How do you differentiate this differential equation? [closed]

Question

I have to differentiate this equation (Gravitational force between N-Bodies)

$\begin{align} \frac{d^2}{dt^2}\vec{r_i}(t)=G \sum_{k=1}^{n} \frac {m_k(\vec{r}_k(t)-\vec{r}_i(t))} {\lvert\vec{r}_k(t)-\vec{r}_k(t)\rvert^3} \end{align}$

where $\vec{r_{i/k}}(t)$ is the position of a body in 3D space and $m_{i/k}$ is its mass

How would you calculate $\frac{d^3}{dt^3}\vec{r_i}(t), \frac{d^4}{dt^4}\vec{r_i}(t) ...$?

Edit: I know that solving for $\vec{r_i}(t)$ is very complicated but is that also the case for the third derivative of $\vec{r_i}(t)$? I'm asking because since $\vec{r_i}(0)$ and $\frac{d}{dt}\vec{r_i}(0)$ are given (and therefore $\frac{d^2}{dt^2}\vec{r_i}(0)$ is also known) one could make a Taylor series with $\frac{d^3}{dt^3}\vec{r_i}(t), \frac{d^4}{dt^4}\vec{r_i}(t)$ and so on

Answer 1

The right hand side of your equation is an explicit function of $r_{i}$ and the $r_{k}$s. One just needs to apply the chain and product rules from introductory calculus. The expression will be super-ugly, and will have explicit $\dot r$'s in it, but I don't see what difficulty you're having with the computation.

Perhaps the only complication is that your denominator is really:

$$\left[\left({\vec r}_{i} - {\vec r}_{k}\right)\cdot\left({\vec r}_{i} - {\vec r}_{k}\right)\right]^{3/2}$$

Answer 2

Even for three bodies it's really complicated

https://en.wikipedia.org/wiki/Three-body_problem

Unless someone else knows different, it would seem like a job for computer simulation.