So now we come to a very famous system of differential equations.
This is called The Three-Body Problem.
So here we will discuss how certain objects in space move, and we will discuss three of these objects
at the same time.
And as an example, we will consider the motion of the Sun, the Earth and the Moon all at the same
time.
It turns out if you would have only two of these objects, for example, only the Sun and the Earth,
it is not so difficult to solve this.
You can even do it analytically, even though it's a bit complicated.
However, as soon as you add a third body or even a fourth and the fifth body, then it becomes impossible
to solved analytically.
So we need numerical methods, and solving such problems is, of course, extremely relevant.
For example, in astronomy and the scientists and the engineers at NASA, for example, solve such equations
all the time to predict the motion of certain objects in space and even of rocket ships.
And yeah, there are missions to the Moon, for example.
You can calculate all of these things by solving the differential equations that we will discuss soon.
And at the end of this lecture, we will even plan a trip to the Moon and we will.
I will show you how we can get in a stable orbit around the Moon.
So let's get started with the three body problem and with our particular example.
So here, if prepared in your template file, already some constants that we are going to use.
So we will deal here with the actual numbers.
So I've taken those four from Wikipedia mostly.
So we have you the gravitational constant, the masses of these objects, and then we have the average
values of their distances and their velocities.
So of course, these will change a bit and we will see this when we solve the equations.
But these values should just serve as a starting point and as the starting conditions follow us simulations.
So later on, we don't have to use these numbers anymore.
We can just use the constants that I have introduced to.
So not just come to the differential equations.
So first of all, let me tell you that we need quite a lot of differential equations since we have a
quite a lot of variables.
So for each of the three bodies, we have the coordinates X, Y and Z.
And since we will solve an equation of motion which describes the force acting on the object.
So this is the mass times, the velocity, sorry, the mass times, the acceleration, of course, and
the acceleration is a second or the derivative.
So it's a second order differential equations.
So this means, as we have learned, we do not only need to the constants here to the coordinates X,
Y and Z, but also the velocities we x y and we see we need those for all three bodies.
So in total, we have 18 variables.
And if we take them later on the fourth body into account, we have even more of them.
However, the differential equations all look very similar.
So pleased to be afraid.
Here are quite a lot of equations, but they all have the same shape and they are just sums of different
terms, and it all comes down to the force law or the law.
Newton's law of Gravitation.
So say you have two objects, which of course have a mass and are positioned at some point in space.
Then they will attract each other due to the mass.
And the force that corresponds to this is this one here.
So that's the gravitational constant, then we have the mass of the two objects, then here at a distance
in space, and this is basically just the direction.
So if you say you have one reference object could be, for example, the Sun in the center of our system
than you have at that position are zero.
And you have the mass of MS 04 The Sun, and then you can calculate the force between the Sun and every
object in space by using here end of the mass of the object that you want to consider and are the position
vector.
And if you say that R0 is in the center at zero, then you just get this equation here, or you could
would rather right here and zero.
So you see, it's one of our square dependence, and here it is.
It's just one of our three because this one here has two dimension of a length.
So we will use this form here and we will add up the forces between the bodies.
So if we have three bodies, then we have three equations.
So this will be the force and times.
The second all a time derivative of the position vector and for the first object is is given by the
gravitational force between object one and two and object one and three, then four object two.
We have two terms, as is the interaction between one two and two three and four object three.
We have the interaction between one three and two three.
So we have two terms for all of these equations here.
And you see this is here a vector.
So this contains actually three equations.
So X, Y and Z coordinate.
And here also for all of these vectors, we have X, Y and Z coordinates.
So what I want to do now is I want to multiply these brackets here so that we get four terms in total.
And then I want to order them by hour one or two or three and this is what I have done here.
And also, I have divided by the mass because you see we have here and one and one and one, so we can
just divide by the mass and then we get these three equations.
And you see here the total of four turns and you see four object one.
We have two terms with the fact of our one, a single term for our two and our three.
And then the other two equations are very similar.
So I recommend that you post the video now or maybe for half a minute and look at the equations.
But I think it's pretty clear what I did here.
I just took this law and then considered these two terms for all of these objects and then just multiplied
and ordered the terms.