I got a question that’s probably very basic but I just can’t figure it out.
I want to derive a system of second order differential equations that describes the motion of a planet in (x,y)-coordinates (cartesian coordinates). I’ve seen ways of doing this by switching to polar coordinates but I want to know how to derive it without switching to polar coordinates. I don’t know where to start.
If we have the following equations: $$ F = \frac{-GMm}{r^2} \text{ and } F = ma $$
We can combine these two and then write it as $$ a = \frac{-GM}{r^2} $$ Where $r = \sqrt{x^2+y^2}$ which will give us $$ a = -\frac{GM}{x^2+y^2} $$
But this doesn’t seem right.
If anyone could help me figure out how to derive this so that I get a system of second order differential equations I would be very grateful.
Thanks!