Circular restricted three body problem

[Deadstar]

Hey folks and happy new year!

I've been learning about the 3 body problem for a wee while now and also lagrange equilibrium points and it has got me experimenting with the same set up. I'm now investigating halo orbits and while I know you don't calculate them using this kind of method, it has given me some ideas to try out. What I've been looking at is particles crossing the x-axis perpendicular to it (as it 'appears' in pictures of halo oribts, which is where I got the idea).

Now for lagrange points, I use the (perhaps standard?) method of solving the equations of motion...

$$\ddot{x} - 2 \.{y} = \frac{\partial U}{\partial x}$$

$$\ddot{y} + 2 \.{x} = \frac{\partial U}{\partial y}$$ (#)

Where $$U = \mu_1 (\frac{1}{r_1} + \frac{r_1^2}{2}) + \mu_2 (\frac{1}{r_2} + \frac{r_2^2}{2}) - \frac{1}{2}\mu_1 \mu_2$$ with n=1.

Setting $$\.{x} = \.{y} = \ddot{x} = \ddot{y} = 0$$.

So, now dealing with the orbits crossing the x-axis for the Earth sun 3 body system... Can I just use a method similar to the above to find the positions of these orbits on the x-axis given a y velocity..?

Does it even make sense to solve this since could we not have a particle going a certain speed at almost any point on the x-axis..?

I've been playing around and not really knowing what I've been doing but... Given that I want the particle to cross the x-axis, y will be zero and in (#), we will have to have the RHS equal to zero and hence $$\ddot{y} = -2 \.{x}$$ which seems a weird thing to presume especially given that I would have had the x velocity equal to zero but the y accel non zero! So then it's just a case of finding an expression for (in my calcs, r1) in terms of x acceleration and y velocity.

Now, if you've understood what I've been rambling about, does what I'm doing even make any sense? I can't help but feel everything is riddled with mistakes but as a mathematician I tend to not consider if what I'm doing makes physical sense and just look at what the numbers tell me...

One more quick thing to ask... I always read about people performing numerical integrations on computers for n-body problems but does this mean actually solving an integral or is it just an expression for performing numerical calculations..?

 

[gneill]

One more quick thing to ask... I always read about people performing numerical integrations on computers for n-body problems but does this mean actually solving an integral or is it just an expression for performing numerical calculations..?

It's performing numerical calculations to 'follow' the trajectory (in the mathematical sense) of a system of differential equations, thus producing a numerical solution for the system. In the case of celestial mechanics, it can produce position, velocity, acceleration versus time numbers all along the trajectory (or just the 'final' position, etc., for a specified time).

Differential equations are 'integrated' to find their solution. Hence numerical integration.