I've been playing around with some physics problems and trying to figure out things by myself, mostly just for fun. I would appreciate if someone corrected my thoughts and gave me some feedback. I'm not actually trying to solve the various problems I mention - I'm just trying to see if I can put state these problems in a somewhat formally correct way and put them in context of the mathematics I know.
Suppose $A$ and $B$ are objects, with $A$ having far larger mass than $B$, and suppose we can ignore other forces acting on these two objects. We can then describe the position of $B$ relative to $A$ at a certain time as a curve
$c \ \colon \langle 0, \infty \rangle \rightarrow \mathbb{R}^3$,
though for simplicity, I'll be working in $\mathbb{R}^2$.
First, consider a condition like $c'' = k c$, for $k$ negative. Not really a physical phenomena, since this is equivalent to something like "the magnitude of the force of $A$ acting on $B$ is a multiple of distance between $A$ and $B$". Still, solving the differential equation the solutions of this condition can be described as linear transformations of $ (\sin (x \sqrt{-k}), \cos(x \sqrt{-k}))$, so that seems quite neat.
Similarly, Newton's law of universal gravitation can here be expressed by the condition $c'' = -c/||c||^3$, and so solving this differential equation would describe all possible orbits/movements of $B$.