- #1
Deadstar
- 99
- 0
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...
[tex]\ddot{x} - 2 \.{y} = \frac{\partial U}{\partial x}[/tex]
[tex]\ddot{y} + 2 \.{x} = \frac{\partial U}{\partial y}[/tex] (#)
Where [tex]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[/tex] with n=1.
Setting [tex]\.{x} = \.{y} = \ddot{x} = \ddot{y} = 0[/tex].
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 [tex]\ddot{y} = -2 \.{x}[/tex] 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..?
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...
[tex]\ddot{x} - 2 \.{y} = \frac{\partial U}{\partial x}[/tex]
[tex]\ddot{y} + 2 \.{x} = \frac{\partial U}{\partial y}[/tex] (#)
Where [tex]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[/tex] with n=1.
Setting [tex]\.{x} = \.{y} = \ddot{x} = \ddot{y} = 0[/tex].
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 [tex]\ddot{y} = -2 \.{x}[/tex] 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..?