So let's get started.
We will discuss now a so-called eigenvalue problem, and for example, I've chosen the coupled oscillators.
It's a pretty famous example and I really like it in this course because we can practice here a lot
of concepts that we have encountered previously as well.
So let's get started.
This is the system that we want to consider.
So here we have three coupled oscillators.
So imagine this to be some bull.
For example, it has a certain mass and here we have some springs and these balls, they are coupled
by these springs.
So we have a total of four springs and these.
This poll here is coupled to the left wall, which is fixed, of course.
And then it's coupled also to the next oscillator, which is on the right.
Then we have the middle oscillator who's coupled to the left and to the right, and we have the right
oscillator who is coupled to the left one and to the edge.
So the wall here, which is fixed as well.
So first of all, we have to, of course, to find the equations of motion.
And for this, I write down the total force acting on these three oscillators.
So, you know, that's mass times velocity, which is cordone, it's double dot.
So our double dot in this case means the total force.
The force that acts on these oscillators is just the typical force that is caused by, such as printing,
which is linear to the change of the variable with respect to some equilibrium position.
So I hope this doesn't sound too complicated.
It's actually pretty simple.
So basically, if you just take this spring and just this oscillator and you leave out the whole rest,
then you have here a force acting on this ball, which is zero in the equilibrium position.
This is what I have plotted here.
So when R one is exactly equal to this line here, then it's equilibrium.
But when it's elongated along the right or long left direction, then you have a force along in the
opposite direction.
So this force is linear in the coordinate, so we have minus the difference of these two coordinates.
So our one minus R L, which we will set to zero, by the way, later on.
So it will basically just be minus K times R one.
And then we have to subtract the equilibrium length, which I have called a.
So this would be just the single oscillator.
And then of course, we have more terms because this oscillator is coupled also to the other one.
So we have to basically calculate the length from this blue ball to this one, which is R1 minus R2.
And then since the force is zero for the equilibrium position, eh, we have to subtract also e and
then for the other oscillators, it's very, very simple and very similar.
We just have to take into account the two springs to the left and to the right.
So here we have a coordinate R2 two minus R1 and R2 minus R3, and here for the right one, it's pretty
similar to the left one.
We have one coupling to the next bowl and one coupling to the wall.
So we have here are a three minus R2, which is this minus this.
So this length and then R3 minus RR, which is the position of the right wall, which is fixed.
So these are now our three equations of motion.
So we could, of course, now just continue.
But I want to drop some constant terms.
So you see, we have actually in our system just three parameters or three degrees of freedom, but
it is safe.
So are one or two or three the coordinates of the three oscillators and we have our L and R and A which
are fixed.
So these do not change.
So it would be very good to get rid of those from the equations of motion.
And we can do this by using a transformation.
So we introduce a new R one, which we could call our tilde, for example.
But I will just write here that we will transform our one minus e to our one.
We will transform our two minds to a two hour two and our three minus three eight to our three.
So basically, this means our new variables are one or two or three are just basically the data to ours.
So the the difference from the equilibrium position.
So yeah, maybe it sounds too difficult here mathematically, but it's actually very simple.
We consider now only the deviation from the equilibrium.
And when we do this, and if you take your time and you use these terms here in these equations and
then you use that.
Are L0 and the hour is four times, four times, actually not four times are, then you will see that
these equations of motion simplify it to these ones here.
And I think if you think about it, it really makes sense.
We have now the changes from the equilibrium.
So we have here a typical force minus K times R one, then minus K times the difference from these two
allegations and so on and so on.
So we have these three equations for one or two or three double dots, and they depend on our one hour
to an hour three and the terms are even mixed.
For example, here are two double dot depends on our two hour one and our three.
And so this means we can write down these three equations, as is just a system of equations or even
as such, a matrix equation here.
So if you know matrix multiplication, for example, this means a one double dot is equal to minus K
over m two times R one minus one time are two plus zero times are three.
So this will be exactly what we have here.
We have minus K two times one and then we have.
Plus, one times Kay Times are, too.
And so here's the minus sign, this turns and plus two minus one, and we have to divide by M due to
this term.
So this will be now the system of equations that we will solve or the matrix equation and we will solve.
And first of all, we will do this numerically and then in the next lecture and it follows.
After this, I will explain to you why this can also be considered an eigenvalue problem and which other
interesting facts we can learn about the system from the eigenvalues of the matrix that we have just
discovered here.