# Solutions to Kepler problem/Motion in central field

It is clear that particals in a repulsive central field given by a point mass at the origin are moving along hyperbolas, e.g. given by the expression ${x^2 \over a^2} - {y^2 \over b^2} = 1$ (after a change of coordinates). This also applies to three dimensions since the motion takes place in a plane due to conservation of angular momentum.

On the other hand, in three dimensions, the motion satisfies the initial value problem $$\left\{\begin{array}{l} {d^2X \over dt^2} = C {X \over |X|^3}, \\ X(0) = x, \dot X(0) = v \end{array}\right.$$ for some given initial position and velocity $(x,v) \in \mathbb{R}^6$.

Unfortunately, while I can derive the trajectories, I have no idea, what can be said about the speed at which the particle moves at a certain time. In particular, I am interested in the quantitative behavior of $|V| = |\dot X|$ for given $x$ and $v$.

Are there any expansions or explicit expressions for the solution $X$ of the above initial value problem? Or is there any expansion or explicit expression for the absolute velocity $|V|$?

EDIT By conservation of energy we know that $$\frac{1}{2}|V|^2 = \frac{1}{2}|v|^2 + C\left(\frac{1}{|x|} - \frac{1}{|X|}\right).$$ On the other hand, the behaviour of $\frac{1}{|X|}$ is well understood according to https://en.wikipedia.org/wiki/Kepler_problem#Solution_of_the_Kepler_problem $$\frac{1}{r} = -\frac{km}{L^{2}} \left[ 1 + e \cos \left( \theta - \theta_{0}\right) \right].$$ Actually, I don't quite understand how $\theta$ can be replaced by a time dependent expression and how the constants depend on the initial data $x$ and $v$...

Putting $k m/ L^2 = 1/p$ where p is semi-latus rectum, the last equation becomes

$$\frac{p}{r} = \left[ 1 + e \cos \left( \theta - \theta_{0}\right) \right]$$

in which $\theta$ is the polar coordinate for classical polar conic equation.If $\theta_0$ is is non-zero, the conic is rotated so that major axis would not be along x- or y- axis.

You would notice that certain values of L,m k and total energy E decide eccentricity e. It will be found that high energy orbits force the orbit along non-returning hyperbolas.

Notice also that this same Newtonian formulation can result in hyperbolas, no need to have a separate repulsive differential equation. e =2 for hyperbolic orbits are sketched in:

https://en.wikipedia.org/wiki/Kepler_orbit

EDIT1:

In order to change to independent time variable from $\theta$ independent variable,

$$u''+ u = -1/p, \theta^{'} = L u^2$$

the latter part expresses conservation of angular momentum.

• I don't know, how this answers my question. It's still mysterious, how the constants depend on $x$ and $v$ and we still only have a representation depending on $\theta$, while I'm interested in the evolution in time. Mar 6, 2015 at 8:38
• Actually the set of derivatives of $x , v$ and the dynamic relations involve the constants. Mar 6, 2015 at 9:41
• It seems to me like the derivation of this solution in en.wikipedia.org/wiki/… is nonsense (it is asserted that $\omega$ is a conserved quantity). But the derivation on en.wikipedia.org/wiki/Kepler_orbit looks good. However, I still don't understand, how you get an expression in the time variable from this. Mar 8, 2015 at 17:34
• @thomas: Please see carefully it says that $m \omega r^2 = L$, the angular momentum is conserved. There is no neat closed form expression for $r = f(t)$ existing, may be possible in elliptic integrals,let me check.. The ellipse $r- \theta$ relation is in closed form of course. $r'' + \mu/r^2 - h^2 /r^3 = 0$ is the radial ODE to be solved for $r-time$ relation. Mar 8, 2015 at 18:31
• Oh, I see. You are right. I understood the $\equiv$-sign as "equals constantly", but they seem to use it in the sense "defined as" (which I usually denote by $:=$). So, possibly there is no expansion for $r=|X|$ in terms of $t$. But my question was actually targeted at some expansion of $|V|$ in terms of $t$, possibly only an estimate. I had some hope that at least that would be possible. Mar 8, 2015 at 23:02