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