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).

Linearity of minimally superintegrable systems in a static electromagnetic field

“…We show that the two nonlinear minimally superintegrable systems are actually linear. In [20] the linear minimally superintegrable system, namely Case A.2, was studied and the eight-dimensional Lie symmetry algebra of the corresponding linear Lagrangian equations was determined in order to derive integrals of motion by means of a geometrical version Noether theorem.…”

“…We show that the two nonlinear minimally superintegrable systems (Case 9.2a and Case 9.2b) hide linearity by means of Lie symmetries. On the contrary, Case 9.2c corresponds to a linear minimally superintegrable system, and therefore is outside the scope of this paper, although the corresponding three linear Lagrangian equations admits an eight-dimensional Lie symmetry algebra that could be used to determine integrals of motion either by means of Noether's theorem as in [20] or by means of Jacobi last multiplier as in [35].…”