# Falling rotating object in higher order potential fields

For which $n$ would an object with a non-zero rotation fall to the center of this field?

$$\alpha >0\\ V(r) = \frac{\alpha}{r^n}$$

(Apparently it should never touch the center if it has non-zero rotation and $n=1$). I am totally stumped, as I cant see why the order should change whether or not it falls into the center. Any ideas/explanations would be greatly appreciated.

• I assume that $x$ is a position vector in 3D and $r=\|x\|$, the magnitude of the position? – fibonatic Oct 4 '16 at 12:00
• Sorry, I fixed it. I dazed out as I wrote it. Thanks for the reminder. – Blitz Oct 6 '16 at 10:10
• So the problem is one-dimensional? – fibonatic Oct 6 '16 at 11:30
• Only in polar coordinates. In cartesian coordinates it's not. – Blitz Oct 6 '16 at 12:35

Consider an object with non-zero rotation, and velocity vector $\bf{v}$. That it has a non-zero rotation means that, at least as long as $r \neq 0$, $\textbf{v}\times\textbf{r} = rv_\perp \neq 0$, where $\bf{r}$ is the radial position vector and $v_\perp$ is the component of $\bf v$ perpendicular to $\bf r$. As a simple consequence of conservation of angular momentum we must have $$v_\perp \propto \frac{1}{r}.$$ This imposes $$\dot{v}_\perp = \frac{dv_\perp}{dt} = \frac{dv_\perp}{dr}\frac{dr}{dt} \propto \frac{1}{r^2}v_r.$$ On the other hand the potential \begin{align} V(\textbf{r}) &\propto \frac{1}{r^n} \end{align} imposes an an acceleration $$\textbf{a} \propto -\nabla V \propto \frac{1}{r^{n+1}}\textbf{e}_r,$$ whence $\dot{v}_r = |\textbf{a}|$. We thus have $$\frac{\dot{v}_\perp}{\dot{v}_r} \propto r^{n-1}v_r.$$ For $n > 1$ we have \begin{align} \lim_{r\to 0}\frac{\dot{v}_\perp}{\dot{v}_r} = 0. \end{align} We can interpret this to mean that as the object is brought closer to the center the acceleration gradually becomes purely radial. If the potential acts attractively this means that the object may eventually collide with the center. However, for $n = 1$ we have \begin{align} \lim_{r\to 0} \frac{\dot{v}_\perp}{\dot{v}_r} \propto \lim_{r\to 0} v_r \neq 0, \end{align} which means that even when the object is brought closer to the center the acceleration will never become radial: the object will never accelerate directly towards (nor directly away from) the center. And since it was originally not moving towards the center, it will never collide with the center.