On the force fields which are homogeneous of degree $-3$

The dynamics defined by a force field which is positively homogeneous of degree $-3$ can always be reduced, by simply constraining it. The dimension of the phase space is reduced by two dimensions, while it may only be reduced by one dimension if the degree of homogeneity is different from $-3$. This remark is an elegant foundation of Appell's projective dynamics. We show how it relates to Kn\"orrer's remark on the correspondence between the Neumann potential on a sphere and the geodesic motion on an ellipsoid.