# Help Needed: Solving the N-Body Problem in 3 Dimensions

• LizardKing23
In summary, the conversation revolves around finding a solution for the n-body problem, specifically in regards to the equations for gravity in three dimensions. The n-body problem involves calculating the orbit of more than two bodies influencing each other through gravity, and it is currently believed to be unsolvable. The equations for calculating three dimensional gravity are presented, along with clarification on their components. The conversation also discusses the similarities to the Coulomb Force Law and the solution to the two body problem.
LizardKing23
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

At the risk of sounding unintentionally arrogant, I am 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.

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

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

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Thank you very much, Shyboy.

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

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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...

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

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?

It stands for "unit vector"

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

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

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.

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## 1. What is the N-Body Problem in 3 Dimensions?

The N-Body Problem in 3 Dimensions is a mathematical problem that involves predicting the motion of a group of particles in a three-dimensional space, taking into account the gravitational forces between them.

## 2. Why is solving the N-Body Problem in 3 Dimensions important?

Solving the N-Body Problem in 3 Dimensions is important in various fields such as astrophysics, aerospace engineering, and computer graphics. It helps understand the motion and interactions of particles in a three-dimensional space, which has practical applications in predicting the movement of celestial bodies, designing spacecraft trajectories, and creating realistic simulations.

## 3. What are the challenges in solving the N-Body Problem in 3 Dimensions?

The main challenge in solving the N-Body Problem in 3 Dimensions is the computational complexity. As the number of particles increases, the number of calculations needed to predict their motion also increases exponentially, making it a computationally intensive task. Additionally, the problem is a non-linear and chaotic system, which makes it challenging to find exact solutions.

## 4. How have scientists attempted to solve the N-Body Problem in 3 Dimensions?

Scientists have used various techniques to solve the N-Body Problem in 3 Dimensions, such as numerical integration methods, particle-mesh methods, and direct summation methods. Each method has its advantages and disadvantages, and researchers continue to develop new algorithms and approaches to improve the accuracy and efficiency of solving this problem.

## 5. What are the potential future developments in solving the N-Body Problem in 3 Dimensions?

With the increasing power of computers, scientists are exploring the use of machine learning algorithms and artificial intelligence techniques to solve the N-Body Problem in 3 Dimensions. These approaches have shown promising results in predicting the motion of particles and could potentially lead to more efficient and accurate solutions in the future.

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