Superintegrability and time-dependent integrals

Abstract

While looking for additional integrals of motion of several minimally superintegrable systems in static electric and magnetic fields, we have realized that in some cases Lie point symmetries of Euler-Lagrange equations imply existence of explicitly time-dependent integrals of motion through Noether’s theorem. These integrals can be combined to get an additional time-independent integral for some values of the parameters of the considered systems, thus implying maximal superintegrability. Even for values of the parameters for which the systems don’t exhibit maximal superintegrability in the usual sense they allow a completely algebraic determination of the trajectories (including their time dependence).