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So welcome back, and now let's discuss more about the physics.

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Previously, we have only found cases where the trajectory would be linear, and this was because these

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starting velocity was zero.

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And this is about to change now.

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So of course, we could just go ahead and change you the starting velocity.

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But I want to keep these results because they are quite nice.

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So let's just copy the content of these three cells.

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And actually, we can just copy all of this in the same cell.

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It's no problem at all.

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And then also the plots.

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So now we have everything in the same cell and then we can go ahead and just change the starting velocity.

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That could just change this one to, for example, this and then I will rerun the whole notebook just

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in case.

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And here you see the trajectory where it moves back and forth on a linear line.

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So just on a linear trajectory on the line.

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So to change this, we change the starting velocity and we will kick it along the y direction.

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I run this now.

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You see, we have now both of these coordinates that are different from zero.

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And you see the trajectory becomes such a spiral.

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It's an elliptical trajectory.

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And I think it's pretty clear from a physical point of view, you just take the ball.

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And if you would just leave it to roll and to relax, that would roll on this line.

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But if you would kick it in along this direction and it would spiral down to the center point of the

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ball.

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OK, so that was pretty nice, I think.

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But so far we could have solved probably both of these examples analytically.

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Now I want to show you a case where this is not possible, where we will get a really difficult trajectory,

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that we're really helpful to have these routines in Python to help us to solve these differential equations.

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So and for this, we will copy once again this whole code, but we will add here an external force.

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And to do this, we have to change the differential equation so that we copy this one as well.

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And we have to add here a term that corresponds to an external force.

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And so just for a second, let me scroll back to the very beginning of the notebook.

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So here we have introduced this external force already.

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We wrote the force.

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The total force on the ball is equal to the damping force, to the gradient of the potential and to

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an external force, which we disregarded so far.

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So no, we will keep it in to take it into account.

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So we will have to add a term to the X and to the Y coordinate that's equal to the external force divided

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by them.

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Let me scroll back down.

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So this means we have to add something here and here and let me just go ahead.

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I will add here something with an amplitude of A0.

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So we have to specify this A0.

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So we could, for example, say, I don't know, let's let's keep it equal to four.

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Then we have to add here a term that could, for example, be patriotic in time.

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So I will add here and p dot sign and then write two times pi times t divided by some periods.

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So to oversee the isolation period.

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So this could, for example, be 50 seconds.

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So I want to keep it here a bit larger than the period of the oscillation, which is maybe, I don't

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know, five seconds.

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So here it's much larger period, so that we can nicely distinguish both of these periods and.

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Now I will do this at an angle, so I will write this down in terms of polar coordinates, I just write

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and pilot call sign of fire.

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And so I would be the angle along which the force is applied periodically.

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So fi is equal to all its, I don't know, 45 degrees.

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So that's the diagonal.

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Of course, we have to transform to radiance, so I divide my 180 degree and I multiply by PI.

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So this would be for the X coordinate and for the Y coordinates.

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We do a very similar thing.

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But due to the polar coordinates, it's now and pitot sign for the fire, but the rest is the same.

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And now we can run both of these cells and we see this is our trajectory.

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So it looks pretty complicated.

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But if we look at the time evolution of these coordinates, it becomes pretty clear what's happening

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in the beginning.

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We have some very brief oscillations.

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This is basically the same thing as this one here.

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So this is what give rise to this and optical spiral trajectory with a period of approximately five

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seconds that we can also see here.

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However, on top of this, we have another very strong oscillation, which happens on a much slower

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timescale.

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This corresponds to the external force that we have just implemented with an amplitude of four, which

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is quite strong, and we have the period t o c, which is equal to 50.

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This is exactly the period that we see here.

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And so you see in the beginning, it starts to spiral here.

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But due to our periodically changing force, in the end it will just be moving on this linear trajectory

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back and forth.

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And you can imagine this if you would take a bow in such a ball, put it here and kick it along this

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direction.

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But now additionally, you take the ball and you tilt it back and forth in this direction.

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Then the ball would first move a bit chaotically, but then after a bit of time, it would just move

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along with the direction in which you are tilting back and forth.

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And this is exactly what we see here, and I think this is really cool example because it's just such

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a simple and intuitive system.

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And still, it gives rise to such a complicated solution which would have never been able to get analytically.

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However, with our methods integrated solve on the IVP from non-paid module, this is pretty easy to

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establish because to be honest, the only thing that we have to do is to define such a function here.

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Where even this one was optional is just basically a single line of code, just this one.

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And then we would just have to call this command and the rest is really just plotting.

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So I think it's pretty easy, pretty useful and just a nice tool.

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And now, for the first time, we have solved a two dimensional system in a physical way.

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But as we have learned, this is a four dimensional system.

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Mathematically speaking and previously, we have already considered the one dimensional harmonic oscillator,

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which was a two dimensional system in terms of mathematics.

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This is at least if you would transform the second order differential equation to to first order differential

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equations, as we have done to be able to use this method.