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So in the previous lecture, we have solved the differential equation, and we have found that the first

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look at the result seems to be reasonable.

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So it makes sense what we get here.

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But also we didn't really see what's going on here in detail.

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We just see that basically Earth and Moon are both rotating on a circular trajectory around the Sun.

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So let's look at all three objects in detail.

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Let's start with the Sun.

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So I copy these three lines here, and we have now the trajectory of the sun and to see a bit more.

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Let me increase here to simulation time to 10 years.

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So let's rerun and let's rerun this.

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And you see, this is no the trajectory of the Sun.

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So the axes are a bit distorted.

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So here you have to multiply this with 10 to the power of seven.

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And here you see we have quantities and the magnitude of 10 to the power of five.

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So these are still some kilometers that the Sun is moving here.

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And at first, this may look surprising because we have to find that the Sun is starting at the center

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of our coordinate system and that the Sun does not have a velocity at the beginning.

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However, of course, the Sun is also affected by the other two objects, so actually it would be better

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if we would plot in the following all of our solutions with respect to the Sun.

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So basically subtracting here the values from the Sun and from the Moon.

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But still, you see this changes in the magnitude of 10 to the power of seven, and here we have magnitude

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tended to part of 11.

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So I will just neglect the small motion here and we will consider it basically to zero.

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But if we want to be precise, you would have to take these two values here and subtract them here from

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these values.

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So now let's go ahead and discuss the Earth.

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And first of all, I want to plot the motion

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basically like this without the Moon.

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And you see, I mean, it's pretty clear the blue trajectory was underneath the moon trajectory.

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So it's really circulating around the sun.

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But what we can also do is we can go one step further and we can plot something else.

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So let me add a new cell here, and I will plot now on the x axis the time T in here.

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And on the y axis, I will plot the distance, sun and Earth.

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So now I will basically plot the blue curve here, and I will plot for the X values solution, but T

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and since we want to plot it in years, I have to divide by this factor here, which are 60 seconds,

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60 minutes, 24 hours, 365 dots, two five days.

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So I go back here and divide by these values.

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So now we have here at a time, in years and for the y axis, I plotted the distance between Earth and

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Sun.

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So I right and p dots within Arc thought no.

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And now we must subtract it two vectors.

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So basically the position vectors.

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So this would be solution dot y and then the first three components for the sun and then minus solution

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thought y three to six for the Earth.

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And now when I run this, it gives me probably an error.

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Yes, because when we scroll down here, there are many error messages, but the lowest one is probably

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the one that's reasonable.

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The problem is that X and Y, which are our two lists, so the T list and the value list must have the

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same dimension.

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And the T list has, of course, 100000.

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And one point as we have defined and our value list here, this one only has a single point now.

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That's the problem of the total in our norm because it basically flattens all of these arrays.

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So these are now three components for one hundred thousand and one points of time, and then that calculates

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the norm of all of these components.

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So we must tell only calculate the norm with respect to these X, Y and Z components and do this forever

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point of time and then construct an area out of this.

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We can do this by writing axes equal to zero.

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If you would write X is equal to one, then we would get a three component vector here, where the norm

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is calculated with respect to all the points of time.

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So like this, I think should work.

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And indeed, we see now that we have to distance on the Earth and the time ten years and from zero to

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ten years, we see all of these fluctuations and obviously the periods of these fluctuations is one

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year.

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So our values are constants are correct and the algorithm is correct, and it takes one year for the

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Earth to move around the sun.

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So in our 10 years, we have a total of 10 installations, so now we can go ahead and discuss the Moon.

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And you see the trajectory of the Moon is the same as the Earth basically on this length scale.

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But of course, if we would zoom in, then we would see that the Earth surrounded the moon circles around

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the Earth.

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So I want to plot first the motion or the moon orbits around the Earth.

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So I basically what should I take, I think a copy this one and then changed his cell so that I don't

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have to start from zero here.

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So we have here a plot that has an aspect ratio of one.

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And here we have the coordinates X and Y and I just right here X of the Moon with respect to Earth and

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Y of Moon with respect to Earth.

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So we have now only a single plot, which we plot in green because we want to have the Moon and I want

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to plot it with respect to the Earth.

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So I write you minus solution called y comma or two component is three.

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And here are eight minus solution y component four.

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And this is what we get.

