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?