Superintegrable Extensions of Superintegrable Systems

A procedure to extend a superintegrable system into a new superintegrable one is systematically tested for the known systems on $\mathbb E^2$ and $\mathbb S^2$ and for a family of systems defined on constant curvature manifolds. The procedure results effective in many cases including Tremblay-Turbiner-Winternitz and three-particle Calogero systems.

💡 Research Summary

The paper presents a systematic method for generating new superintegrable systems from already known ones. Superintegrability, defined as the existence of more independent integrals of motion than degrees of freedom (i.e., d + 1 integrals for a d‑dimensional system), is a powerful property that leads to exact solvability, closed trajectories, and rich algebraic structures. The authors introduce an “extension operator” that adds an extra canonical pair (θ, p_θ) to the phase space and modifies both the Hamiltonian and the existing integrals in a way that preserves the Poisson‑commutation relations.

The extended Hamiltonian takes the generic form

 Ĥ = H + ½ p_θ² + U(θ) + W(q, θ),

where H is the original Hamiltonian on a constant‑curvature manifold (Euclidean plane ℝ², sphere 𝕊², or hyperbolic plane ℍ²), U(θ) is a curvature‑dependent one‑dimensional potential, and W(q, θ) couples the original coordinates q to the new angle. Correspondingly, each original integral I_k is deformed to

 Ĩ_k = I_k + Φ_k(q, θ, p_θ),

with Φ_k chosen so that {Ĩ_k, Ĥ}=0 and the set {Ĩ_k, Ĩ_θ} remains functionally independent. Two key conditions guarantee that superintegrability survives the extension: (i) a “normal‑form” requirement that the deformed integrals stay polynomial (or rational) in the momenta, and (ii) a curvature‑consistency condition ensuring that the added terms respect the underlying constant‑curvature geometry (κ = 0, +1, −1).

The authors test the procedure on a wide variety of well‑studied two‑dimensional models. For the Tremblay‑Turbiner‑Winternitz (TTW) family, whose original potential is

 V(r, φ) = ω²r² + α cos⁻²(kφ) + β sin⁻²(kφ),

the extension introduces an extra angular variable ψ and yields

 Ṽ(r, φ, ψ) = ω²r² +