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This is a conclusion given without proof in the Chinese version of Arnold's Methematical Methods in Classical Mechancs. Contents related to this conclusion are missing from the English version (I guess they were added by the Chinese translator).

Years ago, I have found a proof (on page 121) of a special case: for the trajectory in the force field of power $a$ (i.e. $\mathbf F\propto\hat{\mathbf r}r^a$), if we regard it as a curve in the complex plane and find the $\alpha$th power of the it, we get a trajectory in the force field of power $A$, where $(a+3)\,(A+3)=4$, and $\alpha=(a+3)/2$. The proof utilizes the conservation of energy and a new time parameter $\tau$ with the relation to the old time $t$ being $\mathrm d\tau/\mathrm d t=|z|^{a+1}$.

However, I cannot prove the more general result: conformal map $w(z)$ transforms a trajectory in potential $U(z):=|dw/dz|^2$ to one in another potential $V(w):=-|dz/dw|^2$. I tried using $\mathrm d\tau/\mathrm d t=|w/z|^2$ with no luck.

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This result is proven in V.I. Arnold Huygens and Barrow, Newton and Hooke Appendix 1. This may be done by utilizing the Maupertuis principle: $$\sqrt{2\left(E-U(z)\right)}\left|dz\right|=\sqrt E\sqrt{2\left(E'-V(w)\right)}\left|dw\right|,\quad EE'=-1.$$ This means that the dual trajectories have the same abbreviated action, so one of them being physical would imply that the other is physical too.

A more comprehensive discussion about this problem is available here.

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