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Coulomb's force law and Hooke's law are connected by a transformation called Bohlin's transformation.
$\newcommand{\DD}[2][]{\frac{d^2#1}{d#2^2}}$
There is a symmetry transformation that connects equations of motion
under the two most important force laws of Kepler's and Hooke's law. Here is
the statement from the reference given at the end.
Bohlin's Theorem
Suppose a point in the complex plane moves following Hooke's law
\begin{equation}
\DD[w]{t} = -\frac{k}{w} = - C w
\end{equation}
Square \(w\) and consider a point following trajectory \(z(\tau(t))=[w(t)]^2\),
with \(\frac{d\tau}{dt}=|w|^2\) where a new time \(\tau\) has been chosen in order
to grant law of areas. Then \(z(\tau)\) will satisfy the gravitational law:
\begin{equation}
\DD[z]{\tau} = -\frac{k}{m}\frac{z}{z^3} = - \tilde{C}
\frac{z}{z^3}
\end{equation}
where \(\tilde{C}= 2(|w{\,}^\prime(0)|^2 + |w(0)|^2)\).
Reference
Maria Luisa Saggio, Eur. J. of Physics 34 (2013) 129-137
