# N-body problem

1. ### LizardKing23

13
i am attempting to find a solution for the n-body problem, but i don't know the equations for gravity in three dimensions. if someone could post them for me, i would be most appreciative.

thank you

also, any advice as to how to approach this problem would be appreciated as well

2. ### LizardKing23

13
At the risk of sounding unintentionally arrogant, im going to assume that I need to be more specific when i say 'n-body problem'

The n-body problem is the supposedly unsolvable method of calculating the orbit of more than two bodies influencing each other through gravity. It is currently believed that this is not possible, and that a three-body system is unpredictable, not because of a lack of proficiency in our current math, but because math itself is unable to solve it. I believe this to be wrong, and am interested in attempting to solve the n-body problem. However, in order to start this, i need the equations for calculating three dimensional gravity stuff. you know, like F=G*m1*m2/r^2, except for three dimensions, with x, y and z axes. also any tips that could help me are welcomed.

thank you.

3. ### shyboy

138
Don't worry, you are not looking arrogant here, you are looking like pure mathematician.
The 3D law of gravity is

$$\vec{F_2}=-G\frac{m_1 m_2}{R_{12}^2}\frac{\vec R_{12}}{R_{12}}$$

The Force is directed along the line connecting the bodies and it is attractive.

or

$$\vec{F_2}=-G\frac{m_1 m_2}{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\frac{\vec R_{12}}{R_{12}}$$

Last edited: Apr 16, 2005
4. ### LizardKing23

13
Thank you very much, Shyboy.

Just to clarify, what do the arrows above F and R represent?

Last edited: Apr 16, 2005
5. ### quasar987

4,770
It means they are vector quantities.

I find shyboy has written the law in a peculiar manner. I would have written.. consider two point-masses m1 and m2 respectively, and respectively located by the position vectors $\vec{r_{1}}$ and [itex]\vec{r_{2}}[/tex]

$$\vec{F}_{1\rightarrow 2} = -\frac{Gm_1m_2 \vec{r_{12}}}{|\vec{r_{12}}^3|} = -\frac{Gm_1m_2 \hat{r_{12}}}{|\vec{r_{12}}^2|}$$

Where

$$\vec{r_{12}} = \vec{r_{2}} - \vec{r_{1}}$$

Or, in ugly cartesian coordinates...

$$\vec{F}_{1\rightarrow 2} = \frac{-Gm_1m_2}{\Left[(x_1-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2\Right]^{3/2}} \left( (x_2-x_1)\hat{x} + (y_2-y_1)\hat{y} + (z_2-z_1)\hat{z} \right)$$

6. ### LizardKing23

13
Haha I feel very ignorant here, but given my formal background in physics, i guess I am ignorant. What does the ^ above the R represent, as opposed to the vector arrows?

7. ### quasar987

4,770
It stands for "unit vector"

8. ### LizardKing23

13
so in that equation, the unit vectors represent direction only, since (X2-X1) and so on would represent magnitude. I think Ive got it now. thank you to everyone for your help.

9. ### whozum

Woudl this problem be similar to the Coulomb Force Law's inability to handle multiple moving charges?

10. ### Inquisiter

21
LizardKing23, what are you smoking? I want some too! Or is this some kind of a joke? Do you even know the solution to the two body problem? Hint: it is reducible to a problem of one body moving in a central potential, and the solution of the one body problem involves calculus.

Last edited: Apr 19, 2005