Some questions concerning orbits of objects [closed]

Question

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 $B$.

Answer 1

It pretty much makes good sense to me, but i give you some feedback you asked for:

1.) why the parametrisation starts at 0? In newtonian gravity, since you assume (i guess) that $c''$ is acceleration, the parametrisation should be the absolute time $t$ that has no boundaries.

2.) The equation $c′′=−c/||c||^3$ seems to me little strange. You are equiting vector (acceleration) to a point of Euclidean space. Of course the Eclidean space has structure of vector space, but it depends on choice of origin of your frame (the structure is given by displacement of the point from some fixed point). It would make much more sense to write: $$c′′=−\frac{c(t)-P}{||c(t)-P||^3},$$ where P is the center of the force.

3.) Your approach assumes structure of Euclidean space. If you would like to generalize it to some other manifold, you would need to start thinking more locally and abstractly. F.e. $c(t)-P$ would stop making any sense. Also taking second derivatives of a curve would not be as straighforward as in Euclidean space. $c'$ defines tangent vector to the curve at every point of the curve $v(t)$. But taking derivative of this $v(t)$ would assume you somehow compare two vectors at two different points ffor which you need to transport at least one of them to common location. That can be done if we have connection, which luckily we do have.

Just a food for thought...

4.) Newtonian gravity in the abstract language of differential geometry is shown in the book Gravitation by Misner, Wheeler and Thorne. I don't know if you wish to look into textbook since you are trying to do figure out things for yourself, but it is interesting reading.