As for the three body problem (the formula used, and the related areas)

[julypraise]

Could you let me know which formula is Newton's Gravitation Law used for the three body or n body problem in general?

Suppose there are $n$ objects with the masses $m_{j}$, $j=1,2,3,\dots,n$ and the displacement functions $\mathbf{x}_{j}:\mathbb{R}\to\mathbb{R}^{3}$ with initial conditions of $\mathbf{x}_{j}(0),\dot{\mathbf{x}}_{j}(0)$. Then is the formula

(1) $m_{j}\ddot{\mathbf{x}}_{j}=G\sum_{i\neq j}\frac{m_{i}m_{j}(\mathbf{x}_{i}-\mathbf{x}_{j})}{\left|\mathbf{x}_{i}-\mathbf{x}_{j}\right|^{3}}$

used, or

(2) $m_{j}\ddot{\mathbf{x}}_{j}=G\sum_{i\neq j}\frac{m_{i}m_{j}}{\left|\mathbf{x}_{i}-\mathbf{x}_{j}\right|^{2}}$

used?

If the trend is to use (1), then why is it? And what is the trend in defining the formula of Newton's Gravitation Law when $\left|\mathbf{x}_{i}-\mathbf{x}_{j}\right|=0$?

And is there any textbook (kind graduate or undergraudate textbook level) that teaches this area not by analytical method but by algebraic method, especially focusing on the concept of symmetries? Or should I just find papers to study this area in such a view?

And could you let me know the (mathematical) areas (specifically the names of the areas) that are closely related to this problem?

 

[DrStupid]

(2) is limited to one-dimensional cases and for $\left|\mathbf{x}_{i}-\mathbf{x}_{j}\right|=0$ there is no force

 

[Hassan2]

Could you let me know which formula is Newton's Gravitation Law used for the three body or n body problem in general?

Suppose there are $n$ objects with the masses $m_{j}$, $j=1,2,3,\dots,n$ and the displacement functions $\mathbf{x}_{j}:\mathbb{R}\to\mathbb{R}^{3}$ with initial conditions of $\mathbf{x}_{j}(0),\dot{\mathbf{x}}_{j}(0)$. Then is the formula

(1) $m_{j}\ddot{\mathbf{x}}_{j}=G\sum_{i\neq j}\frac{m_{i}m_{j}(\mathbf{x}_{i}-\mathbf{x}_{j})}{\left|\mathbf{x}_{i}-\mathbf{x}_{j}\right|^{3}}$

used, or

(2) $m_{j}\ddot{\mathbf{x}}_{j}=G\sum_{i\neq j}\frac{m_{i}m_{j}}{\left|\mathbf{x}_{i}-\mathbf{x}_{j}\right|^{2}}$

used?

If the trend is to use (1), then why is it? And what is the trend in defining the formula of Newton's Gravitation Law when $\left|\mathbf{x}_{i}-\mathbf{x}_{j}\right|=0$?

And is there any textbook (kind graduate or undergraudate textbook level) that teaches this area not by analytical method but by algebraic method, especially focusing on the concept of symmetries? Or should I just find papers to study this area in such a view?

And could you let me know the (mathematical) areas (specifically the names of the areas) that are closely related to this problem?

in formula 2, something is missing ( a unit vector along x_i-x_j) because the summation must be a vector. If you correct it, both formulae become the same becaues the unit vector = (x_i-x_j)/|x_i-x_j|