1
00:00:00,210 --> 00:00:06,300
So now, instead of calculating the small angle approximation, we want to calculate the actual differential

2
00:00:06,300 --> 00:00:08,580
equation of the harmonic oscillator.

3
00:00:09,900 --> 00:00:13,260
So what I'll do is I will just copy these two cells.

4
00:00:16,350 --> 00:00:24,960
And we only have two very slightly modify them, so the only thing we have to do is changing here the

5
00:00:24,960 --> 00:00:30,060
differential equation to and three dot sign.

6
00:00:32,420 --> 00:00:33,350
Of theater.

7
00:00:35,710 --> 00:00:43,060
And the starting value, 32, zero zero of zero point two is quite small, so let's consider this to

8
00:00:43,060 --> 00:00:44,660
be a small starting angle.

9
00:00:44,710 --> 00:00:53,050
And that's run it, and this time we almost get the same result as previously.

10
00:00:53,050 --> 00:00:55,270
There's maybe a very brief difference.

11
00:00:55,270 --> 00:01:02,890
So you see here, maybe the red the red line is slightly left of the blue dots, which was maybe also

12
00:01:02,890 --> 00:01:04,300
here the case, but not so much.

13
00:01:04,300 --> 00:01:06,120
But yeah, it's hard to say.

14
00:01:06,130 --> 00:01:13,150
Maybe there's a tiny difference, but this is due to the fact that the small starting angle of 0.2

15
00:01:15,730 --> 00:01:17,800
is really a very small starting angle.

16
00:01:17,800 --> 00:01:21,960
So let's check the zero point two is, of course, in radiant.

17
00:01:21,980 --> 00:01:23,860
So this corresponds to PI.

18
00:01:24,760 --> 00:01:30,070
So we have to divide by and p dot pi and have to multiply one hundred eighty degrees.

19
00:01:30,550 --> 00:01:34,480
This would correspond to 11 degrees, which is still pretty small.

20
00:01:35,650 --> 00:01:38,360
So now let's consider a larger starting value.

21
00:01:39,070 --> 00:01:44,120
So I would just copy all of these three cells here, and you see what we going to do next.

22
00:01:44,140 --> 00:01:47,200
Large starting earlier than damping and then a driven oscillator.

23
00:01:47,950 --> 00:01:52,930
And then all of these cases, we don't have to change really much about the coat.

24
00:01:56,170 --> 00:01:58,840
So let me just copy this.

25
00:02:01,600 --> 00:02:07,660
Of course, we could now go ahead, by the way, and the find a new function, which does not only give

26
00:02:07,660 --> 00:02:12,640
us the solution, but which also does the plot here, I think for the moment.

27
00:02:12,880 --> 00:02:15,790
It's really not really necessary, but you could do it.

28
00:02:15,790 --> 00:02:21,460
If you want to have a more tiny notebook and have a more clean notebook, then maybe it's a good idea

29
00:02:21,460 --> 00:02:22,000
to do this.

30
00:02:23,080 --> 00:02:35,050
So here I will consider a starting value of 2.0, which corresponds to a much larger angle of more than

31
00:02:35,050 --> 00:02:35,890
90 degree.

32
00:02:36,430 --> 00:02:38,530
So here we should really see a difference.

33
00:02:39,100 --> 00:02:42,100
And the rest we could really leave as it is.

34
00:02:42,100 --> 00:02:45,850
So I will also delete this one here because it's really not necessary.

35
00:02:46,690 --> 00:02:52,480
We'll just leave the theater zero zero and.

36
00:02:55,660 --> 00:03:01,510
You see, now we get immediately a large difference between the two calculations.

37
00:03:02,320 --> 00:03:06,950
And this is really not a matter of conversation or convergence.

38
00:03:07,000 --> 00:03:16,090
So what we could check now is what happens if we decrease the step size to zero point zero one and then

39
00:03:16,090 --> 00:03:20,410
maybe in turn we increase the simulation time to 2000.

40
00:03:22,240 --> 00:03:25,020
So you see, the result is still pretty much the same.

41
00:03:25,030 --> 00:03:26,830
There is just the difference now.

42
00:03:27,310 --> 00:03:35,050
The reason is that the analytical solution is just not correct anymore because it was the solution for

43
00:03:35,050 --> 00:03:40,420
another differential equation, which was only approximately correct if zero was small.

44
00:03:41,080 --> 00:03:44,830
And now, since it isn't small anymore, the result will be different.