Lagrangians for spherically symmetric potentials

Two Lagrangians are s-equivalent (s for ‘‘solution’’) if they yield equations of motion having the same set of solutions. We consider Lagrangians s-equivalent to T−V, where T is flat space kinetic energy and V is a spherically symmetric potential. We show that for n=dimension of space ≥3, there are many s-equivalent Lagrangians which cannot be formed from T−V by multiplication by a constant or addition of a total time derivative. In general these s-equivalent Lagrangians lead to inequivalent quantum theories in the sense that the energy spectra are different.