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So we see if we calculate the motion of the moon versus back to the Earth, it's once again a circular

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trajectory.

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However, the lines here are pretty thick.

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And this comes, of course, from many, many constellations here.

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And it turns out that this oscillation is not always on the same point.

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So maybe here or starting values were a bit inaccurate.

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Or maybe our accuracy of the RJ 45 was not high enough.

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So, yeah, we could, of course, improve here the method.

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But the general result stands, and I think it's reasonable that we have this circular motion of the

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Moon around the Earth, even though in the beginning of our simulation, the radius changes a bit so

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we can check this, of course.

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Maybe it has also a different origin.

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I'm just thinking now so we can check this.

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Maybe my expectation was wrong and it has a different origin.

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Let's see.

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I a copy this one from the Sun Earth distance and now we use distance.

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Earth Moon.

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So we can leave everything as it is, we just have to change these indices, and we take three six two

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six nine, which means Earth minus Moon and of course, in green.

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And yes, OK, I first had a wrong assumption.

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It is not ethical line because the radius changes, it is a thick line because the moon moves actually

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on an oval trajectory around the Earth, and this oval trajectory changes all of the time.

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Let me show you what I mean.

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So this is something I wanted to prepare anyway.

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What I want to do next is I want to plot once again this one here, but I want to exaggerate the basically

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the moon orbit radius.

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So it sounds a bit artificial and it is indeed artificial.

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But you will.

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I think you will realize in the second what I'm trying to do here.

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So I will go ahead and basically copy our starting plots.

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Where is it?

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Here it is.

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Our starting plot with all three objects, and I will delete this one, this was not the one I wanted.

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And now I will just change the line for the Moon and I will exaggerate the radius between Earth and

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Moon.

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So I will start with this one with the position of the Earth.

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And then in reference to this Earth Point, I will add the change of solution.

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Why?

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Sorry, thought y components six minus solution y component three the distance of the Moon to the Earth

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and the X component.

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And here we do the same thing for the light component.

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So this will be seven minus four.

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So basically, I start from the position of the Earth and then at the change or the distance between

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Earth and Moon so that I arrive at the Moon.

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So far, it will give us the same result as before.

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However, now I can add here eFFECTOR and I will add to your factor of 100, which artificially just

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for visualization increases the radius between Earth and Moon by a factor of 100.

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And you see, now this is what happens here.

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The moon moves around the Earth, and at the same time, the Earth moves around the Sun, so the moon

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will be shifted.

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And so you get this nice spiral trajectory.

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And then the other plot, this one here, basically, it looked as if there wasn't convergence reached

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yet or that there was some error, but it is just yeah, due to the spiral trajectory.

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OK, so that's pretty nice.

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I think even though, of course, this plot here is not really correct because I have artificially increased

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the radius and it doesn't follow know the correct laws of physics, but we can see how much better what's

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going on here.

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So that's just a nice trick for the visualization and for understanding what's going on.

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OK, so this was the whole first part.

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This the solution of the three body problem.

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We have introduced the equations of motion considering Sun, Earth and Moon, and we have 18 variables,

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18 differential equations, and we have pretty easily solved this.

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The only problem was that we had to tune here the tolerance, basically the accuracy of our methods

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because by using the starting values, we did not get the correct solution.

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The Earth was not moving on a circular trajectory, but was spiraling towards the sun and then crashing

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into the Sun.

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That would, of course, be a very bad simulation.

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However, it was pretty easy to just decrease the tolerance, so make the calculation more strict.

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And this slowed down, of course, the simulation.

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But in the end, we got a nice result, and we have analyzed the motion of the Sun, which is actually

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not fixed because it is pushed around also a bit by the Earth and by the Moon.

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And we have analyzed the motion of the Earth and the distance to the Sun.

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And you see, there's a nice period.

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So you see the Earth is moving a bit on an oval trajectory is slightly oval and the period is one year.

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Exactly.

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And then we have the same thing for the Moon, but with respect to the Earth, and we see here that

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it's also moving on an all the trajectory and that we can exaggerate the moon radius and then get such

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a nice plot as here.

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So now in the next lecture, we will go ahead and further complicate our problem and we will add here

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a fourth body, a rocket or spaceship.

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So stay tuned.

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It's a really exciting topic, and let's go ahead.