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So now we come to a very famous system of differential equations.

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This is called The Three-Body Problem.

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So here we will discuss how certain objects in space move, and we will discuss three of these objects

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at the same time.

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And as an example, we will consider the motion of the Sun, the Earth and the Moon all at the same

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time.

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It turns out if you would have only two of these objects, for example, only the Sun and the Earth,

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it is not so difficult to solve this.

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You can even do it analytically, even though it's a bit complicated.

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However, as soon as you add a third body or even a fourth and the fifth body, then it becomes impossible

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to solved analytically.

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So we need numerical methods, and solving such problems is, of course, extremely relevant.

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For example, in astronomy and the scientists and the engineers at NASA, for example, solve such equations

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all the time to predict the motion of certain objects in space and even of rocket ships.

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And yeah, there are missions to the Moon, for example.

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You can calculate all of these things by solving the differential equations that we will discuss soon.

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And at the end of this lecture, we will even plan a trip to the Moon and we will.

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I will show you how we can get in a stable orbit around the Moon.

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So let's get started with the three body problem and with our particular example.

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So here, if prepared in your template file, already some constants that we are going to use.

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So we will deal here with the actual numbers.

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So I've taken those four from Wikipedia mostly.

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So we have you the gravitational constant, the masses of these objects, and then we have the average

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values of their distances and their velocities.

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So of course, these will change a bit and we will see this when we solve the equations.

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But these values should just serve as a starting point and as the starting conditions follow us simulations.

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So later on, we don't have to use these numbers anymore.

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We can just use the constants that I have introduced to.

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So not just come to the differential equations.

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So first of all, let me tell you that we need quite a lot of differential equations since we have a

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quite a lot of variables.

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So for each of the three bodies, we have the coordinates X, Y and Z.

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And since we will solve an equation of motion which describes the force acting on the object.

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So this is the mass times, the velocity, sorry, the mass times, the acceleration, of course, and

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the acceleration is a second or the derivative.

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So it's a second order differential equations.

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So this means, as we have learned, we do not only need to the constants here to the coordinates X,

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Y and Z, but also the velocities we x y and we see we need those for all three bodies.

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So in total, we have 18 variables.

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And if we take them later on the fourth body into account, we have even more of them.

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However, the differential equations all look very similar.

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So pleased to be afraid.

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Here are quite a lot of equations, but they all have the same shape and they are just sums of different

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terms, and it all comes down to the force law or the law.

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Newton's law of Gravitation.

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So say you have two objects, which of course have a mass and are positioned at some point in space.

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Then they will attract each other due to the mass.

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And the force that corresponds to this is this one here.

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So that's the gravitational constant, then we have the mass of the two objects, then here at a distance

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in space, and this is basically just the direction.

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So if you say you have one reference object could be, for example, the Sun in the center of our system

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than you have at that position are zero.

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And you have the mass of MS 04 The Sun, and then you can calculate the force between the Sun and every

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object in space by using here end of the mass of the object that you want to consider and are the position

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vector.

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And if you say that R0 is in the center at zero, then you just get this equation here, or you could

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would rather right here and zero.

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So you see, it's one of our square dependence, and here it is.

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It's just one of our three because this one here has two dimension of a length.

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So we will use this form here and we will add up the forces between the bodies.

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So if we have three bodies, then we have three equations.

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So this will be the force and times.

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The second all a time derivative of the position vector and for the first object is is given by the

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gravitational force between object one and two and object one and three, then four object two.

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We have two terms, as is the interaction between one two and two three and four object three.

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We have the interaction between one three and two three.

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So we have two terms for all of these equations here.

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And you see this is here a vector.

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So this contains actually three equations.

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So X, Y and Z coordinate.

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And here also for all of these vectors, we have X, Y and Z coordinates.

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So what I want to do now is I want to multiply these brackets here so that we get four terms in total.

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And then I want to order them by hour one or two or three and this is what I have done here.

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And also, I have divided by the mass because you see we have here and one and one and one, so we can

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just divide by the mass and then we get these three equations.

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And you see here the total of four turns and you see four object one.

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We have two terms with the fact of our one, a single term for our two and our three.

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And then the other two equations are very similar.

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So I recommend that you post the video now or maybe for half a minute and look at the equations.

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But I think it's pretty clear what I did here.

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I just took this law and then considered these two terms for all of these objects and then just multiplied

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and ordered the terms.