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So now we can finally start programming in the previous lecture, we have talked about the system and

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have figured out the equation of motion.

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And I think it's pretty important that we figure it out this matrix here because later on, we will

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calculate its eigenvalues and then discuss the meaning of these eigenvalues.

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And then you will understand why we can call this also an eigenvalue problem.

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But actually, you may say now, why should we do this?

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These are just couple differential equations, and we have learned already in the previous lectures

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how we can solve these.

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And this is, of course, totally true.

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And for this reason, we will now, in this particular lecture, solve these equations as we did before

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using our standard routines.

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So don't worry, I will not bore you with another whole section about these differential equations.

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This will just be a single lecture where we will just practice what we've learned before, and I will

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show you that we can pretty easily solve these.

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Now that we know how it's done, so we shouldn't lose any time, let's go ahead and find, first of

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all, the constants case equal to one am as equal to one.

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And then we must, of course, define the system of two differential equations.

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So I call the function f underscore three oscillators and parameters.

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Artie, which does not explicitly enter here, but you remember the integrated solve IBP routine requires

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two time argument and then the coordinate argument.

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So this is how we always did it and we will do it here as well.

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So now we must return some function.

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And of course, we want to return here to equate to the right and side of these equations.

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So basically these three equations here, but hopefully you also remember that it's not so easy if you

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have a second order differential equation.

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So we have here a second order time derivative and not the first order one.

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This means we must not only return to three right and side of the equation, but we must introduce another

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variable first, which would be the first derivative of art.

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And this is, of course, the velocity.

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So we will return the velocity of the first oscillator, which I will call like this b zero.

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Then we have we won and we have the two and then we have here the three right and sides of these equations.

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So to make this more clear, I will first of all, remind you that our variable here and why.

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We could also call it R or whatever we want is just the right insights of these equations.

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But now we do not only have three of these, but we have six of these because we have to consider the

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first order time derivative and the second order.

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So I write R is equal to Y and then the first three entries and V the velocity is equal to Y and then

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the other entries.

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So three to six.

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So remember, this means start from zero and go to two zero one two and here three four five.

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All right.

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And now I write Equation one is equal to no, I will write this one here minus R one.

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So in our python indices, this would be our with the index zero.

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And then we have.

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Actually, we have this twice, as you can see here, so I have to ride two times are zero and then

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minus one times are one.

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And then, for example, I could write in brackets times minus K over M, and then we do a similar thing

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for the other ones.

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But I will, of course, now paste these here.

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Equation with one equation two, Equation three and Equation two and three will, of course, look pretty

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similarly.

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So two and three and the two will be, as you can see here, minus.

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Our zero plus two times our one minus her two and then the last one is, of course, also very easy.

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So the only thing that you have to take care of is that you get the indices right because here in Python,

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we count from zero one two and here we have one two three.

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So please don't be fooled here.

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And then everything will be good.

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OK, I think it should work like this.

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Let's run it.

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OK, no error.

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That's always a good thing.

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And now we can continue and write T start is equal to zero and T and is equal to 100.

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Then I will say the starting values for white would be also a six component vector.

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And maybe we should specify now know of some specific numbers, but we can also go ahead and just simulate

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some random oscillations.

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For example, we could write some amplitude or some some magnitude of the variation, and then we generate

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six random numbers in the range minus one to plus one.

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So this would be minus notes.

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It would be two times the random number.

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So and people do random dot rounds six and then minus one.

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So here it is, would mean now from minus 0.5 to plus 0.5.

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This would mean from minus one to plus one, and this would mean from minus 0.3 to plus zero point three.

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And then we'll also multiply this by our value a, which we had here, which is basically the year the

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equilibrium position.

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Or you could also see the letters constant here to the standard or in equilibrium, the distance between

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two oscillators.

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So we have to define it, of course, as well.

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So like, let's say it's equal to one.

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And now we have our starting conditions and hard times.

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So this means we can now call our routine and stored solution.

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And as previously we write, integrate dots, solve, underscore IVP, then we must specify the function.

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So really, this is nothing new to this.

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Exactly, as we have learned, isn't how we practice it.

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Then from T start to T and his starting parameters would be why start?

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The method would be R.K. forty five and this would already work, but I want to get a equally spaced

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output for added time values or right here.

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T sorry typo t evolve is equal to and point in space.

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And it goes from tee start to tee and in steps of, let's say, zero point one.

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So maybe like this.

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OK, so I run this and once again, no error, which is even better.

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And now we can plot its so we have a plot with an ex label for X or the x axis indicates the time and

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the Y label indicates the coordinates of the three oscillators.

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And now we can go ahead and plot pal t dot plot

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and for the X values, we take the time.

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So this would be a solution and a scroll and a go three or a C dot T.

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And for the Y coordinates, we have a solution underscore three O C, Dot Y and then the zero index.

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So let's try it and we see some wild oscillation.

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So let's add the other two oscillators.

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This would be one and two, and we see they are now on top of each other and it oscillates quickly.

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Maybe it's good to shift them by their equilibrium position.

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So basically, what we what we see now here is the solution for our transformed variables, which would

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be these ones here.

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So this would be the coordinate with respect to the equilibrium position.

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But if you want to transform back to this one here, we have to add a two hour one to a two hour two

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and three a two or three.

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Then we are back in our old coordinate system.

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So this is what I did here and I'll run it.

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And you see, now they are oscillating apart from each other.

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You see, the first one oscillates around one, the second one around to the third one, around three

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one to three.

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So of course, since I have chosen the starting conditions to be random for you, the solution will

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probably look different.

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So also, when I restart and run ourselves, we will get now a bit of a different solution.

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But in all cases, we have here some oscillation, and you see, it's not really just a cosine oscillation.

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So even though we have simple harmonic oscillators for each of these springs.

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So for example, if we would leave out this whole part here and just consider this oscillator with this

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spring, we would get a harmonic oscillator and as a solution, we would get some cosine or sine function.

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However, here we see that there is some oscillation, but it's not harmonic.

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This is because the system becomes very sensible or very, very unstable, you could even say.

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And if you change the starting conditions, and this will pretty much affect also all of the other oscillators,

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and they will oscillate in a pretty funny way.

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However, it was no problem to solve this.

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Using our standard routine solve dot, integrate dot solve underscore IBP as we did before, and now

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we have our numerical solution.

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So now in the next lecture, we can go ahead and analyze this equation of motion once again and solve

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the problem with a different method.

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And this will be something new.

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So please stay tuned, and I hope you are excited for the next lecture.