Are all classical superintegrable systems in two-dimensional space linearizable?

Abstract

Several examples of classical superintegrable systems in two-dimensional space are shown to possess hidden symmetries leading to their linearization. They are those determined 50 years ago in [1], and the more recent Tremblay-Turbiner-Winternitz system [6]. We conjecture that all classical superintegrable systems in two-dimensional space have hidden symmetries that make them linearizable.

Functional Realizations of Lie Algebras as Noether Point Symmetries of Systems

“…As has been already pointed out in many contexts, the non-compact Lie algebra sl(2, R) plays a relevant role within the group-theoretic analysis of differential equations, in particular, concerning the (super)-integrability of plane systems and the linearization analysis [9][10][11]. This fact suggests to consider this Lie algebra more closely in the context of inverse problems, as done in [12].…”

“…Clearly, as the system related to T is linearizable, it admits five additional Noether symmetries, all of which possessing a zero term in ∂ ∂t [11]. Looking now for a potential U (t, q) that preserves the symmetry, it follows at once from the X 3 -invariance that ∂U ∂t = 0, and thus the perturbed system is conservative.…”

“…Looking now for a potential U (t, q) that preserves the symmetry, it follows at once from the X 3 -invariance that ∂U ∂t = 0, and thus the perturbed system is conservative. The invariance by X 1 implies that U must satisfy the first-order differential equation (see (11))…”

“…which shall serve as generic gauge term for the symmetry condition (11). Analyzing the symmetry condition (7), the vector fields X i are the symmetry generators of a Noether point symmetry of a Lagrangian L (t, q,q) whenever the first-order PDE 1 2…”

Exact Traveling Wave Solutions of Nonlinear Evolution Equations: Indeterminant Homogeneous Balance and Linearizability

“…Single NLODEs of various orders have been studied; past efforts are reviewed in a recent paper on the linearization of fifth-order NLODEs with emphasis on Lie symmetry analysis [7]. A type of hidden symmetry has been applied to Newtonian equations for a Hamiltonian system to reduce coupled ODEs to a single second-order linear ODE [8]. The linearization of NLPDEs is even more challenging.…”