T. 39 00:02:57,510 --> 00:03:03,060 So let's see if our routine is able to restore this result. 40 00:03:03,480 --> 00:03:07,380 So this differential equation is just a tiny bit more difficult into to freefall. 41 00:03:07,380 --> 00:03:09,540 So the freefall we had theta. 42 00:03:10,920 --> 00:03:12,810 So basically we had here why? 43 00:03:12,870 --> 00:03:15,270 Second, derivative is given by minus g. 44 00:03:15,990 --> 00:03:19,260 And here we have A. Another term with the function itself. 45 00:03:20,160 --> 00:03:22,710 So what this means is we have to modify our code. 46 00:03:22,890 --> 00:03:27,510 So here these are the two cells from the free fall, which I've copied. 47 00:03:29,160 --> 00:03:31,590 So let's go back down and let's modify them. 48 00:03:32,520 --> 00:03:33,870 So here I will write. 49 00:03:33,990 --> 00:03:37,350 Blink for L is equal to two. 50 00:03:37,830 --> 00:03:43,980 And so over a constant C, which is this one is given by 51 00:03:47,190 --> 00:03:59,250 nine point eight one divided by the length and then the damping I will also consider to be zero damping 52 00:03:59,970 --> 00:04:01,380 B is equal to zero. 53 00:04:01,890 --> 00:04:11,310 So this one would appear to be pendulum geometry and then we can define our function f only e. 54 00:04:13,380 --> 00:04:21,540 So the f only function would in this case, be if we take this on the left and this on the right, it 55 00:04:21,540 --> 00:04:32,790 would be minus B times y one and then another term, which is minus C times y zero. 56 00:04:34,050 --> 00:04:40,170 So bring these two terms on the right, and we have this approximation that the sine is equal to the 57 00:04:40,170 --> 00:04:41,060 theater itself. 58 00:04:41,070 --> 00:04:48,240 And here Cetus called Y is zero and the first derivative is called y one. 59 00:04:49,920 --> 00:04:52,200 Now we must define the starting values. 60 00:04:52,650 --> 00:04:56,850 And here we will once again use T zero as equal to zero. 61 00:04:57,420 --> 00:04:59,520 And then maybe that it's not so. 62 00:04:59,810 --> 00:05:08,840 Using Let me go right here, Theta Zero and Theta one, but it would not be necessary, actually. 63 00:05:11,730 --> 00:05:20,550 And then we can also right here, six zero zero and to one zero. 64 00:05:21,390 --> 00:05:27,120 And I've tested this before and while used to work quite well, our 0.2 and zero. 65 00:05:27,660 --> 00:05:33,210 So you have to remember that three to zero must be small since we are here in the small angle approximation. 66 00:05:33,720 --> 00:05:36,990 So, yeah, make sure that this value is pretty small. 67 00:05:37,410 --> 00:05:39,780 Otherwise, it will just be unreasonable. 68 00:05:40,800 --> 00:05:43,050 And then we can increase the number of and max. 69 00:05:43,290 --> 00:05:47,670 And I think there's one, of course, we have to have to change now. 70 00:05:48,960 --> 00:05:52,200 So as I said, this would not have been necessary to change. 71 00:05:52,200 --> 00:05:54,720 We could have just wrote here why and also why. 72 00:05:55,290 --> 00:06:00,720 But I think it's a bit more reasonable and you can understand it better since I changed it. 73 00:06:02,490 --> 00:06:06,720 So now we can run the cell, calculate the solution and plot it. 74 00:06:07,140 --> 00:06:08,580 So here, right, Peter. 75 00:06:09,420 --> 00:06:12,090 And this time I only want to plot thetr. 76 00:06:12,120 --> 00:06:13,830 I don't care about the first derivative. 77 00:06:15,180 --> 00:06:25,560 So the analytical solution will be test, test, beta test thetr and it will be co-signed. 78 00:06:25,890 --> 00:06:37,640 So will be theta zero times and P dots cosine and P dot, square root and G over. 79 00:06:38,580 --> 00:06:50,520 So nine point eight one over A and then times T times test T, and I hope that this will now get give 80 00:06:50,520 --> 00:06:51,430 the correct result. 81 00:06:52,140 --> 00:06:53,190 And it looks pretty good. 82 00:06:54,150 --> 00:06:58,560 It looks like, yeah, it looks like a perfect cosine function. 83 00:06:59,460 --> 00:07:06,000 And really, there is no wonder because the analytical solution is in fact the cosine function and our 84 00:07:06,000 --> 00:07:10,110 step size was small enough to be able to recreate this result. 85 00:07:10,950 --> 00:07:11,880 So that was pretty good. 86 00:07:13,170 --> 00:07:21,210 We have used our second order method that we have to find here to solve now the pendulum, the harmonic 87 00:07:21,210 --> 00:07:24,330 oscillator in small angle approximation. 88 00:07:25,770 --> 00:07:32,070 However, this small angle approximation is, of course, only an approximation, and this approximation 89 00:07:32,070 --> 00:07:34,890 is only done so that we are able to solve it analytically. 90 00:07:35,580 --> 00:07:37,620 But since we have a nice cold, we don't need it. 91 00:07:37,920 --> 00:07:44,040 We could actually use here the sign of theta and solve the actual equation, and this is what we are 92 00:07:44,040 --> 00:07:46,020 going to do in the next lecture.