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So I am doing a Presentation on the $N$-Body Problem for my Physics final (12th grade) and the main content of the Presentation is going to be the comparison between an analytical and a numerical approach to solve the problem.

I chose the "$N$-Body by Taylor series" method as an analytical solution (since the Sundman method seemed too advanced). However I have not been able to find out if the Taylor series converges or not. I have read that the Taylor series solution too the problems has no real life applications because it normally only works in small time frames. Is it because the function doesnt converge or just because its unpractical working with infinitely many terms?

Keep in mind that I am not an expert on the subject of differential equations and Taylor Series

Edit: So the differential equation I have to solve is: $\begin{align} \frac{d^2}{dt^2}\vec{r_i}(t)=G \sum_{k=1}^{n} \frac {m_k(\vec{r}_k(t)-\vec{r}_i(t))} {\lvert\vec{r}_k(t)-\vec{r}_i(t)\rvert^3} \end{align}$

where: $\vec{r_i}(0)$ and $\frac{d}{dt}\vec{r_i}(0)$ are given therefore $\frac{d^2}{dt^2}\vec{r_i}(0)$ is also known

so the taylor series would be: $\begin{align} \sum_{n=0}^{\infty} \frac{1}{n!} \cdot \frac{d^n}{dt^n}\vec{r_i}(0) \cdot t^n \end{align}$ (correct me if I'm wrong)

Eli
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SIMONE ESPOSITO
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1 Answers1

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This answer may be a bit late for your final project, but it was fun to research.

Spacetime is a smooth manifold. This is not enough to guarantee analyticity. But it just so happens that spacetime also possesses an analytic structure (see Smooth vs analytic spacetimes), so it is an analytic manifold, so the taylor series of $\vec{r}(t)$ is equal to its Taylor Series for all time $t \geq 0$.

So the Taylor Series is equal to $\vec{r}(t)$, but its convergence is very inefficient. Using the Taylor Series, only short time scales can be practically implemented.

Hypernova
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