Global isochronous potentials

We present a geometric characterization of the nonlinear smooth functions $V: R\to R$ for which the origin is a global isochronous center for the scalar equation $\ddot{x}=-V^{\mathrm{\prime }}(x)$. We revisit Stillinger and Dorignac isochronous potentials $V$ and show a new simple explicit family. Implicit examples are easily produced.