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So let's get started.

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We will discuss now a so-called eigenvalue problem, and for example, I've chosen the coupled oscillators.

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It's a pretty famous example and I really like it in this course because we can practice here a lot

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of concepts that we have encountered previously as well.

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So let's get started.

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This is the system that we want to consider.

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So here we have three coupled oscillators.

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So imagine this to be some bull.

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For example, it has a certain mass and here we have some springs and these balls, they are coupled

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by these springs.

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So we have a total of four springs and these.

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This poll here is coupled to the left wall, which is fixed, of course.

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And then it's coupled also to the next oscillator, which is on the right.

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Then we have the middle oscillator who's coupled to the left and to the right, and we have the right

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oscillator who is coupled to the left one and to the edge.

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So the wall here, which is fixed as well.

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So first of all, we have to, of course, to find the equations of motion.

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And for this, I write down the total force acting on these three oscillators.

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So, you know, that's mass times velocity, which is cordone, it's double dot.

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So our double dot in this case means the total force.

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The force that acts on these oscillators is just the typical force that is caused by, such as printing,

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which is linear to the change of the variable with respect to some equilibrium position.

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So I hope this doesn't sound too complicated.

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It's actually pretty simple.

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So basically, if you just take this spring and just this oscillator and you leave out the whole rest,

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then you have here a force acting on this ball, which is zero in the equilibrium position.

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This is what I have plotted here.

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So when R one is exactly equal to this line here, then it's equilibrium.

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But when it's elongated along the right or long left direction, then you have a force along in the

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opposite direction.

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So this force is linear in the coordinate, so we have minus the difference of these two coordinates.

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So our one minus R L, which we will set to zero, by the way, later on.

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So it will basically just be minus K times R one.

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And then we have to subtract the equilibrium length, which I have called a.

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So this would be just the single oscillator.

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And then of course, we have more terms because this oscillator is coupled also to the other one.

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So we have to basically calculate the length from this blue ball to this one, which is R1 minus R2.

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And then since the force is zero for the equilibrium position, eh, we have to subtract also e and

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then for the other oscillators, it's very, very simple and very similar.

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We just have to take into account the two springs to the left and to the right.

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So here we have a coordinate R2 two minus R1 and R2 minus R3, and here for the right one, it's pretty

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similar to the left one.

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We have one coupling to the next bowl and one coupling to the wall.

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So we have here are a three minus R2, which is this minus this.

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So this length and then R3 minus RR, which is the position of the right wall, which is fixed.

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So these are now our three equations of motion.

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So we could, of course, now just continue.

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But I want to drop some constant terms.

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So you see, we have actually in our system just three parameters or three degrees of freedom, but

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it is safe.

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So are one or two or three the coordinates of the three oscillators and we have our L and R and A which

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are fixed.

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So these do not change.

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So it would be very good to get rid of those from the equations of motion.

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And we can do this by using a transformation.

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So we introduce a new R one, which we could call our tilde, for example.

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But I will just write here that we will transform our one minus e to our one.

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We will transform our two minds to a two hour two and our three minus three eight to our three.

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So basically, this means our new variables are one or two or three are just basically the data to ours.

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So the the difference from the equilibrium position.

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So yeah, maybe it sounds too difficult here mathematically, but it's actually very simple.

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We consider now only the deviation from the equilibrium.

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And when we do this, and if you take your time and you use these terms here in these equations and

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then you use that.

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Are L0 and the hour is four times, four times, actually not four times are, then you will see that

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these equations of motion simplify it to these ones here.

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And I think if you think about it, it really makes sense.

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We have now the changes from the equilibrium.

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So we have here a typical force minus K times R one, then minus K times the difference from these two

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allegations and so on and so on.

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So we have these three equations for one or two or three double dots, and they depend on our one hour

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to an hour three and the terms are even mixed.

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For example, here are two double dot depends on our two hour one and our three.

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And so this means we can write down these three equations, as is just a system of equations or even

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as such, a matrix equation here.

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So if you know matrix multiplication, for example, this means a one double dot is equal to minus K

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over m two times R one minus one time are two plus zero times are three.

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So this will be exactly what we have here.

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We have minus K two times one and then we have.

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Plus, one times Kay Times are, too.

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And so here's the minus sign, this turns and plus two minus one, and we have to divide by M due to

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this term.

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So this will be now the system of equations that we will solve or the matrix equation and we will solve.

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And first of all, we will do this numerically and then in the next lecture and it follows.

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After this, I will explain to you why this can also be considered an eigenvalue problem and which other

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interesting facts we can learn about the system from the eigenvalues of the matrix that we have just

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discovered here.