1 00:00:00,150 --> 00:00:07,800 Last thing I want to show you is that we can very easily transform our whole system to describe a different 2 00:00:07,800 --> 00:00:12,840 geometry, which would be such a ring of coupled oscillators. 3 00:00:13,500 --> 00:00:18,710 So basically, we just take our straight line, which was fixed at the left and at the right edge, 4 00:00:18,720 --> 00:00:21,780 and we put these two edges together to form a circle. 5 00:00:22,860 --> 00:00:30,180 And if we approximate this by using such a circle here and basically we say our whole problem has just 6 00:00:30,180 --> 00:00:37,080 one dimension that goes along the circle here and basically just one coordinate on one path on which 7 00:00:37,080 --> 00:00:38,400 these coordinates can move. 8 00:00:39,000 --> 00:00:46,230 Then we can very easily describe this because then our equations of motion are exactly the same as before 9 00:00:46,320 --> 00:00:48,240 here, just for six oscillators. 10 00:00:48,690 --> 00:00:55,380 The only thing that's different is this matrix element and this one, because it means that the first 11 00:00:55,380 --> 00:01:01,440 and the last oscillator coupled with each other because there is no spring between the first and the 12 00:01:01,440 --> 00:01:02,070 last one. 13 00:01:02,490 --> 00:01:05,459 And except for this, everything is the same as before. 14 00:01:06,450 --> 00:01:14,130 So basically, we just have to change our equations of motion and add here these two missing terms. 15 00:01:14,130 --> 00:01:19,470 For example, here for the first one, we have to add to your term that coupled to the last one. 16 00:01:19,890 --> 00:01:23,580 And for the last one, we have to couple a term with the first one. 17 00:01:24,390 --> 00:01:28,800 And so this will be our new oscillator and the numerical solution. 18 00:01:28,800 --> 00:01:36,450 You see, I run this already will be we can calculate with exactly the same method as before, and this 19 00:01:36,450 --> 00:01:39,090 will be one of the possibilities that we will get. 20 00:01:39,990 --> 00:01:43,650 And you see, still these oscillators they oscillate, of course. 21 00:01:44,100 --> 00:01:47,400 But the thing is, you see like some background drift. 22 00:01:47,880 --> 00:01:54,810 And the thing is, since we have randomly chosen the starting conditions and maybe the velocity of all 23 00:01:54,810 --> 00:02:01,650 of these oscillators is on average non-zero, the whole ring will start to circle. 24 00:02:01,650 --> 00:02:06,300 And if you imagine this, you take just some balls coupledom with springs. 25 00:02:06,870 --> 00:02:14,610 Then, of course, the whole bowl and a whole circle of these oscillators can start to move in a circle. 26 00:02:14,640 --> 00:02:18,420 It's, of course, allowed because none of these oscillators is fixed. 27 00:02:18,960 --> 00:02:22,200 So this is allowed and it can happen and it will happen. 28 00:02:23,250 --> 00:02:28,020 So, for example, if I rerun this, you see, now the drift is along the opposite direction. 29 00:02:28,560 --> 00:02:33,900 It is a bit random because it depends, of course, on the starting conditions that we have chosen randomly. 30 00:02:36,090 --> 00:02:43,140 And we can, of course, also calculate the EIGEN frequencies by generating this matrix here as before. 31 00:02:43,140 --> 00:02:48,210 And then we just have to explicitly define these two elements here. 32 00:02:49,020 --> 00:02:53,580 First row, last column and vice versa, and the value will be minus one. 33 00:02:54,510 --> 00:02:56,970 And then we can calculate the eigenvalues. 34 00:02:57,660 --> 00:03:01,710 And you see, here are just some numbers for the eigenvalues. 35 00:03:02,190 --> 00:03:08,010 But one thing that's maybe a bit confusing at first and remarkable and even gives us an error. 36 00:03:08,040 --> 00:03:13,800 Once we calculate the square root to get to the eigen frequencies, is that one of these eigenvalues 37 00:03:13,800 --> 00:03:17,220 is negative, as I wrote down here. 38 00:03:17,880 --> 00:03:20,670 It's not really a negative number, it's just zero. 39 00:03:20,700 --> 00:03:26,220 So you see, this is minus one times 10 to the power of minus 16, so it is essentially zero. 40 00:03:27,240 --> 00:03:32,010 So the reason why it's negative here is just some numerical artifact, you could say. 41 00:03:32,370 --> 00:03:36,870 And this gives us the problem when calculating the square root, which gives us 10, not the number 42 00:03:36,870 --> 00:03:37,530 as an error. 43 00:03:38,340 --> 00:03:40,800 However, we know that this one is actually zero. 44 00:03:41,220 --> 00:03:43,230 So the square root would also be zero. 45 00:03:44,070 --> 00:03:47,970 So what does it mean when an eigen frequency is zero? 46 00:03:48,570 --> 00:03:53,070 It means that the EIGEN mode is not an oscillation, but the translation. 47 00:03:54,120 --> 00:04:00,750 So the system has an IGen mode, which does not correspond to any oscillation, but to a collective 48 00:04:00,750 --> 00:04:03,240 motion of all the oscillators in space. 49 00:04:03,820 --> 00:04:09,780 And this is exactly what we see here with this background drift motion, which is now allowed since 50 00:04:09,780 --> 00:04:14,190 none of these oscillators is fixed and they are just coupled with each other. 51 00:04:14,520 --> 00:04:18,060 But of course, the whole ring can rotate collectively. 52 00:04:19,649 --> 00:04:24,390 All right, this was just a very brief example of what else we can solve with our methods and you've 53 00:04:24,390 --> 00:04:30,120 seen it's actually very much related and very simple to implement because the only thing that was different 54 00:04:30,570 --> 00:04:35,670 were two of these elements in The Matrix and the rest was just the same. 55 00:04:35,970 --> 00:04:42,570 However, even if you change such small things as periodic boundary conditions, you see that you can 56 00:04:42,570 --> 00:04:43,590 get physically. 57 00:04:43,710 --> 00:04:51,090 Also new phenomena like this drift motion, which is resembles by this eigen frequency, which is zero 58 00:04:51,090 --> 00:04:54,330 and here as an artifact gives even an error message.