Last thing I want to show you is that we can very easily transform our whole system to describe a different
geometry, which would be such a ring of coupled oscillators.
So basically, we just take our straight line, which was fixed at the left and at the right edge,
and we put these two edges together to form a circle.
And if we approximate this by using such a circle here and basically we say our whole problem has just
one dimension that goes along the circle here and basically just one coordinate on one path on which
these coordinates can move.
Then we can very easily describe this because then our equations of motion are exactly the same as before
here, just for six oscillators.
The only thing that's different is this matrix element and this one, because it means that the first
and the last oscillator coupled with each other because there is no spring between the first and the
last one.
And except for this, everything is the same as before.
So basically, we just have to change our equations of motion and add here these two missing terms.
For example, here for the first one, we have to add to your term that coupled to the last one.
And for the last one, we have to couple a term with the first one.
And so this will be our new oscillator and the numerical solution.
You see, I run this already will be we can calculate with exactly the same method as before, and this
will be one of the possibilities that we will get.
And you see, still these oscillators they oscillate, of course.
But the thing is, you see like some background drift.
And the thing is, since we have randomly chosen the starting conditions and maybe the velocity of all
of these oscillators is on average non-zero, the whole ring will start to circle.
And if you imagine this, you take just some balls coupledom with springs.
Then, of course, the whole bowl and a whole circle of these oscillators can start to move in a circle.
It's, of course, allowed because none of these oscillators is fixed.
So this is allowed and it can happen and it will happen.
So, for example, if I rerun this, you see, now the drift is along the opposite direction.
It is a bit random because it depends, of course, on the starting conditions that we have chosen randomly.
And we can, of course, also calculate the EIGEN frequencies by generating this matrix here as before.
And then we just have to explicitly define these two elements here.
First row, last column and vice versa, and the value will be minus one.
And then we can calculate the eigenvalues.
And you see, here are just some numbers for the eigenvalues.
But one thing that's maybe a bit confusing at first and remarkable and even gives us an error.
Once we calculate the square root to get to the eigen frequencies, is that one of these eigenvalues
is negative, as I wrote down here.
It's not really a negative number, it's just zero.
So you see, this is minus one times 10 to the power of minus 16, so it is essentially zero.
So the reason why it's negative here is just some numerical artifact, you could say.
And this gives us the problem when calculating the square root, which gives us 10, not the number
as an error.
However, we know that this one is actually zero.
So the square root would also be zero.
So what does it mean when an eigen frequency is zero?
It means that the EIGEN mode is not an oscillation, but the translation.
So the system has an IGen mode, which does not correspond to any oscillation, but to a collective
motion of all the oscillators in space.
And this is exactly what we see here with this background drift motion, which is now allowed since
none of these oscillators is fixed and they are just coupled with each other.
But of course, the whole ring can rotate collectively.
All right, this was just a very brief example of what else we can solve with our methods and you've
seen it's actually very much related and very simple to implement because the only thing that was different
were two of these elements in The Matrix and the rest was just the same.
However, even if you change such small things as periodic boundary conditions, you see that you can
get physically.
Also new phenomena like this drift motion, which is resembles by this eigen frequency, which is zero
and here as an artifact gives even an error message.