Working off my previous question, I think I've finally managed to describe what I'm wondering about in a quite clear, non-vague way.
Suppose we have an object $B$ tied to an object $A$ by a rope, with the mass of $A$ far larger than $B$, so as far as figuring out how $B$ moves with respect to $A$, we can essentially ignore all the forces besides the one of the rope acting on $B$. Suppose that in addition we can also shorten/pull the rope as much as we like.
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1 A side question - I suppose at this point, we don't even have to think of there being a rope - there's two objects, and we can change the force between to any arbitrary amount (but it's always an attractive force), is that right? That sounds like a pretty much equivalent way of describing this situation.
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Suppose we can parametrize the movement of $B$ with respect to $A$ by
$c(t) = r(t) (\cos(\varphi(t)),\sin(\varphi(t)))$,
and similarly we can describe the force acting on $B$ as
$c''(t) = k(t) (\cos(\varphi(t)),\sin(\varphi(t)))$,
where $k(t)<0$.
2 Now suppose we know the values of $r'(t)$ and for simplicity assume $r'$ is constant. Notice that $r' = c' \cdot c$. Suppose we also know $c(0)$ and $c'(0)$ (the possible value of $c'(0)$ is restricted by $r'(0)$, so we assume this is satisfied - i.e. $c' \cdot c = r'$ is satisfied), but we don't know any other values of $c$ or $c'$, and we wish to approximate $c$.
We could solve the differential equation corresponding to this problem, but suppose we don't really want to do that for whatever reason - maybe we don't really know $r'(t)$ for $t$ outside of $\langle 0,a \rangle$ for some $a$, and we would like to keep approximating $c(k \delta)$ for $k>0$ as we receive these values.
The obvious way of approximating $c$ is to simply take the first degree approximation of $c(0) + t c'(0)$. For the approximation of $c(1 \delta)$ we already know the necessary values of $c(0)$ and $c'(0)$, but what about the approximation of $c(k\delta)$ for $k>1$? We know $c((k-1)\delta)$, but how do we approximate $c'((k-1)\delta)$? Consider the case $k=2$.
We could approximate $c'(1 \delta)$ the following way : we take the velocity $c'( 0 \delta)$ and we add a certain multiple of $c(1\delta)$ to it, so that $(c'(0) + s c(1 \delta)) \cdot c(1 \delta) = r'(1 \delta)$. We basically add just enough of $c(1\delta)$ to $c'(0 \delta)$ so that this approximation satisfies $c'(1\delta) \cdot c(1 \delta) = r'(1\delta)$.
Intuitively, I think of this as "the velocity is the same as in the previous point, but the force acting on the point changes the velocity in the direction of the central object $A$ just enough so that $r'$ has the value it's supposed to have".
This works pretty well from what I've tried, and this approximation is even a good approximation in terms of it being a good Taylor approximation - this $c'(1\delta)$ approximation $approx(1\delta)$ can be formally described as:
$approx(1 \delta) = c'(0) + [ r'(1\delta) - (c'(0) \cdot c(1 \delta))] c(1 \delta)$,
and
$\dfrac{[ r'(1\delta) - (c'(0) \cdot c(1 \delta))] c(1 \delta)}{\delta} = \dfrac{ c(1\delta) \cdot ( c'(1\delta) - c(0))}{\delta} c(1\delta)$,
where $\dfrac{ c(1\delta) \cdot ( c'(1\delta) - c(0))}{\delta} \rightarrow c''(0)$,
so
$\dfrac{ c'(1 \delta) - approx(1\delta)} {\delta} \rightarrow 0$.
However I'm not convinced of it being true on an intuitive level, and I feel like I'm missing some very simple explanation of why this approximation works well. I can see some intuition why it should work as I've mentioned above, but I don't really see it as very convincing. To some extent it seems like it couldn't be any other way - if $r'$ is set, well of course the change of velocity will be a an addition of $c(1\delta)$ multiple so that $r' = c' \cdot c$ - the difference in velocity has to be a multiple of $c(1\delta)$ because that's the way the force is pointing, and so it has to be exactly this multiple and no other.
Still, it feels like it might be simpler than I'm making it - in some sense I would expect that this problem could be solved from a more "velocity" based perspective of some type - i.e. the object has an initial velocity, and I know its velocity in a certain direction $D$ at any time, and that there is only one force acting and it's in the same direction $D$ - therefore... something? Maybe I'm looking for a general solution for problems with those sort of initial conditions... but I don't really know.
Perhaps my own explanation is already good enough - in this case, I would appreciate if someone assured me that what I've written seems correct etc.
