- #1

greg_rack

Gold Member

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I have started studying differential equations on my own, sticking to my last high-school year's textbook, along with a few physics applications of ODEs.

Online I came across the n-body problem, which then took me to the basic two-body problem!

I'm here to ask you a few things about the two-body vector differential equation of motion:

$$\ddot{\textbf{r}}+\frac{\mu}{r^3}\textbf{r}=0$$

First: which type of DE is it? My textbook covers(approximately) linear DEs, explaining just a few resolution methods for first and second order homogeneous and non-homogeneous ones.

Is this non-linear, and so not solvable with "usual" and easier methods? If yes, why is it so?

Second: the textbook I found says: "

*though this equation looks easy, its complete solution is not*".

Why?

Lastly, after the aforementioned consideration, it provides the steps to find a "

*partial solution that will tell us the size and shape of the orbit*", which is a function ##r##(rel. position of the two bodies) of the form of a conic section: this "partial solution" thus tells us -only- the shape and size of the orbit, but doesn't provide us with a function of time for the movement.

How is that? What is a "partial solution", and what would be needed to find a "complete" one?

Online resources look all very confusing and over-complicated, and I'm having a really hard time deciphering those... hope you can help me, and my questions aren't too silly :)