A remark on rational isochronous potentials

Abstract

We consider the rational potentials of the one-dimensional mechanical systems, which have a family of periodic solutions with the same period (isochronous potentials). We prove that up to a shift and adding a constant all such potentials have the form $U(x)=1/2{\omega }^2{x}^2$ or $U(x)=1/8{\omega }^2{x}^2+c^2{x}^{-2}.$

5 pages, contribution to a special issue of JNMP dedicated to F. Calogero, slightly revised version

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