Deriving a second-order system of differential equations that describes the motion of a planet (in cartesian coordinates)

Question

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!

Answer 1

The one part you've missed is that $a$ is actually a vector - typically you'd write the DE as

$$\frac{d^2 \mathbf{r}}{dt^2} = - \frac{GM}{r^2} \hat{\mathbf{r}}$$

where $\mathbf{r} = (x, y)$ is the position vector, $r = |\mathbf{r}|$ is its magnitude, and $\hat{\mathbf{r}} = \frac{\mathbf{r}}{r}$ is a unit vector pointing in the same direction.

Then the only thing left you need to write this entirely in terms of $x$ and $y$ is to substitute $r = \sqrt{x^2 + y^2}$.