Another "Strange Central Force"?

Inspired by the recent challenge problem, I tried to simulate the motion of a particle under different "strange central forces" - those described by

$F = - a r^k$ , where $a$ is a positive constant of appropriate physical dimension, and $k$ is a real number. It turns out that the trajectory of the body is always a closed non self-intersecting curve only for $k = -2$ and $k = 1$ . Is this a known result? How can we prove this?

#Physics

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$